Category Archives: Sciences

The Starry Nights of Vincent Van Gogh (1) : Café Terrace at Night, in Arles

“In which space do our dreams live? What is the dynamism of our nightlife? Is the space of our sleep really a rest area? Is it not rather an incessant and confused movement? On all these problems we have little light because we do not find, when the day comes, only fragments of night life.

In these texts written from 1942 to 1962 (gathered in Le Droit de rêver, PUF, collection “Quadrige”, 2010), Gaston Bachelard celebrates the difficult synthesis of imagination and reflection that seems to him to guarantee, for writers as for artists such as Baudelaire and Van Gogh’s, fidelity to dreamlike values. “A Van Gogh’s yellow is like an alchemical gold, a gold butine like a solar honey. It is never simply the gold of the wheat, the flame, or the straw chair; it is a gold forever individualized by the endless dreams of genius. It no longer belongs to the world, but it is the good of a man, the heart of a man, the elementary truth found in the contemplation of a lifetime.

Coucher de soleil sur champ de blé près d’Arles (Sunset on wheatfield near Arles), 1888

In the series of notes that I begin here, I will analyze in detail the extraordinary reports that Vincent Van Gogh (1853-1889) maintained with the vision of the Provençal sky.

On February 20, 1888, aged 35, Vincent, the man from dark-heavened Northern Europe, moved to the old city of Arles, in the South of France. Although he arrived in the city by a snowy day, he discovered the Provençal light, brighting day and night. Stunned by the transparency of the firmament, he writes to his brother Theo: “The deep blue sky was spotted with clouds deeper blue than the fundamental blue of an intense cobalt, and others of a blue clearer, like the blue whiteness of the milky ways. In the background, the stars sparkled, clear, green, yellow, white, lighter pink, diamond-like diamonds. ” From then sprout in him the crazy project of painting the sky.

Van_Gogh : Self portrait as an artist, Arles1888

On April 12, he wrote to his friend the painter Émile Bernard: “A starry sky, for example, well — it’s a thing that I’d like to try to do, just as in the daytime I’ll try to paint a green meadow studded with dandelions“. He hesitates however and procrastinates, intimidated by the subject. On June 19, he expressed his hesitation to Émile Bernard: “But when will I do the starry sky, then, that painting that’s always on my mind? Alas, alas, […] the most beautiful paintings are those one dreams of while smoking a pipe in one’s bed, but which one doesn’t make. But it’s a matter of attacking them nevertheless, however incompetent one may feel vis-à-vis the ineffable perfections of nature’s glorious splendours.

On 9th (or 10th) of July 1888 he confesses to Theo: “But the sight of the stars always makes me dream in as simple a way as the black spots on the map, representing towns and villages, make me dream“.

From word to deed takes place between 9 and 14 September. In fact, he begins on the 9th a long letter addressed to his sister Willemien: “I definitely want to paint a starry sky now. It often seems to me that the night is even more richly coloured than the day, coloured in the most intense violets, blues and greens. If you look carefully you’ll see that some stars are lemony, others have a pink, green, forget-me-not blue glow. And without labouring the point, it’s clear that to paint a starry sky it’s not nearly enough to put white spots on blue-black.

From left to right, self-portrait of Emile Bernard, friend of Vincent, photographic portraits of his brother Theo and his sister Willemien.

He did not post it and resumed his letter on the 14th. In the meantime he painted his first starry night, the painting is called Cafe Terrace at night (currently at the Kröller-Muller Museum in Otterlo, the Netherlands):

I started this letter several days ago, up to here, and I’m picking it up again now.  I was interrupted precisely by the work that a new painting of the outside of a café in the evening has been giving me these past few days. On the terrace, there are little figures of people drinking. A huge yellow lantern lights the terrace, the façade, the pavement, and even projects light over the cobblestones of the street, which takes on a violet-pink tinge. The gables of the houses on a street that leads away under the blue sky studded with stars are dark blue or violet, with a green tree. Now there’s a painting of night without black. With nothing but beautiful blue, violet and green, and in these surroundings the lighted square is coloured pale sulphur, lemon green. I enormously enjoy painting on the spot at night“.

Terrasse de café le soir
And on September 16th, he describes his painting to Theo more briefly: “The second [painting of this week] shows the outside of a café, lit on the terrace outside by a large gas-lamp in the blue night, with a patch of starry blue sky.represents the outside of a cafe illuminated on the terrace by a large gas lantern in the blue night. with a corner of starry blue sky. […] The question of painting night scenes or effects, on the spot and actually at night, interests me enormously.
We know exactly where the painting was executed: Place des hommes, now renamed Place du Forum. The map of Arles in Van Gogh’s time, shown below, shows its location, as well as other intramural sites where Vincent settled to paint La Maison jaune (The Yellow House) in September 1888), the Pont métallique de Trinquetaille (the Metallic Bridge of Trinquetaille) in October 1888) and Nuit étoilée sur le Rhône (Starry Night on the Rhone), on which I will return at length in the following post.
The café, which at that time was called the Terrace, has since been renamed Café Van Gogh. Fortunately, the historic site has not been ransacked by modern constructions as is so often the case elsewhere, and even today the walker immediately recognizes the layout of the streets and buildings painted by Vincent, day and night.
Café Van Gogh nowadays. Night photography shows the light pollution characteristic of urban lighting, violent and useless projectors erasing all traces of the firmament. A huge loss of cosmic feeling …
Now a question that arises is: are the stars he has represented on the canvas randomly arranged, or do they correspond to a real configuration of the night sky?
In the preparatory study for the painting shown below, the sky is just sketched with wiggling lines, without any star. It is quite possible that Vincent made this study during the day .
Preparatory study for the painting, September 1888. Dallas Museum of Art, The Wendy and Emery Reves Collection.
However, in view of van Gogh’s epistolary statements, everything suggests that he wanted to show a certain realism in the pictorial transposition of the firmament seen at night. Since, according to the letter that Vincent sent to his sister Willemien, we know the date of execution (between 9 and 14 September) within a few days, it is possible to check using a reconstitution software astronomical what portion of sky was represented by Vincent, seen from the Forum Square in a direction very close to the South (this is the orientation of the street).
Let’s use the excellent Stellarium software. Position us at the GPS coordinates of the Forum Square, namely 43 ° 40 ‘39.7 “N 4 ° 37’ 37.6” E, set the date from September 9, 1888 at about 10 pm, let us look south and let the map scroll to find a stellar configuration possibly close to that of the table, between 20 and 30° of declination (such is the height of the stars represented in the table).
I once read an article (which I lost references) claiming that it is the legs of the constellation Scorpio, with the stars α (the brilliant Antares), σ, β, δ and Scorpion π. The problem is that between 9 and 14 September, the constellation Scorpio is only above the horizon until 17h UT, after it passes below and can not be seen, even at the beginning of the night which in September falls well later. Also at that time the Moon was at its first crescent in the legs of Scorpio. This is not the correct identification.
Let’s now examine the map of the sky seen between 9 and 14 September 1888 around 22h in the southern extension of the Forum Square: we see the stars of the constellation Aquarius up to magnitude 5, with its characteristic configuration shaped from Y.
Stars of the Aquarius constellation visible to the naked eye between 9 and 14 September 1888 at 22h in the direction of the Cafe de la Terrasse.
 I added the profile of the buildings hiding part of the field of view, traced the characteristic lines connecting the most brilliant stars, and compared with Vincent’s painting:
The identification seems pretty convincing … It also reinforces the epistolary statements in which Vincent expressed his concern to represent a real sky and not imaginary.
This will be even more spectacular in the two famous starry nights painted in Arles in 1888 and Saint-Rémy in 1889. I will analyze them in the same way in the following posts, with the key to very unexpected surprises …

 

40 Years of Black Hole Imaging (3): from Kerr black holes to EHT

Sequel of the previous post 40 Years of Black Hole Imaging (2) : Colors and Movies 1989-1993

Generalizations to Kerr Black Holes

Unfortunately Marck’s simulations of black hole accretion disks remained mostly ignored from the professional community, due to the fact that they were not published in peer-reviewed journals and, after their author prematurely died in May 2000, nobody could find the trace of his computer program…

Then, unaware of Marck’s results, several researchers of the 1990’s were involved in the program of calculating black hole gravitational lensing effects in various situations. Stuckey (1993) studied photon trajectories which circle a static black hole one or two times and terminate at their emission points (« boomerang photons »), producing a sequence of ring-shaped mirror images. Nemiroff (1993) described the visual distortion effects to an observer traveling around and descending to the surface of a neutron star and a black hole, discussing multiple imaging, red- and blue-shifting, the photon sphere and multiple Einstein rings. He displayed computer-generated illustrations highlighting the distortion effects on a background stellar field but no accretion disk, and made a short movie now available on the internet (Nemiroff 2018), two snapshots of which are shown in figure 1.

Figure 1. Trip to a black hole by Robert Nemiroff, 1993.

The first simulations of the shape of accretion disks around Kerr black holes were performed by Viergutz (1993). He treated slightly thick disks and produced colored contours, including the disk’s secondary image which wraps under the black hole (figure 2). The result is a colored generalization of the picture by Cunningham and Bardeen (1973) shown in 40 Years of Black Hole Imaging (1).

Figure 2. Primary and secondary images of a simple accretion disk model around a Kerr black hole, seen by a faraway observer. Colors indicate combined gravitational and Doppler shifts (from Viergutz 1993).

More elaborate views of a geometrically thin and optically thick accretion disk around a Kerr black hole were obtained by Fanton et al. (1997). They developed a new program of ray tracing in Kerr metric, and added false colors to encode the degree of spectral shift and temperature maps (figure 3). Zhang et al. (2002) used the same code to produce black-and-white images of standard thin accretion disks around black holes with different spins, viewing angles and energy bands (figure 4).

Figure 3. False color contour maps showing how the monochromatic radiation emitted by a Keplerian accretion disk would be seen at infinity for various values of the inclination angle to the plane of the disk (top to bottom : 5°, 45°, 85°). The left column refers to a non-rotating black hole, the right one to a rapidly rotating black hole with a=0.998 M. The white zones stand for the regions with zero redshift. Left-hand side of the disk is approaching the observer and blueshifted (from Fanton et al. 1997).
Figure 4. Disk images of accretion disks extending up to 20 Schwarzschild radii for different spins of Kerr black holes, viewed in different energy ranges and inclination angles (from Zhang et al. 2002).

Ben Bromley et al. (1997) calculated integrated line profiles from a geometrically thin disk about a Schwarzschild and an extreme Kerr black hole, in order to get an observational signature of the frame-dragging effect (Figure 5).

Figure 5. Image of a geometrically thin disk around an extreme Kerr (maximally rotating) black hole seen at an inclination of 75°. The inner and outer radii of the Keplerian (circularly rotating) disk are at 1.24 M and 6 M. The colors encode the apparent light frequency, the white strip divides redshifted and blueshifted regions. The asymmetric appearance of the inner disk edge results from the frame-dragging effect of black hole rotation (from Bromley et al. 1997).

In 1998 Andrew Hamilton started to develop for a student project at the University of Colorado a “Black Hole Flight Simulator”, with film clips that have been shown at planetariums, also available on the Internet. The first depictions were very schematic, but the website was constantly implemented. It now offers journeys into a Schwarzschild or a Reissner-Nordström (i.e. electrically charged) black hole with effects of gravitational lensing on a stellar background field, as well as animated visualizations of magneto-hydrodynamic simulations of a disk and jet around a non-rotating black hole (Hamilton 2018).


Journey into and through an electrically charged (non realistic)  Reissner-Nordström black hole, from Andrew Hamilton, 2010

From Idea to Reality

A turning point in the history of black hole imaging came when the possibility of viewing in practice the shadow of SgrA* with VLBI radio astronomy techniques was first discussed (Falcke et al. 2000, Doeleman et al. 2001). Heino Falcke, Fulvio Melia and Eric Agol (who curiously did not quote my 1979 article) developed a general relativistic ray-tracing code that allowed them to simulate observed images of Sgr A* for various combinations of black hole spin, inclination angle, and morphology of the emission region directly surrounding the black hole (figure 6).

Figure 6. Images of an optically thin emission region surrounding the galactic black hole SgrA*. The black hole is maximally rotating (a = 0.998) in the top row and non-rotating in the bottom row. The emitting gas is assumed to be in free fall (top) or on Keplerian shells (bottom) with a viewing angle 45°. The left column shows the ray-tracing calculations in general relativity, the other columns are the images seen by an idealized VLBI array at 0.6 mm and 1.3 mm wavelengths, taking account of the interstellar scattering (from Falcke et al. 2000).

In 2001, Ben Bromley, Fulvio Melia and Siming Liu provided maps of the polarized emission of a Keplerian disk to illustrate how the images of polarized intensity from the vicinity of SgrA* would appear in future VLBI observations (Figure 7).

Figure 7. Polarization maps at three wavelengths (1.5 mm, 1 mm, 0.67 mm from top row to bottom row) calculated for the galactic black hole candidate SgrA*. The left most column shows how the radio maps might look seen from a close observer, the other columns show how the map might look from Earth with our vision blurred by gas in interstellar space (from Bromley et al. 2001)

Indeed, in parallel with but rather independently from the theoretical simulations reviewed here, the work to image SgrA* by VLBI experiments had begun also back in the 1970’s, after the discovery of the compact radio source Sgr A* at the center of the Milky Way and its identification as the likely emission of gas falling onto a supermassive black hole (Balick and Brown 1974). And as soon as it was realized that the shadow of SgrA* could really be photographed in the forthcoming years, the program of imaging black holes with or without accretion disks and/or stellar background field developed at a much accelerated rate. Several dozens of papers with more or less elaborate visualizations bloomed out, so many that I’ll stop my illustrated history of black hole imaging at this turning point.

