Johannes Kepler – the first scientific crystallographer

Johannes Kepler's (1571−1630) best-known contribution to science is his three laws of planetary motion. He was a pioneer in model building, which he extended into the third dimension, though his famous planetary model proved to be a non-discovery. He discussed the stacking of equal spheres when examining snowflakes and advanced the notion that internal structure determines the external shape. This was relevant to a fundamental issue in crystallography, and he has been considered the first scientific crystallographer.

Science has made great use of Kepler's works over the centuries, but little is known of the man himself. Galileo did not give him much recognition, and Newton did not mention him in most parts of his Principia or did so only in passing [1]. Kepler's principal languages were German and Latin, and his penetration into English literature was slow. Then, in the 19th century, two monographs in English were devoted to Kepler [2, 3]. The first complete biography of Kepler was published in German by the great Keplerian scholar Max Caspar and translated into English in 1959 [4]. Arthur Koestler's monumental book, The Sleepwalkers, also appeared in 1959 [5]. Part of it is a biography of Johannes Kepler, which was also published separately as a little volume entitled The Watershed [6]. These publications broke the ice and did a great deal to make Kepler better known. Since then, there has been a growing amount of literature about Kepler.

Kepler's three laws of planetary motion have earned him a solid place in the domains of physics and astronomy. Just as a reminder:

1. The orbit of a planet is an ellipse, with the Sun at one focus.
2. The line joining a planet to the Sun sweeps out equal areas in equal times.
3. The squares of the periods of the planets are proportional to the cubes of their mean distances from the Sun.

Kepler's work relevant to crystallography is documented in the Historic Atlas of Crystallography [7] and more extensively in our volume about the role of symmetry in scientific discoveries, giving emphasis to model building [8].

Closely packed spheres by Kepler [9].

Kepler's treatise, De Nive Sexangula (The Six-Cornered Snowflake), was published in 1611 in Latin [9], followed by an English translation 355 years later.

The cover of De Nive Sexangula and J. M. Wacker von Wackenfels' portrait by Aegidius Sadeler in the Scientific Library of the City of Trier, Germany (in the public domain; file: Johann Matthäus Wacker von Wackenfels.jpg - Wikimedia Commons).

Kepler wrote this 24-page treatise as a New Year's gift to his patron, J. M. Wacker von Wackenfels (1550−1619), thereby making his name immortal in science history. Wacker von Wackenfels was a scholar and author interested in philosophy and history. He originated from modest circumstances and worked himself up the societal ladder through scholarship, hard work and advantageous marriages. He received support for his studies and earned hereditary nobility and a position in the high court of the Holy Roman Empire.   

Kepler was the first credited with discussing the problem of packing equal spheres. In this work, he discussed the structure of the honeycomb and, when trying to figure out why snowflakes have hexagonal shapes, he considered the question in three dimensions. This contrasts with Descartes, who, when discussing snowflakes a quarter of a century later, considered them only in a plane [10].

In De Nive Sexangula [9], Kepler realized that external shape is correlated with internal structure. The external polyhedral shape grows naturally because of the internal regular arrangements of the building elements. In contrast, a symmetrical polyhedral shape does not necessarily mean an ordered internal structure. This is obvious today, yet even some widely used introductory chemistry texts state that crystals and amorphous bodies differ in their shapes: it is easy to cut a piece of glass to look perfect, but it will still remain amorphous. Alan L. Mackay called our attention to the importance of this knowledge for the courtesans in sixth-century India, according to the Kama Sutra by Vatsyayana [11]. They had to learn mineralogy as they were often paid in precious stones. They had to be able to distinguish genuine crystals from paste. 

Kepler felt the excitement of discovery when looking at snowflakes in cold, dry weather. They were all hexagonal in their full shape and symmetry, but they came in a great variety, and no two looked the same. As he was considering the origin of these perfect shapes, he concluded that the uniform arrangement of their elemental building particles must be the reason for these shapes. These building particles were water molecules, but of course, he could not know this. However, he correctly assigned the formation of shapes to very dense internal packing. Furthermore, he correctly described cubic closest packing, hexagonal packing and simple cubic packing. He could not explain why the snowflakes have sixfold symmetry but conjectured that hexagonal packing would provide the maximal close packing. Thomas C. Hales referred to Kepler's conjecture as "No packing of congruent balls in Euclidean three-space has a density greater than that of the face-centered cubic packing". Hales published his proof of Kepler's conjecture in 2005 in the Annals of Mathematics [12] and dedicated his work to the memory of the Hungarian geometer László Fejes Tóth.