As already suspected a long time ago, the gravitational dynamics of stars orbiting the Galactic Center SgrA, as observed for more than 20 years, give a good estimate for the centeal black hole mass : 4.4 millions solar masses. Credit : Keck Observatory.

On the observational side, successive radio imaging observations progressively reduced the size of emission region if SgrA*. A breakthrough was to extend VLBI to 1mm wavelength, where the scattering effects are greatly reduced and angular resolution is matched to the shadow of the galactic black hole. Then the collective effort was named the “Event Horizon Telescope” as the natural convergence of many historical and parallel works done by several independent teams in the world (Doeleman et al. 2009). The later measurement of the size of the 6 billion solar mass black hole in M87 gave a second source suitable for shadow imaging (Doeleman et al. 2012).

Optical image of the giant elliptical galaxy M87 taken by the Hubble Space Telescope. Its core emits an enormous jet of relativistic plasma. At its very center, M87 harbours the second-largest black hole as seen from Earth, M87*, with a mass of 6.6 billion Suns but over 2000 times farther away than Sagittarius A*.

Now the Event Horizon Telescope Consortium involves 20 universities, observatories, research institutions, government agencies and more than a hundred scientists who hope to make black hole imaging a reality as soon as 2019. The first telescopic image of M87* was delivered on April 10th, 2019.

Sheperd Doeleman, director of the Event Hoziron Telescope, at the press conference of April 10th 2019 in which the first telescopic image of black hole M87* was shown.

The path from idea to reality can take very a long time. Imaging black holes, first with computers, now with telescopes, is a fantastic adventure. Forty years ago I couldn’t hope that a real image would be reachable in my lifetime and that, thanks to contributions by so many dedicated colleagues, my dream would become true.

In May 2019 I was invited to give the keynote talk at the 3rd Black Hole Initiative Conference at Harvard University and I could warmly congratulate the EHT team. The young commputer scientist Katie Bouman led the development of one of the various algorithms for imaging black holes. We were glad to meet each other, the young and the old !

With Katie Bouman on 21 May 2019 at the Black Hole Initiative Conference, Harvard University

Here is the video of my talk :

Technical References for the 3 posts

Abramowicz, M., Jaroszynski, M., Sikora, M. : Relativistic accreting disks, Astron. Astrophys. 63, 221 (1978).

Balick, B., Brown, R.L. : Intense sub-arcsecond structure in the galactic center, Astrophys. J. 194, 265-270 (1974).

Bardeen, J. M. 1973, Timelike and null geodesics in the Kerr metric, in Black Holes (Les Astres Occlus), ed. C. Dewitt & B. S. Dewitt, New York: Gordon and Breach, pp.215–239.

Bromley, B., Chen, K., Miller,W. : Line Emission from an Accretion Disk around a Rotating Black Hole: Toward a Measurement of Frame Dragging, Astrophys.J. 475, 57 (1997).

Bromley, B., Melia, F., Liu, S. : Polarimetric Imaging of the Massive Black Hole at the Galactic Center, Astrophys.J. 555, L83-86 (2001).

Carter B. : Axisymmetric Black Hole Has Only Two Degrees of Freedom, Physical Review Letters 26, 331 (1971).

Carter B. , Luminet, J.-P. : Les Trous Noirs, Maelströms cosmiques, La Recherche 94, 944 (1978).

Carter B. , Luminet, J.-P. : Pancake Detonation of Stars by Black Holes in Galactic Nuclei, Nature 296, 211 (1982).

Chatzopoulos, S., Fritz, T. K., Gerhard, Gillessen, O., S. , Wegg, C. , Genzel, R. , Pfuhl, O. : The old nuclear star cluster in the Milky Way: dynamics, mass, statistical parallax, and black hole mass, MNRAS, 447, 948 (2015)

Cunningham, C. T. : The effects of redshifts and focusing on the spectrum of an accretion disk around a Kerr black hole, Astrophys. J., 202, 788 (1975)

Cunningham, C.T., Bardeen J.M. : The optical appearance of a star orbiting an extreme Kerr black hole, Astrophys. J.173 L137-142 (1972).

Cunningham, C.T., Bardeen J.M. : The optical appearance of a star orbiting an extreme Kerr black hole, 1973, Astrophys. J., 183, 237

Davelaar, J., Bronzwaer, T., Kok, D., Younsi, Z., Moscibrodzka, M., Falcke, H.: Observing supermassive black holes in virtual reality, Computational Astrophysics and Cosmology 5,1 (2018). https://doi.org/10.1186/s40668-018-0023-7

Delesalle, L. , Lachièze-Rey, M. , Luminet, J.-P. : Infiniment Courbe, TV documentary, 52 mn, France: CNRS/Arte, 1994.

Doeleman, S.S., et al. : Structure of Sagittarius A* at 86 GHz using VLBI closure quantities, Astron. J., 121, 2610-2617 (2001).

Doeleman, S.S., et al. : Event-horizon-scale structure in the supermassive black hole candidate at the Galactic Centre, Nature, 455, 78 (2008).

Doeleman, S.S., et al. : Imaging an Event Horizon : submm-VLBI of Super massive Black Hole, The Astronomy and Astrophysics Decadal Survey, Science White Papers, no. 68 (2009).

Doeleman, S.S., et al. : Jet-Launching Structure Resolved Near the Supermassive Black Hole in M87, Science 338 (6105), 355 (2012).

Doeleman, S.S. : Seeing the unseeable, Nature Astronomy 1, 646 (2017)

Falcke, H. : Imaging black holes : past, present and future, Journal of Physics : Conf. Series 942, 012001 (2017)

Falcke, H., Melia, F., Agol, E. : Viewing the Shadow of the Black Hole at the Galactic Center, Astrophys. J. Lett. 528, L13–L16 (2000).

Fanton C., Calvani M., de Felice F., Cadez A. : Detecting Accretion Disks in Active Galactic Nuclei, Publ. Astron. Soc. Japan 49, 159-169 (1997).

Fukue, J., Yokoyama, T. : Color Photographs of an Accretion Disk around a Black Hole, Publications of the Astronomical Society of Japan 40, 15–24 (1988).

Goddi, C., Falcke, H., et al.: BlackHoleCam: fundamental physics of the galactic center. Int. J. Mod. Phys. D 26, 1730001-239 (2017).

Hamilton, A.: Falling into a black hole. http://jila.colorado.edu/~ajsh/insidebh/intro.html (1998-2018). Accessed 2019-02-26.

Hills, J.G. : Possible power source of Seyfert galaxies and QSOs, Nature 254, 295 (1975).

James, O., von Tunzelmann, E., Franklin, P., Thorne, K.S.: Gravitational lensing by spinning black holes in astrophysics, and in the movie Interstellar. Class. Quantum Gravity 32(6), 065001 (2015).

Johnson, M. D. , Bouman, K. L., Blackburn, L. L. , Chael, A. A., Rosen, J. , Shiokawa, H. , Roelofs, F. , Akiyama, K. , Fish, V. L. , Doeleman S. S. : Dynamical Imaging with Interferometry, Astrophys. J., 850(2), 172 (2017).

Kerr, R.P. : Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics, Physical Review Letters 11, 237 (1963).

Kormendy, J., Ho, L. : Supermassive Black Holes in Inactive Galaxies, Encyclopedia of Astronomy and Astrophysics, ed. P.Murdin, article id. 2635. Bristol : Institute of Physics Publishing (2001).

Luminet J.-P.: Seeing Black Holes : from the Computer to the Telescope, Universe 4(8), 86 (2018) [arXiv :1804.03909].

Luminet, J.-P. : Black Holes, Cambridge University Press, 1992.

Luminet, J.-P. : Interstellar Science, International Review of Science vol.1 n°2 (march 2015) [arXiv : 1503.08305].

Luminet, J.-P.: Image of a Spherical Black Hole with Thin Accretion Disk, Astron.Astrophys. 75, 228 (1979).

Marck J.-A., Luminet, J.-P. : Plongeon dans un trou noir, Pour la Science Hors-Série Les trous noirs (July 1997) 50–56.

Marck, J.-A. : Color animation of a black hole with accretion disk, https://www.youtube.com/watch?v=5Oqop50ltrM (1991) (put online in 2011 by J.-P. Luminet).

Marck, J.-A. : Colored images of a black hole accretion disk for various angles of view, unpublished (1989).

Marck, J.-A. : Flight into a Black Hole, videocassette 11 mn, Meudon : CNRS Images (1994).

Marck, J.-A. : Short-Cut Method of Solution of Geodesic Equations for Schwarzschild Black Hole, Classical and Quantum Gravity 13(3) 393–402 (1996).

Marrone, D. P., Moran, J. M., Zhao, J.-H., Rao, R. : An Unambiguous Detection of Faraday Rotation in Sagittarius A*, Astrophys. J. Lett. 654, L57 (2007).

Nemiroff, R. : Visual distortions near a neutron star and black hole, Am. J. Phys. 61, 619 (1993) [astro-ph/9312003].

Nemiroff, R. Trip to a Black Hole https://www.youtube.com/watch?v=ehoOkyHtBXw (2018)

Nerval, G. de : Le Christ aux Oliviers, in Les Chimères, Paris, 1854. Free translation by J.-P. Luminet.

Ohanian, H. C. : The Black Hole as a Gravitational ‘Lens’, Am. J. Phys. 55, 428-432 (1987).

Page, D.N., Thorne, K.S. : Disk Accretion onto a Black Hole. I. Time-Averaged Structure of Accretion Disk, Astrophys. J. 191, 499-506 (1974).

Palmer L., Pryce M. & Unruh W. : Simulation of starlight lensed by a camera orbiting a Schwarzschild black hole, unpublished (1978).

Pounds, K. A. et al. : An ultra-fast inflow in the luminous Seyfert PG1211+143, Monthly Notices of the Royal Astronomical Society 481(2), 1832-1838 (2018).

Pringle, J. E., Rees, M. J. : Accretion Disc Models dor Compact X-ray Sources, Astron. Astrophys. 21 , 1 (1972).

Riazuelo, A. : Seeing relativity I. Ray tracing in a Schwarzschild metric to explore the maximal analytic extension of the metric and making a proper rendering of the stars, (2018) [ArXiv :1511.06025]

Rogers, A. : The Warped Astrophysics of Interstellar, https://www.wired.com/2014/10/astrophysics-interstellar-black-hole/.

Sargent, W. L., Young, P. J., Lynds, C. R., et al. : Dynamical evidence for a central mass concentration in the galaxy M87, Astrophys. J. 221, 731–744 (1978).

Schastok, J. , Soffel, M. , Ruder, H., Schneider, M. : Stellar Sky as Seen From the Vicinity of a Black Hole, Am. J. Phys., 55, 336-341 (1987).

Shaikh, R. : Shadows of rotating wormholes, Phys. Rev. D 98, 024044 (2018) [arXiv :1803.11422]

Shakura, N.I., Sunyaev, R. A. : Black holes in binary systems. Observational appearance. Astro. Astrophys. 24, 337-355 (1973)

Stuckey, W. M. : The Schwarzschild Black Hole as a Gravitational Mirror, Am. J. Phys. 61(5), 448-456 (1993).

Thorne K.: The Science of Interstellar, Norton & Company (november 2014).

Thorne, K, private communication, 24/10/2014

Viergutz S U. : Image Generation in Kerr Geometry. I. Analytical Investigations on the Stationary Emitter-Observer Problem, Astron. Astrophys. 272, 355–77 (1993) ; Viergutz, S.U. : Radiation from arbitrarily shaped objects in the vicinity of Kerr Black Holes, Astrophys. Space Sci. 205, 155 (1993).

 

40 Years of Black Hole Imaging (2): Colors and movies, 1989-1993

Sequel of the previous post 40 Years of Black Hole Imaging (1) : Early Work 1972-1988

First Flight into a Black Hole

In 1989-1990, while I spent one year as a research visitor at the University of California, Berkeley, my former collaborator at Paris-Meudon Observatory, Jean-Alain Marck, both an expert in general relativity and computer programming, started to extend my simulation of 1979. The fast improvement of computers and visualization software (he used a DEC-VAX 8600 machine) allowed him to add colors and motions. To reduce the computing time, Marck developed a new method for calculating the  geodesics in Schwarzschild space-time, published only several years later (Marck 1996). In a first step Marck started from my model of 1979 and calculated static images of an accretion disk around a Schwarzschild black hole according to various angles of view, see Figure 1 below.

Figure  1. False-Coloured images of a black hole accretion disk for various angles of view by J.-A.
Marck &  J.-P. Luminet , 1989 (unpublished).

In 1991, when I went back to Paris Observatory, I started the project for the French-German TV channel Arte of a full-length, pedagogical movie about general relativity (Delesalle et al. 1994). As the final sequence dealt with black holes, I asked Marck to introduce motion of the observer with the camera moving around close to the disk, as well as to include higher-order lensed images and background stellar skies in order to make the pictures as realistic as possible. The calculation was done along an elliptic trajectory around a Schwarzschild black hole crossing several times the plane of a thin accretion disk and suffering a strong relativistic precession effect (i.e. rotation of its great axis), see figure 2 below.

Compared to my static, black-and-white simulation of 1979, the snapshot reproduced in Figure 3 below shows spectacular improvements:

Figure 3. Colored image of a black hole accretion disk as seen by a moving observer at 7°
above the disk’s plane. The observer uses a camera equipped with filters to convert into
optical radiation the emitted electromagnetic radiation. The arbitrary coloring encodes the
apparent luminosity of the disk, the brightest and warmest parts being colored yellow, the
colder parts red. The transparency of the disk was enhanced in order to show the secondary
image through the primary, as well as some background stars. Compared with figure 8 there
are additional distortions and asymmetries due to the Doppler effect induced by the motion of
the observer himself. As a result the region of maximum luminosity has no more the shape of a
crescent (from Marck 1991)

The full movie is  available on my youtube channel :

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My books (4) : The Wraparound Universe

Until now I published as an author 30 books in my native language (French), including 14 science essays, 7 historical novels  and 9 poetry collections (for the interested reader, visit my French blog  here.
Although my various books have been translated in 14 languages (including Chinese, Korean, Bengali…), only 4 of my essays have been translated in English.

The fourth one was :

The Wraparound Universe

316 pages – AK Peters Ltd, 2008 – ISBN 978 1 56881 309 7 – ISBN 0 521 40906 3 (paperback)

WraparoundWhat shape is the universe? Is it curved and closed in on itself? Is it expanding? Where is it headed? Could space be wrapped around itself in a formation that produces ghost images of faraway galaxies? Such are the questions posed by Jean-Pierre Luminet, which he then addresses in clear and accessible language. An expert in black holes and the big bang, he leads us on a voyage through the surprising byways of space-time, where possible topologies of the universe, explorations of the infinite, and cosmic mirages combine their mysterious traits and unlock the imagination.

Praise for The Wraparound Universe

– “Luminet’s deep understanding of the history of cosmology combines with his scientific knowledge and expository skills to produce a delightful introduction to the much-debated question of the shape of the universe. Directed at an intelligent layperson, The Wraparound Universe combines geometrical insights with astronomical observations leading to the idea of a universe that is finite yet has no boundary.” (Jeff Weeks, author of The Shape of Space)
– “This book is well written and nicely illustrated, giving a clear exposition of the mathematical and astronomical ideas involved as well as the various ways of observationally testing this possibility.” (George Ellis, Cape Town University, Templeton Prize)

Available on amazon : click here

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New Scientist, CERN Courier, etc : Click here

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Geometry and the Cosmos (3) : from Ptolemy’s circles to Inflationary Cosmology

Sequel of the previous post Geometry and the Cosmos (2): From the Pre-Socratic Universe to Aristotle’s Two Worlds

Ptolemy’s Circles

Any model of the universe must incorporate the mechanisms determining the motion of the planets and other celestial bodies. From Plato and Aristotle to Kepler, astronomers could not imagine the universe governed by shapes other than circles and spheres, the only geometric forms that could possibly represent divine perfection. This constraint forced them to devise extremely complex systems which would “fit the facts”, in other words account for the apparent movements of the planets and stars as observed from the earth while conforming to the ideological demands of the concept of universal harmony.

Despite the ingenuity of astronomers like Euxodus (see previous post), their circular systems did not accurately describe the complex movements they had observed: the planets accelerated and decelerated and even occasionally went back the way they had come. Moreover, they did not account for the changes in brightness of the planets, which suggested variations in their distance from the earth that were incompatible with the idea that they travelled in circles centred on or near the earth.

How could Aristotelian cosmology be reconciled with astronomical observation? The most elaborate attempt to do so was made by Ptolemy (Claudius Ptolemaeus) in the second century AD. In his Syntaxis Mathematicae, better known by its Latin title Almagest, the Alexandrian thinker succeeded in explaining the motion of each celestial body by a system of extremely elaborate mathematical constructs.

Left : a depiction of Ptolemy at work in a medieval manuscript. Right : A Greek manuscript of the Almagest, IXth century.

Ptolemy adopted the concept of a stationary earth and celestial bodies which could move only in circles. But he multiplied the number of circles and offset them one against the other, proposing complex and ingenious interactions between them. The circle in which a planet moves, called its epicycle, no longer had the earth at its centre as in Eudoxus’ theory, but itself revolved around another circle, called the deferent (or eccentric circle if its own centre is offset from the earth’s position). This theory enabled Ptolemy to “fit the facts” without departing too far from Aristotelian philosophical principles and it survived for 1,500 years — longer than any other idea in the history of science – until the discovery of elliptical orbits by Kepler.

The Motion of the Outer Planets According to Ptolemy. Ptolemy’s theory of planetary morion was first mentioned in his Planetary Hypotheses, which survives only in an Arabic translation, and fully developed in his Almagest. The original Greek title, Syntaxis, means “compendium”, but the work seems to have been known as megiste, “the greatest”, whence the Arabic al-Majesti and subsequently the Latin title Almagestum. The most influential Latin translation was made in 1175 by Gerard of Cremona. The page above, from a later edition of Book X, shows a kinematic model of the motion of the outer planets – Mars, Jupiter and Saturn. The earth remains still while the planets move in a regular pattern relative to an equant point offset from the centre of the planetary sphere. Claude Ptolemy, Almagestum, translated into Latin by Gerard of Cremona, 13th century. Vatican, Biblioteca Apostolica Vaticana, Lat. 2057
Ptolemy’s Epicycles.
The Ptolemaic system was based on three geometric patterns: the epicycle, the eccentric circle and the equant. The epicycle had been invented in the third century BC by Apollonius of Perga, a brilliant mathematician whose most famous work is a treatise on conical sections (ellipses, parabolas and hyperbolas), and developed by Hipparchus a century later.
(a) Epicycle : a planet P rotates in a small circle (epicycle) whose centre (C) is simultaneously moving along the circumference of a large circle, known as the deferent, with the earth (T) at its centre.
(b) Eccentric Circle : the earth is offset from the centre (O) of the deferent. This model breaks the Aristotelian rule which states that the earth must be at the centre of the cosmos.
(c) Equant : despite its complexity, the eccentric circle model does not provide a sufficiently accurate explanation of the apparent motion of the planets. Ptolemy therefore postulated an equant point (E) about which the centre of the epicycle (C) rotates. Both the geometric centre of the deferent (O) and the centre of motion are now offset from the earth’s position (T).
(d) Final Model : the Ptolemaic model plots the motion of the planets according to this sytem, but it was so complicated that it was not fully understood by Western civilisation until the 15th century.
Ptolemy proudly defended its complexity: “We must as far as possible apply the simplest hypotheses to the movements of celestial bodies but, if these are inadequate, we must find others which explain them better.” (Almagest, XII, 2) – a statement which placed him firmly in the vanguard of modern scientific thinking.

 Nevertheless, the system of epicycles and eccentric circles suggested that the earth was not exactly at the centre of the cosmos and Islamic astronomers raised several objections to this infringement of Aristotelian harmony. It was the existence of an equant point offset from the earth that particularly preoccupied later scientists. Copernicus, for example, in his De Revolutionibus announced his intention to rid the celestial model of this “monstrosity”. Continue reading

40 Years of Black Hole Imaging (1): Early work 1972-1988

Introduction

Black holes are to many the most mysterious objects in space. According to the laws of General Relativity, they are by themselves invisible. Contrarily to non-collapsed celestial bodies, their surface is neither solid nor gaseous ; it is an immaterial border called the event horizon, beyond which gravity is so strong that nothing can escape, not even light.

Seen in projection on a sky background, the event horizon would have the aspect of a perfectly circular black disk if the black hole is static (the so-called Schwarzschild solution) or of a slightly distorted one if it is in rotation (the Kerr solution). Due to strong gravitational lensing, such a « bare » black hole could leave an observable imprint on a starry background. However, in typical astrophysical conditions, whatever its size and mass (ranging from stellar to galactic scales) a black hole is rarely bare but is dressed in gaseous material. Swirling in a spiral motion, the gas forms a hot accretion disk within which it emits a characteristic spectrum of electromagnetic radiation. Giant black holes, such as those currently lurking at the centers of galaxies, can be also surrounded by a stellar cluster, whose orbital dynamics is strongly influenced. As a matter of fact, , if a black hole remains by itself invisible, it “switches on” in its characteristic way the materials it attracts, and distorts the background starry field by gravitational lensing.

Thus, as soon as the basics of  black holes astrophysics developed in the 1970’s,  the scientists logically wondered what could look like a black hole. Many of you certainly saw didactic or artistic representations of a black hole in popular science magazines, in the form of a black sphere floating in the middle of a circular whirlwind of brilliant gas. So striking they are, these images fail to report the astrophysical reality. This one can be correctly described by means of numerical simulations, taking into account the complex distortions that the strong gravitational field prints in spacetime and light rays trajectories.

Since the first numerical simulations performed 40 years ago, tantalizing progress has been done to detect black holes through electromagnetic radiation from infalling matter or gravitational waves. The first telescopic image by the Event Horizon Telescope of the nearest giant black hole SgrA*, lurking at he center of our Milky Way galaxy, is expected for 2018.

The aim of this series of posts is to retrace the rich history of black hole imaging.

Preliminary steps

Black hole imaging started in 1972 at a Summer school in Les Houches (France). James Bardeen, building on earlier analytical work of Brandon Carter, initiated research on gravitational lensing by spinning black holes. Bardeen gave a thorough analysis of null geodesics (light-ray propagation) around a Kerr black hole. The Kerr solution had been discovered in 1962 by the New Zealand physicist Roy Kerr and since then focused the attention of many searchers in General Relativity, because it represents the most general state of equilibitum of an astrophysical black hole.

The Kerr spacetime’s metric depends on two parameters : the black hole mass M and its normalized angular momentum a. An important difference with usual stars, which are in differential rotation, is that Kerr black holes are rotating with perfect rigidity : all the points on their event horizon move with the same angular velocity. There is however a critical angular momentum, given by  a = M (in units where G=c=1) above which the event horizon would « break up » : this limit corresponds to the horizon having a spin velocity equal to the speed of light. For such a black hole, called « extreme », the gravitational field at the event horizon would cancel, because the inward pull of gravity would be compensated by huge repulsive centrifugal forces.

James Bardeen computed how the black hole’s rotation would affect the shape of the shadow that the event horizon casts on light from a background star field. For a black hole spinning close to the maximum angular momentum, the result is a D-shaped shadow.

Apparent shape of an extreme Kerr black hole as seen by a distant observer in the equatorial plane, if the black hole is in front of a source of illumination with an angular size larger than that of the black hole. The shadow bulges out on the side of the hole moving away from the observer (at right) and squeezes inward and flattens on the side moving toward the observer (at left).

The reference is Bardeen, J. M. 1973, Timelike and null geodesics in the Kerr metric, in Black Holes (Les Astres Occlus), ed. C. Dewitt & B. S. Dewitt, (New York: Gordon and Breach) p.215–239

At the time, C.T. Cunningham was preparing a PhD thesis at the University of Washington in Seattle, under the supervision of Bardeen. He began to calculate the optical appearance of a star in circular orbit in the equatorial plane of an extreme Kerr black hole, taking account of the Doppler effect due to relativistic motion of the star, and pointed out the corresponding amplification of the star’s luminosity. He gave formulas but did not produced any image.
The reference is Cunningham, C.T. and Bardeen J.M., The optical appearance of a star orbiting an extreme Kerr black hole, ApJ 173 L137-142 (1972).

One year later Cunningham and Bardeen published a more complete article with the same title. For the first time a picture was shown of the primary and secundary images of a point source moving in a circular orbit in the equatorial plane of an extreme Kerr  black hole. They calculated as functions of time the apparent position and the energy flux of the point source as seen by distant observers.

Apparent positions of the two brightest images as functions of time for two orbital radii and an observer art a polar angle 84°.024. The small, dashed circle in each plot gives the scale of the plot in units of M. The direct image moves along the solid line, the secundary image along the dashed line. Ticks mark the positions of the images at 10 equally spaced times.

In the upper diagram showing the distorted image of a circle of radius  20M, we clearly see that, whatever the observer’s inclination angle, the black hole cannot mask any part of the circle behind. We also see that the black hole’s spin hardly affects the symmetry of the primary image (although the asymmetry is stronger for the secundary image).
The exact reference is Cunningham, C.T. and Bardeen J.M., The optical appearance of a star orbiting an extreme Kerr black hole, 1973, ApJ, 183, 237. The article can be uploaded here.

In 1975, Cunningham calculated the effects of redshifts and focusing on the spectrum of an accretion disk around a Kerr black hole. He gave formulas and drawed graphics but no image.
The reference is  Cunningham, C. T., The effects of redshifts and focusing on the spectrum of an accretion disk around a Kerr black hole, ApJ, 202, 788 (1975)

In 1978 Leigh Palmer, Maurice Pryce and William Unruh carried out,  for pedagogical purpose, a simulation of starlight lensed by a camera orbiting a Schwarzschild black hole, using an Edwards and Sutherland Vector graphics display at Simon Fraser University. They showed a film clip in a number of lectures in that period, but unfortunately they did not publish their simulation, so that I can’t reproduce here any image.

First calculations for a black hole accretion disk

The same year and quite independently, as a young researcher at Paris-Meudon Observatory specialized in the mathematics of General Relativity, I wondered what could be the aspect of a Schwarzschild black hole surrounded by a luminous accretion disk. Continue reading

Geometry and the Cosmos (2) : From the Pre-Socratic Universe to Aristotle’s Two Worlds

 Sequel of the previous post Geometry and the Cosmos (1): Kepler, from polyedra to ellipses 

The Pre-Socratic Universe

Since He [Zeus] himself hath fixed in heaven these signs,
The Stars dividing; and throughout the year
Stars he provides to indicate to men
The seasons’ course, that all things may duly grow.
Aratus, Phaenomena, I, 18.

Although Kepler was the first to determine the motion of the planets by mathematical laws, his search for a rational explanation to the universe was anticipated by numerous earlier thinkers. Even before the time of Socrates a number of philosophers had broken away from accepted mythology and postulated the idea of universal harmony. From the sixth century BC increasingly rational and mathematical ideologies based on the laws of physics began to compete with the traditional belief that the world was controlled by gods with supernatural powers. Most of these thinkers attempted to describe natural phenomena in mechanical terms, with reference to the elements of water, earth and fire. The Ionian philosophers in particular developed new ideas about the heavens, whose signs were used by many of their compatriots to navigate between the islands. Their fundamental notion was that the universe was governed by mechanical laws, by natural principles which could be studied, understood and predicted.

It was Thales of Miletus who propounded one of the first rational explanations of the world, according to which the earth was separate from the sky. Anaximander and Anaximenes, both also natives of Miletus on the coast of Asia Minor, put forward different ideas, which nevertheless derived from the same rationale: they proposed the existence of cosmological systems, explained natural phenomena in terms of a small number of “elements”, and invented new concepts – Anaximander’s “equilibrium” and Anaximenes’ “compression” – which can be regarded as the first recognition of the force of gravity.

The Expanding Universe. According to Empedocles of Acragas (now Agrigento, in Sicily), the universe was held in balance by forces of harmony and conflict, the attractive force of love and the repulsive force of hate alternatively prevailing. This idea of balance can be seen as a mythical precursor of modern astronomical theories whereby the tendency for structures to become compressed by their own gravitational forces is offset by the expansion of the universe, which constantly dilutes all matter.
In Lemaître’s so-called “hesitating universe”, a cosmological model he devised in 1931 from Einstein’s field equations, the evolution of the cosmos is divided into three disctinct phases : two periods of rapid expansion are separated by a period of “stagnation”, representing a sort of equilibrium between the forces of gravitational contraction and expansion.

According to Heraclitus of Ephesus, the day was caused by exhalations from the sun, while the night was the result of dark emissions from the earth. The stars and the planets were bowls of fire which, when turned over, gave rise to eclipses and the phases of the moon. The moon itself, pale and cold, moved in the rarefied air above the earth, whereas the sun, our nearest star, was bright and hot.

Meanwhile, the Greeks were amassing measurements which would enable them to plot the stars more accurately. This required specialised instruments – gnomons to measure the sun’s shadow, compasses to fix the positions of the stars in the sky, etc. – as well as a system of notation which anyone could understand (previously the study of astronomy had been restricted to priests): how many fingers’ width above the horizon was such and such a star; where was due north, and so on. As well as mining the extensive archive of observations made by the Egyptians and Babylonians, the Greeks developed their own system of records. The pre-Socratic thinkers refined and analysed the basic ideas of their predecessors from Miletus with the result that the mechanistic view of the world gradually lost currency and a belief in underlying harmony became de rigueur. As early as 450 BC Anaxagoras of Clazomenae was accused of impiety for referring to the sun as a mass of hot metal, to the moon as a second earth and to the stars as burning stones – views no longer considered seemly. Continue reading

Geometry and the Cosmos (1) : Kepler, from polyhedra to ellipses

Introduction

The regularity of so much celestial activity has led many cultures to base their models of the universe on concepts of order and harmony. Around the Mediterranean it was the Pythagoreans who first expressed the idea that the universe is characterised by proportion, rhythm and numerical patterns. Plato’s hypothesis was of an organised cosmos whose laws could be deciphered, explained in geometric terms.

The history of physics is nothing other than the story of man’s desire to uncover the hidden order and harmony of things. The most ambitious physicists have attempted to unify apparently discrete phenomena: Galileo with terrestrial and celestial laws; Newton with gravity and the movement of celestial bodies; Maxwell with magnetism and electricity; Einstein with space and time; today’s physists with gravitation and microphysics.

But, as Heraclitus said as long ago as 500 BC, “Nature loves to hide.” Indeed advances in geometry and mathematics have led to new theories of the cosmos which we are unable to comprehend. They provide only abstract images, which do not allow us to visualise the structure of atoms or the dynamics of space-time or the topology of the universe in any direct sense.

It is this fundamental belief in celestial harmony – for which successive generations have found various elaborate expressions: just proportion, equation of the part and the whole, symmetry, constancy, resonance, group theory, strings -that has underlain the development of physics for the past 2,500 years.

Melancholy, or the Spirit of Man in Search of the Secret of the Universe. This etching, dating from 1514 according to the numbers in the square in the top right corner, depicts man contemplating the nature of the world in a state of melancholy, which in medieval times was associated with black bile and with the planet Saturn. The winged man prefigures Johannes Kepler’s interrogations as he calculates how to express the underlying harmony of the cosmos using spheres and polyhedra. The bright light in the sky is the great comet that was observed in the winter of 1513-14. As it shines on the scales (depicting the astronomical sign Libra) it symbolises the end of an earthly cycle, if not the end of time itself. The ladder with seven rungs represents the belief held by the Byzantines that the world would not exist for more than seven thousand years. It is the end of the Middle Ages; Diirer (1471-1528) is to be one of the prime movers of the Renaissance.

 

Geometry and the Cosmos

“Geometry, which before the origin of things was coeternal with the divine mind and is God himself […], supplied God with patterns for the creation of the world.”
Johannes Kepler, The Harmony of the World, 1619.

The 17th century German astronomer Johannes Kepler was undoubtedly the first to integrate man’s fascination with harmony into an overall vision of the world which can properly be called scientific. For Kepler, as for the natural philosophers of ancient Greece, the cosmos was an organised system comprising the earth and the visible stars. His avowed intention was to investigate the reasons for the number and sizes of the planets and why they moved as they did. He believed that those reasons, and consequently the secret of universal order, could be found in geometry. Kepler wanted to do more than create a simple model or describe the results of his experiments and observations; he wanted to explain the causes of what he saw. This makes him one of the greatest innovators in the history of science and it led him in particular to formulate laws of planetary motion which are still valid today.

Despite his innovative methods, Kepler wrote two studies of the cosmos in the style of the ancient Greeks: Mysterium Cosmographicum (The Secret of the Cosmos) in 1596 and Harmonices Mundi (The Harmony of the World) in 1619. At this turning point between ancient and modern thinking Kepler was steeped in a tradition which connected cosmology explicitly with the notion of divine harmony. But what Kepler sought to express was not the numerical mysticism of the Pythagoreans; his starting point was geometric patterns, which he saw as “logical elements”. His profound desire to devise a rational explanation for the cosmos led him to establish procedures which resembled those of modern science. Continue reading

The Rate of Expansion

There, where worlds seem, with slow steps,
Like an immense and well-behaved herd,
To calmly graze on the ether’s flower.
Giovanni Pascoli, Il Ciocco

A question often asked by the general public interested in cosmology about the expansion of the Universe is the distance scales on which it effectively acts. Before commenting on this, let me recall first some historical facts.

Georges Lemaître in 1927

In 1927, Georges Lemaître published a revolutionary article in the Annales de la Société scientifique de Bruxelles entitled “Un univers homogène de masse constante et de rayon croissant, rendant compte de la vitesse radiale des nébuleuses extragalactiques” (“A homogeneous universe of constant mass and increasing radius, accounting for the radial velocity of extragalactic nebulae.” As the title suggests, Lemaître showed that a relativistic cosmological model of finite volume, in which the Universe is in perpetual expansion, naturally explains the redshifts of galaxies, which at that point were not understood. In particular, the article contained a paragraph establishing that forty-two nearby galaxies, whose spectral shifts had been measured, were moving away at speeds proportional to their distances.

Lemaître gave the numerical value of this proportionality factor: 625 km/s per megaparsec, which means that two galaxies separated by 1 megaparsec (or 3,26 million light-years) moved away from each other at an apparent speed of 625 km/s, and that two galaxies separated by 10 megaparsecs moved apart at a speed ten times greater.

The paragraph of Lemaître’s paper in which he derives the law of proportionality between recession velocity and distance, later called the Hubble law.

This unit of measurement, the kilometer per second per megaparsec, shows clearly that the speed of recession depends on the scale. In 1377, in his Book of the Heavens and the World, the scholar Nicole Oresme had noted that, at dawn, one would not notice anything if the world and all living creatures had grown by the same proportion during the night. In Lemaître’s theory, on the contrary, the recession velocity between two points in space grows faster with greater separation, which renders it perceptible.

Eddington and Lemaître

Lemaître’s article, published in French, passed unnoticed until 1931, when it was finally read by Arthur Eddington, who published an English translation. Unfortunately, this version omits the paragraph in which Lemaître established his law of proportionality, see this article for all the details. Meanwhile, in 1929 the great American astronomer Edwin Hubble had published the experimental results he obtained with his collaborators and described a general law, according to which the speed of recession of a galaxy is proportional to its distance. This law, identical to Lemaître’s, with the same proportionality factor, would from now on carry the name of “Hubble’s law.” It forms the experimental basis for the theory of the expansion of the Universe, of which the big bang models are the fruit. Continue reading

Galaxies in Flight

This post is an adaptation of a chapter of my book  “The Wraparound Universe” with many more  illustrations.

 

Galaxies in Flight

                     The spawning galaxy in flight is a rainbow trout which goes
back against
the flow of time towards the lowest waters, towards the dark retreats of duration.
Charles Dobzynski (1963)

Since the time of Newton, we have known that white light, passing through a prism, is decomposed into a spectrum of all colors. Violet and blue correspond to the shortest wavelengths or, equivalently, to the largest frequencies; red corresponds to the largest wavelengths and to low frequencies. In 1814, the German optician Joseph von Fraunhofer discovered that the light spectrum from stars is streaked with thin dark lines, while that from candlelight has bright stripes. These phenomena remained puzzling until 1859. It was then that the chemist Robert Bunsen and the physicist Gustav Kirchhoff analyzed the light created from the combustion of different chemical compounds (burned with the now-famous Bunsen burner) and saw that each of them emitted light with its own characteristic spectrum.

Fraunhofer lines in the solar spectrum

At nearly the same time, Christian Doppler discovered in 1842 that moving the source of a sound produced shifts in the frequency of sound waves, a phenomenon experienced by anyone listening to the siren of an ambulance passing by. The French physicist Armand Fizeau noticed the same phenomenon with light waves: depending on whether a source of light was moving closer or farther away, the received frequencies are either raised or lowered with respect to the emitted frequencies. The shift becomes larger as the speed of displacement is increased. If the source is getting closer, the frequency grows, and the light becomes more “blue”; if it moves away, the frequency lowers and the wavelengths stretch out, becoming more “red,” with respect to the spectrum of visible light. Since this shift affects the whole spectrum by the same amount, it is easily quantified by looking at the dark or bright stripes, which are shifted together, either towards the blue or towards the red, and it furnishes an incomparable means of measuring the speed of approach or retreat for light sources.

Shortly after this discovery, astronomers began an ambitious program of spectroscopy, with the aim of measuring the speed of the planets and stars by using their spectral shifts. Continue reading

Expansion and the Infinite

This post is an adaptation of a chapter of my book  “The Wraparound Universe” with many more  illustrations.

Expansion and the Infinite

Space alike to itself
that it grows or denies itself
Stéphane Mallarmé

The Universe is expanding. What does this really mean? Most people imagine an original huge explosion, as the term “big bang” suggests, and the metaphor is constantly used in popular accounts. Some speakers even have the tendency to mime a gesture of expansion with their hands, as if they were holding a piece of space or an immaterial balloon in the process of inflating. The public imagines some matter ejected at prodigious speeds from some center, and tell themselves that it would be better not to be there at the moment of explosion, so as not to be riddled through with particles.

A misleading view of the expanding universe commonly used in popular science

None of all this is accurate. At the big bang, no point in the Universe participated in any explosion. Put simply, if one considers any point whatsoever, we notice that neighboring points are moving away from it. Is this to say that these points are animated by movement, given a speed? No, they are absolutely fixed, and nevertheless they grow apart.

To unravel this paradox, it is necessary to make more precise what one exactly means when speaking of a fixed point. The position of a point is fixed by coordinates: one number for a line (the miles along a highway), two numbers for a surface (latitude and longitude), and three for space in general (length, width, and height). A point is said to be fixed if its coordinates do not change over the course of time. In an arbitrary space, curved or not, the distance between two points is given by the so-called metric formula, which depends on the coordinates and generalizes the Pythagorean theorem. In principle, therefore, the distance between two points does not vary. In an expanding space, on the other hand, this distance grows, while the points do not move, even by a millimeter, meaning that they strictly conserve the same coordinates. These fixed coordinates are known as “comoving” coordinates. In relativistic cosmology, galaxies remain fixed at comoving positions in space. They may dance slight arabesques around these positions, under the influence of local gravitational fields, but the motion which moves them apart from each other resides in the literal expansion of the space which separates them. Continue reading

Non-Euclidean Geometries

This post is an adaptation of a chapter of my book  “The Wraparound Universe” with many more  illustrations.

************************************************************************

Thus we may perhaps, one day, create new Figures
that will allow us to put our trust in the Word,
in order to traverse curved Space, non-Euclidean Space.
Francis Ponge[1]

The oldest known fragment of Euclid’s Elements as part of the Oxyrhynchus papyri, dated from the Ptolemaic period and belonging to the famous Alexandrian Library

 

In book I of the Elements,[2] Euclid poses the five “requests” that, according to him, define planar geometry. These postulates would become the keystone for all of geometry, a system of absolute truths whose validity seemed irrefutable. One of the reasons for this faith is that these postulates seem obvious: the first of them stipulates that a straight line passes between two points, the second that any line segment can be indefinitely prolonged in both directions, the third that, given a point and an interval, it is always possible to trace out a circle having the point for its center and the interval as its radius, the fourth that all right angles are equal to each other. The fifth postulate is however less obvious:

As the sum of the interior angles α and β is less than 180°, according to the fifth postulate the two straight lines extended indefinitely, meet on that side.

“If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.”

 

 

Although the statement does not refer explicitly to parallel lines, the the fifth postulate is currently called “Parallel postulate”. This can be better understood given the more popular version of the fifth postulate due to the Scottish mathematician John Playfair (1748-1819), who demonstrated that it was equivalent to the one given by Euclid : “Given a straight line and a point not belonging to this line, there exists a unique straight line passing through the point which is parallel to the first“.

A picturesque English edition of Euclid’s Elements by Oliver Byrne, 1847.

 

Since the “parallel postulate” was more complicated than the others, the mathematicians following Euclid would try, for many centuries, to prove it from the four preceding ones, all in vain. In the nineteenth century, there occurred one of the great sudden revolutions in the history of mathematics (and also in human thought, as will be seen by what follows): two new geometries which do not satisfy the fifth postulate, but which are perfectly coherent, were discovered. In one of these geometries, called spherical geometry, no parallel line satisfying the conditions can be traced. This is the case for the surface of a sphere; the straight lines become great circles, whose planes pass through the center of the sphere, and since all great circles intersect each other at two diametrically opposed points (in the manner of the terrestrial meridians, which meet at the poles), no “straight line” can be parallel to another.[3] In the other geometry, called hyperbolic geometry, through any given point there passes an infinite number of lines parallel to another straight line. Continue reading

TOTAL SOLAR ECLIPSE 2017 : THE ROLE OF SOLAR ECLIPSES IN ASTROPHYSICS

In view of the total solar eclipse of 2017 Aug 21 through the United States, this is a reminder of the role of solar eclipses in the developments of astronomy and astrophysics.
It is taken from a chapter of my book Glorious Eclipses, presented elsewhere in this blog.

Summary

Eclipses of the Sun and Moon have never ceased to provide us with a host of lessons about the nature of the universe around us. The first of these lessons concerned the celestial bodies directly involved in eclipses: namely the Earth, Moon, and Sun. Indeed, back in antiquity, the proof that the Earth was round, and the first measurements of the respective sizes and distances of the Moon and Sun were deduced from the observation of eclipses. In the 19th century, it was the normally invisible atmosphere of the Sun that was revealed thanks to eclipses. Far from being the perfectly round, and sharply defined ball of hot gas that it appears to the eye – appropriately protected by suitable filters, of course – the Sun is found to be a sprawling giant, overflowing with energy, plasma, and particles, that extends its influence throughout the whole Solar System. Eclipses also provoked the discovery of helium, the second most abundant element in the Sun, and in the universe as a whole. In a more surprising manner, in the 20th century, Einstein’s General Relativity, a fundamental theory about space, was tested experimentally for the first time, thanks to an eclipse. It is on this new vision of the universe, which explains gravitation in terms of the ‘curvature of space-time’, that all our current knowledge of the origin, the structure, and the evolution of the universe, depends, by way of the fascinating concepts of an expanding universe, the Big Bang, and black holes.

During a total eclipse, the Sun’s invisible empire appears : eruptions and prominences rise above the level of the photosphere, and colour the Sun’s inner atmosphere, known as the chromosphere.
The Earth is round
According to Aristotle, lunar eclipses prove that the Earth is round. Indeed, if the Earth were square, or triangular, its shadow projected onto the disk of the Moon at the time of an eclipse would not appear circular. Aristotle’s geometrical argument is shown in several ancient astronomy texts, including Cosmographia, by Petrus Apianus and Gemma Frisius, 1581.

The first demonstration of an astrophysical nature resulting from eclipses is the one given by Aristotle concerning the fact that the Earth is round. The astronomical views of this Greek philosopher are well-known to us, thanks to his two works, known to us as Meteorology and On the Heavens, dating from the 4th century BC. Like other thinkers of his day, Aristotle believed that all heavenly bodies were spherical, because to him heavenly bodies were a reflection of divine perfection, and the sphere is the most outstandingly perfect geometrical figure. But this argument was not a physical demonstration, because, naturally, Aristotle did not have any experimental means of confirming the spherical nature of the planets and stars.
As far as the Moon was concerned, the philosopher adopted an explanation attributed to the Pythagoreans, namely that the observed appearance of the Moon throughout its various phases corresponded to a spherical body, half of which is illuminated by the Sun. As for the spherical nature of the Earth, the proof given by Aristotle is quite original: he notes that an eclipse of the Moon is caused by the shadow of the Earth, and that the circular shape to the edge of the shadow seen on the Moon’s surface implies that our world is spherical

Sizes and distances of the Moon and Sun

The golden age of Greek astronomy flourished at Alexandria. Since its foundation under the reign of Ptolemy Soter (3rd century BC), the Alexandrian school brought together brilliant mathematicians and geometers, such as Euclid, Archimedes, and Apollonius. Similarly, the greatest ancient astronomers Aristachus of Samos, Eratosthenes, and Hipparchus, as well as Ptolemy (2nd century BC), all worked there.

Aristarchus (310-230 BC) is nowadays known for having been the first to voice the heliocentric theory, i.e., that it is the Sun that reigns at the centre of the world system, not the Earth as was believed at the time. His statement does not appear in any known work, but it was reported by Archimedes and by Plutarch. The only work of Aristarchus that has come down to us relates to the sizes and distances of the Sun and the Moon.
The Alexandrian astronomer completely reopened this question, which had been discussed since the 4th century BC. The Pythagoreans had positioned the heights of the celestial bodies according to musical intervals. Eudoxus, the brilliant disciple of Plato, had estimated the diameter of the Sun as nine times that of the Moon. As for Aristarchus, he devised an ingenious geometrical method of calculating the distance ratios of the Sun and Moon.

On the Sizes and Distances of the Sun and the Moon is the only surviving work usually attributed to Aristarchus.

 

Aristarchus of Samos tried to calculate the relative diameter of the moon and sun, as deduced from the line subtending the arc that divides the light and dark portions of the moon during an eclipse.

 He found that the Sun lay at a distance between 18 and 20 times that of the Moon. (In fact, it is 400 times as far.) By an argument based on the observation of eclipses, he determined the diameter of the Moon as one third of that of the Earth, which is very close to the actual value. He also announced that the diameter of the Sun is seven times that of the Earth. Even though Aristarchus considerably underestimated the size of the Sun, because it is actually 109 times as large as the Earth, he had grasped the essential fact that the daytime star was much larger than the Earth. It was precisely this result that led him to the heliocentric hypothesis. He did, in fact, argue that under these circumstances, it was logical to believe that the Earth and the other celestial bodies revolved around the Sun, rather than the reverse. Aristachus was before his time. The world had to wait until 1543 and the work by Copernicus, before the heliocentric theory was again put forward, this time with success.

A century after Aristachus, and again at Alexandria, Hipparchus developed a complete theory of the Moon.   He defined the   lengths of the synodic month (or lunation, the period in which the Moon returns to the same position relative to   the   Sun);   the   draconitic month   (the period for the Moon to return to the same position   relative to the nodes of its   orbit);   and   the anomalistic month (the period for the Moon to return to perigee or apogee). The immense improvements that   Hipparchus brought to theories of the apparent motion of the Moon and Sun enabled him to have far more success than his predecessors in dealing with the problem of predicting eclipses, which had always   been   of   immense   interest.

Hipparchus   considerably   extended Aristarchus’ method: by observing the angular diameter of the shadow of the Earth at the Moon’s distance during a lunar eclipse, and comparing it with the known apparent diameters of the Sun and Moon (about half a degree), he obtained the ratio of the Earth-Moon and   Earth-Sun distances, giving one when   the   other   is   known.   Pappus, another famous   astronomer of the Alexandrian   school,   recounts   that Hipparchus     made     the     following observation of: “An eclipse of the Sun, which in the area of the Hellespont was precisely an exact eclipse of the whole Sun; such that none of it was visible,   but at Alexandria,   in   Egypt, about 4/5 of its diameter were hidden. By means of the foregoing arguments, [Hipparchus] showed that,   measured in   units   where   the   radius   of   the Earth has the value of 1, the smallest distance to the Moon is 71, and the larger 83. Whence the average of 77.

The total solar eclipse mentioned is that of 20 November 129 BC. The actual value of the Earth-Moon distance is 60,4 terrestrial radii.

During a total eclipse of the Sun, the electrified plasma hugs the lines of the sun’s magnetic field, just as iron filings follow those of a magnet. The corona displays narrow filaments over the poles, and extends in a more homogeneous manner in the equatorial regions.
The empire of the Sun

Continue reading

The Edge Paradox

The world has no outside, no beyond, since it contains and embraces everything
Guillaume d’Auvergne (De Universo, 1231)

If the Universe is finite, it seems necessary for it to have a center and a frontier. The center poses hardly any conceptual difficulty: it suffices to place the Earth there, like the geocentric systems of Antiquity (appearances lead one in this direction), or the Sun, as Copernicus did in his heliocentric system. The notion of an “edge” of the Universe is on the other hand more problematic.

Archytas, born in 428 BC in Tarentum (Italy) and died in 347, was a philosopher, mathematician, astronomer, statesman, and strategist. He belonged to the Pythagorean school and was famous for his scientific abilities.

In the fifth century BCE, the Pythagorean Archytas of Tarentum described a paradox that aimed to demonstrate the absurdity of having a material edge to the Universe. His argument would have a considerable career in all future debates on space: if I were at the extremity of the sky, could I extend my hand or stick out a staff? It is absurd to think that I could not; and if I could, that which is found beyond is either a material body, or space. I could therefore move beyond this once again, and so on. If there is always a new space towards which I can extend my hand, this clearly implies an expanse without limits. There is therefore a paradox: if the Universe is finite, it has an edge, but this edge can be passed through indefinitely.

This line of reasoning was taken up by the atomists, such as Lucretius, who gave the image of a spear thrown to the edge of the Universe, and afterwards by all the partisans of an infinite Universe, such as Nicholas of Cusa and Giordano Bruno. Continue reading

Willem Blaeu, a prolific cartographer and globemaker

Detail of a geographic map by W. Blaeu
SUMMARY
A portrait of W. Blaeu by Jeremias Falck, engraving from the Digitale bibliootheek voor de Nederlandse letteren.

Willem Janszoon Blaeu (1571-1638) founded one of history’s greatest cartographic publishing firms in 1599. Mostly renowned as a cartographer, he also made terrestrial and celestial globes, various instruments such as quadrants, a planetarium and a tellurium. He invented mechanical devices for improving the technics of printing. As an astronomer, a former student of Tycho Brahe, Willem Blaeu made careful observations of a moon eclipse, he discovered a variable star now known as P Cygni, and carried out a measurement of a degree on the surface of the earth (as his countryman Snell did in 1617).

THE LIFE AND WORK OF WILLEM BLAEU

The Blaeu family has its origin in the island of Wieringen, where about 1490, Willem Jacobszoon Blauwe – the grandfather of Willem – was born. From his marriage with Anna Jansdochter sprang six children. The second son, Jan Willemsz. (1527- before 1589) was the father of Willem Blaeu, and continued the family tradition by practicing the prosperous trade of herring packer. From his second marriage with Stijntge, Willem Jansz. Blaeu was born at Alkmaar or Uitgeest.

At an early age, Willem Blaeu went to Amsterdam in order to learn the herring trade, in which he was destined to succeed his father. But Willem did not like this work very much, being more inclined to Mathematics and Astronomy. He did not attend a university and worked first as a carpenter and a clerk in the Amsterdam mercantile office of his cousin Hooft.

Tycho Brahe and assistants making an astronomical observation at Uraniborg

However, in 1595 he became a student of Tycho Brahe (1546-1601). The celebrated Danish astronomer demanded a high standard of his pupils. Some were invited by him, others were undoubtedly taken on special recommendation. We may therefore presume that young Blaeu had reached a good standard of education and technical skill, since he was considered worthy to become a student of the great astronomer. Blaeu lived on the Island of Hven over the winter of 1595/1596, at Brahe’s famous observatory in Uraniborg. Thanks to this exact knowledge acquired from Brahe, Blaeu was able to make tables for sun declination ; especially he also learned from Brahe to make globes and instruments like the quadrants.

As it is well-known, Tycho Brahe had his own cosmic system, a sort of compromise between the Ptolemaic and Copernican. Willem Blaeu, although a supporter of the Copernican system, remained cautious during the rest of his career. In his books he mentioned the Copernican model as one of the existing theories, besides the Ptolemaic and Tychonic. It will not only save him for confrontations with religious people, but this attitude was also beneficial for his sales.

The Tychonic system of the world depicted in Andreas Cellarius atlas “Harmonia macrocosmica” (1660).

 After his return from Hven in 1596, Blaeu settled in Alkmaar. Very little is known of his stay here. He married, probably in 1597, Marretie or Maertgen, daughter of Cornelis from Uitgeest. Here too, his eldest son Joan was born. Continue reading

A brief history of space (4/4)

Sequel of the preceding post A Brief History of Space (3/4) : From Descartes to Schwarzschild

Cosmology developed rapidly after the completion of general relativity by Albert Einstein, in 1915. In this theory, the Universe does not reduce to a space and a time which are absolute and separate; it is made up of the union of space and time into a four dimensional geometry, which is curved by the presence of matter.

Albert Einstein (here in 1910) developed the theory of relativity and was awarded the 1921 Nobel prize for physics. Image by © Hulton-Deutsch, Collection/CORBIS

It is in fact the curvature of space-time as a whole which allows one to correctly model gravity, and not only the curvature of space, such as Clifford had hoped. The non-Euclidean character of the Universe appeared from then on not as a strangeness, but on the contrary as a physical necessity for taking account of gravitational effects. The curvature is connected to the density of matter. In 1917, Einstein presented the first relativistic model for the universe. Like Riemann, he wanted a closed universe (one whose volume and circumference were perfectly finite and measurable) without a boundary; he also chose the hypersphere to model the spatial part of the Universe.

Einstein static universe in a space-time diagram.

At any rate, Einstein’s model made the hypothesis of a static Universe, with the radius of the hypersphere remaining invariable over the course of time. In truth, the cosmological solutions of relativity allow complete freedom for one to imagine a space which expands or contracts over the course of time: this was demonstrated by the Russian theorist Alexander Friedmann, between 1922 and 1924.

At the same time, the installment of the large telescope at Mount Wilson, in the United States, allowed for a radical change in the cosmic landscape. In 1924, the observations of Edwin Hubble proved that the nebula NGC 6822 was situated far beyond our galaxy. Very rapidly, Hubble and his collaborators showed that this was the case for all of the spiral nebulae, including our famous neighbor, the Andromeda nebula: these are galaxies in their own right, and the Universe is made up of the ensemble of these galaxies. The “island-universes” already envisaged by Thomas Wright, Kant and Johann Heinrich Lambert were legitimized by experiment, and the physical Universe seemed suddenly to be immensely enlarged, passing from a few thousand to several dozen million light-years at the minimum. Beyond this spatial enlargement, the second major discovery concerned the time evolution of the Universe. In 1925, indications accumulated which tended to lead one to believe that other galaxies were systematically moving away from ours, with speeds which were proportional to their distance. Continue reading

A brief history of space (3/4) : from Descartes to Schwarzschild

Sequel of the preceding post A Brief History of Space (2/4) : From Ptolemy to Galileo

At the beginning of XVIIth century, the way was open for new cosmologies, constructed on the basis of infinite space. Until then, the notion of space was conceived in the cosmological and physical order of nature, and not as the “background” of the figures and geometric constructions of Euclid. In other terms, physical space was not mathematicized. It became so thanks to René Descartes (1596 – 1650), who had the idea of specifying each point by three real numbers: its coordinates. The introduction of a universal system of coordinates which entirely criss-crossed space and allowed for the measurement of distances was a reflection of the fact that, for Descartes, the unification and uniformization of the universe in its physical content and its geometric laws was a given. Space is a substance in the same class as material bodies, an infinite ether agitated by vortices without number, at the centers of which were held the stars and their planetary systems.

A portrait of René Descartes

This new conception of the cosmos upset philosophical thought and led it far from the initial enthusiasm of the atomists and Giordano Bruno: “The absolute space which inspired the hexameters of Lucretius, the absolute space which that had been a liberation for Bruno, was a labyrinth and an abyss for Pascal.”[5] As for the scholars, they did not allow themselves to be discouraged by these moods and irresistibly moved towards the infinite universe.

The Descartes system of the world using vortices

The tendency toward the radical geometrization of an infinite space, initiated by Descartes, was consummated by the Englishman Isaac Newton (1642-1727). Newton postulated an absolute space, encompassing not only the background space of mathematics and the physical space of astronomy, but also that of metaphysics, since space was the “sensorium of God.” Physical space, finally identified with geometrical space, was necessarily Euclidean (the only one known at the epoch), without curvature, amorphous and infinite in every direction. At the heart of this immobile framework, Newton explained celestial mechanics in terms of the law of universal attraction, from now on considered responsible for gravitation and the large scale structure of the Universe. With Newton, cosmology took root for more than two centuries in the framework of an infinite Euclidean space and an eternal time.

Newton around 1700

All the problems are not resolved in Newtonian cosmology, far from it. On the question of the distribution of stars in space, for example, Newton believed that they must occupy a finite volume since, he argued, if they occupied an infinite space, they would be infinite in number, the force of gravitation would be infinite, and the universe would be unstable. Newton moreover supposed that the stars were uniformly spread within a finite mass|like a galaxy, for example. But a problem of instability remained: since each celestial body is attracted by every other one, at the least movement, at the least mechanical perturbation, all the bodies in the universe would fall towards a unique center, and the universe would collapse. Newton’s universe is therefore only viable if it does not admit motion on the large scale: its space is rigid and its time immobile. Continue reading

A Brief History of Space (2/4) : from Ptolemy to Galileo

Sequel of the preceding post A Brief History of Space (1/4).

The cosmology of Aristotle, as perfected by Ptolemy and reintroduced thanks to arabic translations and commentaries, was adapted to satisfy the demands of the theologians. Notably, that which is situated beyond the last material sphere of the world acquired the status of, if not physical, at least ethereal or spiritual space. Baptized “Empyrean”, it was considered to be the residence of God, the angels and the saints. The medieval cosmos was not only finite, but quite small: the distance from the Earth to the sphere of the fixed stars was estimated to be 20,000 terrestrial radii, because of which Paradise, at its edge, was reasonably accessible to the souls of the deceased. The Christian naturally found his place at the center of this construction.

In this medieval system of the world designed by Apianus in 1520, the whole universe is finite, bounded by a spherical layer containing the fixed stars, beyond which lies the Empyreum, the house of God and Saints.
In this medieval system of the world designed by Apianus in 1520, the whole universe is finite, bounded by a spherical layer containing the fixed stars, beyond which lies the Empyreum, the house of God and Saints.

A Hierachical universe according to Dante's Divine Comedy
A Hierarchical universe according to Dante’s Divine Comedy

 

Nicholas of Cusa
Nicholas of Cusa

This model of the universe imposed itself until the seventeenth century, without nevertheless impeding the resurgence of atomist ideas. After the rediscovery of the manuscript of Lucretius De rerum natura, the German cardinal Nicholas of Cusa (1401-1464) argued in favor of an infinite Universe, of a plurality of inhabited worlds, and of an Earth in motion. However, his arguments remained primarily metaphysical: the universe is infinite because it is the work of God, who could not possibly be limited in His works.

When the Polish canon Nicolaus Copernicus (1473-1543) proposed his heliocentric system, in which the Sun is at the geometric center of the world while the Earth turns around it and around itself, he kept the idea of a closed cosmos, surrounded by the sphere of fixed stars. Even if this is two thousand times further away than in the Ptolemeian model, the universe nevertheless remained bounded.

A romantic depiction of Copernicus
A romantic depiction of Copernicus and his new model of the universe

We must wait several decades more for the first cracks to appear in the Aristotelian edifice. In 1572, a new star was observed by the Dane Tycho Brahe (1546-1601), who showed that it was situated in the sphere of fixed stars, that is to say in the celestial region until then presumed to be immovable. In 1576, the Englishman Thomas Digges (1545-1595), a staunch Copernican, maintained that the stars were not distributed on a thin layer, at the surface of the eighth and last sphere of the world, but extended endlessly upwards. Digges nevertheless was not proposing a physical conception of infinite space: for him, the sky and the stars remained Empyrean, God’s realm, and in this regard did not truly belong to our world.

copernicus-diag Syst-Digges

An epistemological rupture[1] was triggered by two Italian philosophers. In 1587, Francesco Patrizi (1529-1597) produced Of Physical and Mathematical Space[2], where he put forth the revolutionary idea that the true object of geometry was space in itself, and not figures, as had been believed since Euclid. Patrizi inaugurated a new understanding of infinite physical space, in which it obeyed mathematical laws and was therefore accessible to understanding. But it is above all his contemporary Giordano Bruno (1548-1600) who is attributed with the true paternity of infinite cosmology. brunoThe first book of his De immenso is entirely dedicated to a logical definition of infinite space. Bruno argued from physical, and no longer exclusively theological, basics. His cosmological thought was inspired by the atomism of Lucretius, the reasoning of Nicholas of Cusa, and the Copernican hypothesis. From the latter, Bruno retained heliocentrism and the ordering of the solar system, but rejected the cosmological finitism. A precursor to Kepler and Newton, he also refuted the cult of sphericity and of uniform circular motion for describing celestial motion. His bold and original writings were not understood by his contemporaries, most notably Galileo. Above all they were firmly opposed by the Church. In fact, the true philosophical subversion of the end of the sixteenth century did not reside so much in the heliocentric affirmation of Copernicus as in that of the infinite multiplicity of worlds. Camped at the front ranks of the anti-Aristotelian battle, Bruno, carried away by his passion for infinity, refused to abjure and was burned at the stake in Rome.

Johannes Kepler (1571-1630), another great artisan of the astronomical revolution, tried at first to construct a universal model founded on the use of particular geometric figures: the regular polyhedra. He failed at this attempt; the ordering of planetary orbits as calculated did not correspond to the new experimental data collected by Tycho Brahe. After discovering the elliptical nature of the planetary trajectories, Kepler overturned the Aristotelian dogma of circular and uniform motion as the ultimate explanation of celestial movement. He nevertheless refused to follow Bruno in his arguments for the infinitude of the universe. He considered this notion to be purely metaphysical and, since it was not founded on Kepler_Epitomeexperiment, denuded of scientific meaning: “In truth, an infinite body cannot be understood by thought. In fact the concepts of the mind on the subject of the infinite refer themselves either to the meaning of the word `infinite,’ or indeed to something which exceeds any conceivable numeric, visual or tactile measurement; that is to say to something which is not infinite in action, seeing that an infinite measurement is not conceivable.”[3] Kepler supported his argument by expressing for the first time an astronomical paradox that seemed to be an obstacle to the concept of infinite space, and which would be extensively discussed: the “paradox of the dark night.” Just like the edge paradox, this problem would not be satisfactorily resolved until the middle of the nineteenth century, although by completely different arguments.

Starting in 1609, the telescope observations of Galileo (1564-1642) furnished the first direct indications of the universality of the laws of nature. On the question of spatial infinity, however, Galileo, like Kepler, adopted the prudent attitude of the physicist: “Don’t you know that it is as yet undecided (and I believe that it will ever be so for human knowledge) whether the Universe is finite or, on the contrary, infinite?” [4]

systemegalilee001

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[1] See Alexander Koyre, From the Closed World to the Infinite Universe, New York: Harper Torchbook, 1958.

[2] De spacio physico et mathematico ; See R. Brickman, “On Physical Space, Francesco Patrizi”, Journal of the History of Ideas, vol.4, 224 (1943).

[3] De stella nova, 1606. Unfortunately there is no English translation for this masterpiece.

[4] Letter to Ingoli, quoted in A. Koyré, From the Close World to the Infinite Universe, The John Hopkins Press 1957, p. 97.

A Brief History of Space (1/4)

This post is based on a chapter of my book  “The Wraparound Universe” but is much more illustrated.  The chapter is divided into 4 parts, here is the first one.

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That which keeps quiet beyond everything, is this in fact simply what I name Space? . . . Space! An idea! A word! A breath!
Jean Tardieu

There is no space or time given a priori; to each moment in human history, to each degree of perfection of our physical theories of the Universe, there corresponds a conception of those fundamental categories of thought known as space, time, and matter. To each new conception, our mental image of the Universe must adapt itself, and we must accept that “common sense” was found lacking. For example, if space is limited by a boundary, what is there beyond it? Nothing? It is difficult to imagine that, by voyaging sufficiently far in a given direction, one could reach a point beyond which nothing more exists, not even space. It is just as troubling to think of an infinitely large Universe. What would be the meaning of any measurable, that is to say finite, thing with respect to the infinite?

A possible representation of Anaximander learning Pythagoras on his left, detail of Raphael's famous painting The School of Athens.
A possible representation of Anaximander learning Pythagoras on his left, detail of Raphael’s famous painting The School of Athens.

These types of questions were formulated in the sixth century BCE, in ancient Greece, where they rapidly became the object of controversy. The first schools of scholars and philosophers, called “presocratic” (although they were spread over two centuries and were quite different from each other), each attempted in their way to rationally explain the “world,” meaning the ensemble formed by the Earth and the stars, conceived as an organized system. For Anaximander, from the school of Miletus, the world where observable phenomena take place was necessarily finite. Nevertheless, it was plunged within a surrounding medium, the apeiron, corresponding to what we today consider as space. This term signifies both infinite (unlimited, eternal) and indefinite (undetermined). For his contemporary, Thales, the universal medium was made of water, and the world was a hemispheric bubble floating in the middle of this infinite liquid mass.

We meet up again with this intuitive conception of a finite material world bathing in an infinite receptacle space with other thinkers: Heraclitus, Empedocles, and especially the Stoics, who added the idea of a world in pulsation, passing through periodic phases of explosions and deflagrations.

Atomism, founded in the fifth century by Leucippus and Democritus, advocated a completely different version of cosmic infinity. It maintained that the Universe was constructed from two primordial elements: atoms and the void. Indivisible and elementary, (atomos means “that which cannot be divided”), atoms exist for all eternity, only differing in their size and shape. They are infinite in number. All bodies result from the coalescence of atoms in motion; the number of combinations being infinite, it follows that the celestial bodies are themselves infinite in number: this is the thesis of the plurality of worlds. The formation of these worlds is produced within a receptacle without bounds: the void (kenon). This “space” has no other property than being infinite and accordingly matter has no influence on it: it is absolute, given a priori.

Part of a fresco in the portico of the National University of Athens representing Anaxagoras.
Part of a fresco in the portico of the National University of Athens representing Anaxagoras.

The atomist philosophy was strongly criticized by Socrates, Plato, and Aristotle. Moreover, by affirming that the universe is not governed by gods, but by elementary matter and the void, it inevitably entered into conflict with the religious authorities. In the fourth century BCE, Anaxagoras of Clazomenae was the first scholar in history to be accused of impiety; however, defended by powerful friends, he was acquitted and was able to flee far from the hostility of Athens. Thanks to its two most illustrious spokesmen, Epicurus (341-270 BCE), who founded the first school that allowed female students and Lucretius (first century BCE), author of a magnificent cosmological poem, On the Nature of Things, atomism continued to flourish until the advent of Christianity. It was however marginalized over the course of the first centuries of the christian era, and would not again be part of mainstream science until the seventeenth century. Continue reading

My books (3) : Celestial Treasury

Until now I published as an author 30 books in my native language (French), including 14 science essays, 7 historical novels  and 9 poetry collections (for the interested reader, visit my French blog  here.
Although my various books have been translated in 14 languages (including Chinese, Korean, Bengali…), only 4 of my essays have been translated in English.

The third one was :

Celestial Treasury: From the music of the spheres to the conquest of space.

Translated from French by Joe Laredo
Cambridge University Press, 2001 — ISBN 0 521 80040 4

FigCiel-engThroughout history, the mysterious dark skies above us have inspired our imaginations in countless ways, influencing our endeavours in science and philosophy, religion, literature and art. Celestial Treasury is a truly beautiful book showing the richness of astronomical theories and illustrations in Western civilization through the ages, exploring their evolution, and comparing ancient and modern throughout. From Greek verse, mediaeval manuscripts and Victorian poetry to spacecraft photographs and computer-generated star charts, the unprecedented wealth of these portrayals is quite breathtaking.

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PRESS REVIEWS

• Review in Astronomy & Geophysics, 2001 December (Vol.42), 6.32
Big and beautiful

This is such a book as would have the most hardened reviewer reaching for the overworked superlatives. Impressive in size and sumptuous in production, for what is actually quite a reasonable price in present-day terms, it contrives to set forth much of the aesthetic attraction of astronomy both ancient and modern.

Originating as an exhibition catalogue and drawing material from many libraries in Europe, the authors have marshalled a stunning array of historical and modem imagery under the general headings of “The harmony of the world”, “Uranometry”, “Cosmogenesis”, and “Creatures of the sky”. Originally published in French as Figures Du Ciel, a title which implies a much more restricted scope than it actually bas — the English title is far more appropriate — it is here elegantly translated by Joe Laredo. Not the least of its virtues is that as the original edition was jointly published by the Bibliothèque Nationale, the authors have been able to obtain readier access to the treasures of that institution than many other researchers find possible, especially since the move.

Many of the illustrations from conventional astronomical rare books are familiar, though the hand-colouring of different copies makes a fascinating comparison, but others are less so — apart from the unique manuscript sources, the authors have made appropriate use of decorative embossed book covers, illustrations from l9th and 20th century books, especially early science fiction, early space art and even comic books. It can be a trifle disconcerting to find, for example, a modern map of the cosmic microwave background radiation juxtaposed with a l4th century manuscript, but such comparisons can be quite reasonable as long as they are not taken too literally; I feel, though, the series of illustrations comparing the illustrations of the days of creation from the Nuremberg Chronicle with stages in cosmic evolution and the development of life is a little forced. There are one or two isolated nods towards world views outside the main stream leading down from Classical via Arabic to modem western science, but the Hindu Triad and the brief nods towards Chinese, Aztec and Babylonian astronomy seem lonely and isolated and might have been better omitted if there was not room to treat them more fully.

Although the innumerable illustrations are the most prominent feature of the book, the authors’ impeccable credentials as high officials of the CNRS and as successful popularizers of astronomy lend the text authority and style. Occasionally, as used to be said of Sir James Jeans, they get lost in descriptions of immensity and hugeness; but then, in the words of the late Douglas Adams, “Space is big, really big!”. The authors have carefully described the significance of the thought behind the historic images, and the whole book will make a marvellous crib for captions and exhibitions, as well as being ideal fodder for picture researchers. One might pick up small factual disagreements and pedantic quibbles, or take issue with certain aspects of the book production; the truncated and varying sized pages seem to add little but confusion, and I am not clear why the key map from Doppelmaier’s New Celestial Atlas (p108) has been truncated. Lt is also a great shame that a proper index of subjects could not have been added rather than just one of names.

But despite any venial criticism of minutiae, the whole book is a striking demonstration of my own conviction that the most valuable use of historical imagery is to provide an accessible entry point to the subject; such beautiful images, intelligently explained, can engage the interest and commitment of the mathematically challenged in a way that the Schwarzschild Radius or the Chandrasekhar Limit will never do. A book that anybody with the slightest interest in the subject would be delighted to find waiting after the annual visit of the red-coated gentleman with the sub-orbital reindeer!

P D Hingley.

• Review in The Los Angeles Times, March 17, 2002

by Margaret Wertheim

The Sky’s the Limit

Artists and scientists, Robert Oppenheimer wrote, “live always at the ‘edge of mystery’–the boundary of the unknown” and for no group of scientific practitioners is this characterization more apposite than cosmologists, they who dare to envision the universal whole.

Few areas of inquiry bring human minds so constantly into contact with the event horizons of current understanding, so posing the greatest challenges. As a creative response to the ineluctable desire to know how and from whence we arise, cosmological theorizing, for all its claims to truth, is an exercise of the grandest sort in myth-making. That at least is the thesis underlying Marc Lachièze-Rey and Jean-Pierre Luminet’s sumptuous “Celestial Treasury.”

Ostensibly a history of (primarily Western) cosmological thinking, “Celestial Treasury” advances a far more radical agenda. Rather than presenting their subject as a progressive history, onward and upward from pagan darkness to the light of contemporary scientific genius, Luminet and Lachièze-Rey subversively interweave ancient and modern ideas, continually, if gently, alerting the reader to profound resonances between past and present.

For all our superior observational technology, our sophisticated theoretical frameworks and our fiendishly complex mathematics, we are not so far from our forebears as we often like to think.

Consider the ancient Greeks’ idea that everything in the physical world is composed of four basic elements: earth, water, air and fire. Antiquated baloney, you might think, but Luminet and Lachièze-Rey point out that contemporary physics rests on a not-dissimilar premise.

Today the four “elemental” constituents said to be responsible for all phenomena are the four fundamental forces: gravity, the electromagnetic force and the strong and weak nuclear forces (which hold together the nuclei of atoms). These forces, they write, “have an identical function to the elements of the classical world.”

In our drive to know the universe, it is the imagination that engages first, long before the analytical or empiricist spirit kicks in. Johannes Kepler, the great precursor to Isaac Newton and the founding father of modern astrophysics, envisioned the universe as God’s play: As he saw it, the aim of the astronomer was to learn to play God’s game. To do that, the mind must be open to the “facts,” but critically it must also be creatively susceptible. As Albert Einstein once declared: “The gift of fantasy has meant more to me than any talent for abstract, positive thought.”

Throughout history, the creative impulse has been a central engine of cosmological theorizing. Take, for example, the Greek and medieval view that the dance of the planets and stars must be explained by a combination of strictly circular motions. Just as a windup ballerina can be made to perform a complex dance, even though her mechanism consists only of circular gears, so cosmologists for 2,000 years believed the motions of the heavenly bodies could be described by an intricate celestial clockwork.

The apotheosis of this imaginative mechanizing was the dizzyingly elaborate system of the Alexandrian astronomer Claudius Ptolemy. So complex was Ptolemy’s system that in the 13th century Alfonso the Great, seeing the labors of his astronomers, is said to have remarked that had he been present at the Creation he would have given the Lord some hints about simplification. The Ptolemaic conception of the cosmos dominated both Arab and European views of the heavens until the 17th century, when Kepler, Newton and others radically re-envisioned the universe, replacing the cosmic gears with a quasi-infinite network of stellar masses held in place by the force of gravity.

But be not so quick to judge Ptolemy’s vision. Luminet and Lachièze-Rey (an astronomer and astrophysicist, respectively) note that in principle a Ptolemaic-style system could account for the heavenly dance with a high degree of accuracy. In the 19th century, the French mathematician Jean Baptiste Joseph Fourier demonstrated that, in fact, any periodic motion can be described by a combination of circular motions.

Moreover, physics today retains a love affair with the circle. Current favorite contender for a unified theory of the four forces is string theory, which holds that all particles can be understood as the various vibrational states of microscopic circular loops, or “strings.”

Throughout history, cosmological ideas have refracted again and again through our mental prisms, metamorphosing into new variations on old themes. One of the great joys of this book is seeing the ways in which certain tropes keep returning, as if they hold some peculiar power of enchantment over the human mind.

Perhaps my favorite example is the continually recurring fascination with the Platonic solids: a unique set of five forms whose crystalline symmetry has held artists and astronomers, mystics and mathematicians in thrall for thousands of years. As with the cube, whose faces are all squares, the Platonic solids are perfectly regular polyhedra, having all their faces the same. There are just five such forms possible: along with the cube (which has six sides), are the tetrahedron (four sides), the octahedron (eight sides), the icosahedron (20 sides) and the dodecahedron (12 sides). Since their discovery, these five forms have been imbued with almost mystical power.

Plato paired the first four with the four basic elements: Earth was paired with the cube, water with the icosahedron and so on. The fifth, the dodecahedron, he equated with the supposed fifth element, or quintessence, the mysterious substance of which the celestial bodies were said to be composed.

In the 17th century, Kepler thought he had found in these five forms the secret of the planets’ arrangement in the solar system. He turned out to be wrong, but, bizarrely, the idea of a polyhedral arrangement to the cosmos has resurfaced within the framework of general relativity, which allows for some truly extraordinary topologies, including ones in which space takes on a pseudo-crystalline structure.

One such arrangement is an infinite lattice of dodecahedrons. “Celestial Treasury” includes an exquisite computer image of this enigmatic spatial structure from the Geometry Center at the University of Minnesota.

In making their case for cosmological resonances through the ages, Luminet and Lachièze-Rey critically rely not just on words but also on pictures. The uniqueness of this book lies in its juxtaposition of historical images with those generated by contemporary astrophysics, such as the contrasting of Kepler’s polyhedral model with the Minnesota computer model.

Likewise, illustrations from medieval manuscripts of the six days of biblical creation sit side by side with computer simulations of black holes and the origins of space time; Renaissance visions of stellar vortexes are paired with photographs of spiral galaxies taken by the Hubble Space Telescope.

Replete with extended foldouts and delicately detailed inserts, “Celestial Treasury” is a stunningly beautiful survey of the science, mythology and iconography of the cosmos through the ages. This is the most gorgeous coffee-table cosmology book in years.

Such lavish production bespeaks its origins: The book is an offshoot of a 1998 exhibition entitled “Figures du Ciel” at the Bibliothèque Nationale de France, and it is from that library’s extensive collection that most of the older images are taken.

As with two recently ended and superb exhibitions in our own city–“Treasures of the Great Libraries of Los Angeles” at the UCLA Hammer Museum and “Devices of Wonder” at the Getty–“Celestial Treasury” demonstrates that science can be an engine not only of knowledge but also of aesthetic inspiration. Beneath the radar of pedagogical impulse, science, like art, stirs our imaginations.

Margaret Wertheim is the author of “The Pearly Gates of Cyberspace: A History of Space From Dante to the Internet.”

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EXCERPT FROM CHAP. 1 “THE HARMONY OF THE WORLD”

 

The Fabric of the World

Any macroscopic conception of the universe is also a conception of its microscopic structure. The question whether matter can be broken down indefinitely is an ancient one. The Greeks in particular put forward ideas as to its composition: in keeping with their belief in a rational, unified and harmonious universe, not only did there have to be a limited number of fundamental elements, but there must also be laws governing their combination and transformation.

The Milesian philosophers showed their preoccupation with coherence and their aspiration towards universality by asserting the predominance of one element over all others in the universe: for Thales it was water, for Anaximenes air, for Heraclitus fire and for Anaximander apeiron — a forerunner of Aristotle’s ether.

The search for a single primordial element is a recurring feature of natural philosophy. In the fifth century bc Parmenides of Elea and his pupil Zeno hypothesised that the universe consisted of a single substance, “Oneness”, which was motionless and of infinite mass, enveloping all things without any space between them. In the 12th century ad the English prelate Robert Grosseteste, who is widely considered to be one of the founding fathers of empirical science, regarded light as the most elementary substance and the primary constituent of the world. This idea, which was in keeping with Christian doctrine (God created light, from which everything else followed), is also close to modern thinking, especially that of physicists who are seeking a unified theory of matter and the interaction between bodies.

In the mid-fifth century bc the philosopher and soothsayer Empedocles (who claimed divine ancestry) offered a synthesis of his predecessors’ ideas: he proposed that the primordial material was a combination of the four eternal and incorruptible elements, earth, water, air and fire. In his Origin of the Elements he describes how these elements temporarily combine to form the various sublunary bodies: “Birth is not the beginning of life, and death is not the end; the components of our bodies are merely assembled and disassembled. Birth and death are but the names given to this process by man.”

Elements and Polyhedra

The four element system was adopted by Plato, whose Timaeus describes the sublunary world as subject to a series of transformations – birth-corruption-death – of the four elements brought into being by the Creator. Plato then develops his ideas on representing the cosmos in geometric terms. This sublunary world is far less harmonious than the superlunary world; consequently each of the four elements comprising it must be represented by a shape somewhat less symmetrical, less pleasing and perfect than a sphere, i.e. one of five “Platonic solids”, nowadays known as regular polyhedra. Earth is shown by a cube, water by an icosahedron, air by an octahedron and fire by a tetrahedron (or pyramid). Plato thought carefully about the relationship between each element and its representative shape: for example, a cube is the most difficult shape to move, so it is associated with earth, the heaviest element; an icosahedron has more sides than any other Platonic solid (five triangles meet at each point), giving it a virtually round, fluid structure which is most clearly associated with water; and so on.

Being corruptible, these four elements cannot exist in the sky. There is, however, a fifth regular polyhedron, the dodecahedron, vhich consists of 12 pentagons and is therefore the three-limensional equivalent of the pentagon, considered to be a “magic” nape. To Plato each perfect solid represented the essence of its :orresponding element so, when his contemporary Theaetetus pointed out to him that the dodecahedron was the fifth regular polyhedron (there are only five), Plato postulated a fifth essence in prder to unify his geometric model of the world. In the Middle Ages pis fifth essence was to be known as “quintessence”, but in ppinomis, a work published after Timaeus and generally attributed to Plato, it is called aither, meaning “eternal flux”. Of the five jerfect solids the dodecahedron is the nearest to a sphere, the symbol of celestial perfection.

In general Aristotle appropriated the Platonic model of the :osmos, but he did not adopt the idea of a correspondence between he elements and the regular solids. His main concern was to iistinguish between two worlds, between two kinds of activity. The mperfect sublunary world was governed by “lower activity”: everything is composed of corruptible matter and tends to revert to he natural state of its predominant element. For example, if a stone s dropped, it naturally falls towards the centre of the earth, since :arth is its predominant element. Fire, on the other hand, rises into he air. All natural movement is directed either upwards or iownwards, either towards or away from the centre of the earth [then considered to be at the centre of the cosmos). As for the four dements, they are distinguished by their basic characteristics: earth s cold and dry, air is warm and humid, etc. If these characteristics ire changed, one element can be transformed into another. In the iublunary world such transformations are continually taking place, .vhich accounts for its constantly changing nature.

The Aristotelian picture is completed by the “upper activity” of he superlunary world of planets and stars, whose “physicality”, mquestionably real and tangible as it is, consists of the fifth element, quintessence.

Plato’s polyhedra, which Kepler himself used to account for the orbital motion of the planets, were central to the way the world was represented in succeeding centuries. In the Renaissance, artists such as Piero della Francesca and Paolo Ucello were fascinated by them. phis Divine Proportion of 1509, which was illustrated by Leonardo la Vinci, the mathematician and theologian Luca Pacioli used them 😮 define his laws of just proportion, applicable to music, architecture, calligraphy and other arts.

The Nature of the Ether

After Aristotle the nature of the fifth element changed repeatedly and was the subject of constant debate. In the fourth century BC one of his successors, Theophrastus, rejected the idea of a fifth element altogether, likening the sky to a sort of “ethereal fire” (Manilius, Astronomicon Poeticon). Three hundred years later Xenarchus also dismissed Aristotle’s ether and argued that there could not be only two basic geometric shapes (a straight line and a circle). It was not until the 16th century that the ether was redeemed. Descartes mentions a “subtle substance” filling all space and accounting for his “vortices” and consequently all cosmic interaction. The 17th century Dutch physicist Christiaan Huygens applied the idea to light, which he regarded as a disturbance of an omnipresent “luminiferous ether”. For Newton light was composed of particles and independent of Huygens’ undulatory theory. On the other hand he postulated a kind of “gravitational ether”, a substance occupying all space and capable of transmitting the force of gravity. The idea of light being composed of waves was revived in the 19th century, when it was thought to be transmitted by vibrations in the ether. According to James Clerk Maxwell, electromagnetic waves were also propagated by the ether.

The exact nature of the ether has been the subject of repeated controversy. Does it have physical properties? How does it relate to space, to quantum fields, to a vacuum? Does it move relative to the earth? This last question was the subject of a series of experiments (the most famous being the Michelson-Morley experiment in 1887) and hypotheses leading to the theories of relativity which revolutionised man’s concept of the ether. According to the general theory of relativity, it is the distortion (or curvature) of space-time that conveys gravitational interaction and ripples of space-time that convey energy. The curvature of space-time has exactly the same characteristics as Newton’s “gravitational ether”: it is simultaneously physical and geometric. The other great theory of modern physics, quantum physics (more precisely quantum field theory), presents an alternative view of the ether as the “quantum vacuum”, which has energy and fluctuations but whose nature is still unclear. The concepts of “gravitational ether”and quantum vacuum are central to their respective disciplines, yet the two are irreconcilable. This is the impasse facing modern physics. Its resolution may yet come from further investigation into the nature of the ether, the vacuum, space..

Atoms

The extraordinary popularity of Aristotle’s system of elements meant that the alternative view of matter being composed of atoms, although at least as logical and persuasive, did not develop until the 17th century.

An atomistic structure was proposed as early as the fifth century bc, a generation before Socrates, by Leucippus of Miletus (or Elea) and his disciple Democritus of Abdera, whose work is best known through Aristotle’s Metaphysics. While Leucippus is usually credited with few books, including The Great World System, Democritus was a prolific author known to have written at least 52 books, although some of them are quite short. His Concise World System is a continuation of Leucippus’ work.

Leucippus and Democritus were reacting against the theories of Parmenides and Zeno of Elea, rejecting their idea of an infinite, static, all-embracing substance. They argued that our senses detect movement and that consequently there must be empty space. Their conclusion was a bipartite structure: “The first principles of the universe are atoms and empty space; everything is merely thought to exist.” The four elements of Empedocles have no place in their theory; nor does any such thing as quintessence. Everything is composed   of   atoms   (from the Greek atomos, meaning “indivisible”), which cannot be further broken down because they contain no empty space (for things to be broken apart there must be space between them) and are incredibly compact and heavy. Atoms are also eternal; in other words they have always existed and they cannot change or die. In his treatise De Generatione et Corruptione Aristotle records Democritus’ and Leucippus’ view that these “indivisible bodies” are “are infinite both in number and in the forms which they take, while the compounds differ from one another in their constituents and the position and arrangement of these.”31 The less space there is in a material, the denser it is. Fire, for example, is simply matter whose atoms are widely separated: being of low density it escapes the material that produces it. Celestial bodies are made from the least dense material.

Atomistic theory, which was pursued by Epicurus and Lucretius, was completely overshadowed by Aristotelianism but revived in the 17th century, particularly by the French philosopher Pierre Gassendi, and has become the basis of modern physics, even though our ideas about atoms are quite different from those of the ancient Greek philosophers.

The atom of contemporary science, far from being a compact mass, consists mostly of empty space in which tiny particles revolve around an extremely dense nucleus. The ancient atom in fact corresponds more closely to our idea of an elementary particle. These particles, which constitute the fundamental “building blocks” of every known chemical element, are indivisible masses surrounded by space. The fact that they exist throughout the universe has been proved by spectral analysis of the most distant stars and galaxies.

Harmony and Particles

The regular solids used by Plato and Kepler to represent the elements and the planetary orbits are absolutely symmetrical. To create a concrete model of the planets’ orbits, which were puzzling astronomers, Kepler used geometric shapes that he knew were symmetrical, thereby expressing his vision of cosmic harmony. Today physicists are faced with similar problems: how should they classify the numerous particles which experiments suggest are elementary or fundamental? How should they interpret the similarities and differences between them? How should they analyse their actions and reactions? Like Kepler they have looked to their stock of tools for expressing harmonious or symmetrical relationships for an answer. This they have found in the mathematical theory of groups, which permits the classification of geometric symmetries.

According to modern group theory to each regular polyhedron corresponds a “polyhedral group” (all possible displacements it can be subjected to without changing shape). To a sphere corresponds a higher dimensional group, but even this is just a particular case of a general class of transformations, the “symmetry groups”. In particle physics certain particles are associated with other particles or families of particles, forming “gauge theories”. Gauge theories allow the behaviour of particles – especially the way they interact -to be precisely described. The result is a “harmonious” classification of particles and their interactions, e.g. U(l), SU(2), SU(3).

These theories are undoubtedly more successful than Kepler’s geometric explanation of planetary orbits; yet no physicist would pretend to understand any better than Kepler did where such harmony comes from. He at least was able to interpret it as the will of God!

A Polyhedral Universe

Regular polyhedra appear again in relation to the structure of matter. Although Kepler was forced to abandon them in favour of ellipses as a method of describing the structure of the solar system, he retained his fascination for these almost perfect shapes. In looking at the group of semi-regular polyhedra (rhomboids, prisms, etc.) which incorporates the group of regular solids but also includes non-convex shapes, Kepler discovered “stellation”. In his De Niva Sexangula (The Six-Cornered Snowflake) of 1610 he used five- and six-pointed star shapes to represent the structure of snow crystals, thereby laying the foundation of modern crystallography. In crystallography, polyhedral symmetry reigns supreme. In the 18th century alchemy gave way to the less far-reaching but more rational science of chemistry, which is concerned with the geometric structure of molecules and crystals, underlying that of matter itself. Many molecules have extraordinary structures. The numerous possible arrangements of carbon atoms and other similar atoms, which are rich in symmetry, are often types of polyhedron. An entire branch of organic chemistry, which plays such an important part in modern chemistry, is based on the benzene molecule (C6H ), a beautiful hexagonal shape. Not long ago chemists discovered an even more remarkable molecule: fullerene (C60), which consists of a football-shaped polyhedron surrounded by sixty carbon atoms. First simulated in 1985 this molecule has already generated a new branch of applied chemistry and the number of possible applications for fullerene is still rising. It has recently been detected – as the ion C6+0 – in outer space, where it absorbs light from distant stars, and is the largest molecule (more precisely the largest chemical complex) known to exist in space. It is estimated that such molecules account for some two per cent of all the carbon in the universe.

Polyhedra even have an unexpected relevance to modern cosmology, which is investigating the possibility that space itself is in some way polyhedral and that the cosmos as a whole has a crystallographic structure. According to the general theory of relativity, space has a geometric structure characterised by curvature and topology. This idea fascinated several of the founders of 20th century cosmology such as De Sitter, Friedmann and Lemaitre; it then rather lost favour before regaining popularity in recent years. In “topologically multi-connected” models, which can initially seem confusing, space is represented by a “fundamental polyhedron”. The simplest of these models use cubes or parallelepipeds (shapes consisting of six parallelograms) to create a “toroidal” space, but there is an almost infinite number of variations. The common feature of these fundamental polyhedra is that they are symmetrical, so that one face can be related to another: the corresponding points on each face are therefore “linked” in such a way that physical space is the result of a complex “folding” process. The fundamental geometric symmetry of the universe is matched by the symmetry of the polyhedron.

From the point of view of the celestial observer, these “folded” models introduce a radically new perspective. The usual interpretation of the sky is that it consists of a straightforward projection of the space, which is vast if not infinite: each point of light that we can see corresponds to a specific star, galaxy or other celestial body – the further away the fainter. This is not at all the case with a multi-connected model, according to which each actual celestial body is represented by a whole series of “ghosts” so that what we see in the sky is not the universe as it really is, but several different images of the universe, from different angles and distances, superimposed upon one another!

Symmetry in Modern Physics

Symmetry is one of the most fundamental concepts in geometry, whose principal concern is to find “pure” shapes – the equivalent of the physicist’s search for fundamental elements. One of the simplest symmetrical shapes is the sphere, which is symmetrical with respect to any straight line passing through its centre. Others are the regular polyhedra (the Platonic solids), which are symmetrical with respect to a finite number of lines passing through their centre.

Symmetry is so prevalent in nature – from the human body to atoms and crystals – that it is difficult to imagine it not being central to our understanding of the world and its creation. Although symmetry was studied by the French mathematician Evariste Galois in the early 1830s and by the German Emmy Noether around 1916, its importance was not fully understood until the development of group theory later in the 20th century. Symmetry is also omnipresent in the arts. The (subjective) notion of beauty is, however, often associated with a slight asymmetry. The most beautiful faces are not exactly symmetrical; the best architects mix the symmetrical with the unexpected. Similarly physicists study symmetry breakdowns and show how these are as fundamental to nature as symmetry itself.

In recent times physics has become increasingly concerned with geometry, emphasising the “Platonic” nature of modern science. A striking example is quantum theory and in particular the theory of elementary particles; in attempting to describe the structure of atoms, in other words the invisible universe, particle physicists have resorted to abstractions based on geometric concepts. Just as Pythagoras himself would have approved of the quantum numbers representing the various levels of placement of electrons around an atomic nucleus, so Plato would have been delighted by the shape of the mathematical wave functions describing the hydrogen atom.