Statue of Sir Walter Raleigh by William McMillan in Greenwich (photograph by the Hargittais) [13] and portrait of Thomas Harriot, 1602, by an unidentified painter from the collection of the University of Oxford (in the public domain; file: ThomasHarriot.jpg - Wikimedia Commons).

Kepler's discovery is considered a foundation of modern crystallography, though he may not have been the first to recognize how the densest packing of congruent spheres is accomplished. As often happens in science, there was another person, Thomas Harriot (ca 1560−1621), who, in England in 1599, came up with a similar observation of the arrangement of densest packing. He was an astronomer, mathematician and ethnographer; he improved navigational techniques and created advanced maps and participated in expeditions. He was the scientific advisor on one of Sir Walter Raleigh's (1552 or 1554−1618) expeditions around 1587. Raleigh posed Harriot the question about the most economical means of storing cannon balls in his ship. This was thus a question about the densest packing of equal-size spheres. Both the question poser and the one who responded deserve our note. Harriot corresponded with Kepler about optics, but there is no information on whether their exchange contained any reference to densest packing.  

Kepler's treatise inspired Alan L. Mackay to respond in 1981, 370 years later, with his paper De nive quinquangular [14]. Mackay predicted then what were later to be called 'quasicrystals'. For a beautiful computer-automated drawing of a 'Pentagonal snow crystal' by Robert H. Mackay in 1975, see [15]. Philip Ball celebrated the fourth centenary of Kepler's publication about the six-cornered snowflake with an essay in Nature [16]. There and elsewhere, see for example [17, 18], are presented the attempts over centuries to understand and explain the hexagonal shapes of snowflakes. Shafranovskii called attention to Kepler's attempt to create a pattern of fivefold symmetry [19].

Kepler graduated as a theologian, but his first position, at the age of 20, was as a teacher of mathematics and astronomy in Graz, Austria. He was unhappy with this job and found an escape in thinking about Copernicus' new ideas concerning the solar system, the six planets and their distances from the Sun. Then, while drawing some figures on the blackboard, a new idea suddenly came to his mind. It was like a lightning bolt, and he was so shaken that he noted the date when this happened: 9 July 1595. The idea never left him for his entire life. He started with a large circle with a triangle in it and another circle within the triangle. He suddenly realized that the ratio of the two circles was the same as that of the orbits of Saturn and Jupiter, the two outermost planets. He continued inscribing into the next interval between Jupiter and Mars a square, between Mars and Earth a pentagon, and between Earth and Venus a hexagon. He calculated the ratios of the radii of the subsequent circles and compared them with the results of Copernicus. He was soon disappointed because they did not match. At that point came yet another lightning bolt, the realization that he should expand his considerations into the third dimension, just as the planets exist in three dimensions, in space rather than merely in the plane. So, he used spheres rather than circles and polyhedra rather than polygons. 

Kepler's planetary model based on regular solids [20], and a Hungarian postage stamp with Kepler's portrait and his model.

The five regular polyhedra of Plato were the perfect solution, matching the six planets that were known at the time. Except for the orbits of Saturn and Mercury, his results mostly agreed with those of Copernicus. To eliminate the discrepancies, he gave some thickness to his orbits, and this approach improved the agreement. This is also how he arrived at the great discovery of the elliptical orbits of the planets. Kepler wrote a small book, Mysterium Cosmographicum [20]. This was in 1597, when he was 26 years old. This book already contains the seeds of all of Kepler's later discoveries. It also shows him as a modern scientist who knew that his model could be right only if it were consistent with observation. It was in this connection that he introduced the concept of observational error. Later in his life, when he realized that his model still had problems, he extended his considerations to general harmonies. He augmented his geometrical model with the principles of harmony. By doing this, he returned to ancient notions, which was a retreat, but he had no other recourse, as at the time, there was no other recourse. Kepler expressed the complementary nature of geometry and harmony in his Harmonices Mundi (1619) [21]. 

Kepler's planetary model was incorrect. Some considered it childish. Others recognized its fertility for a host of other scientific endeavors. This was not a unique case in science history when an incorrect model paved the way to correct results. Aaron Klug quoted the philosopher A. N. Whitehead, who co-authored Principia Mathematica with Bertrand Russell: "It is more important that an idea be fruitful than that it be correct." When Klug compiled his Nobel lecture for publication, he included a picture representing his original idea of nucleation, which turned out to be erroneous. Klug wanted to include it because it showed how science was a process of establishing the truth (emphasis by Klug) [22]. Klug's notion reflected what Kepler himself noted: "The roads by which men arrive at their insights into celestial matters seem to me as worthy of wonder as those matters in themselves" [23]. 

In conclusion, let us quote Koestler concerning this matter ([6], p. 61):

"That some of his answers were not right does not matter. As in the case of Ionian philosophers of the heroic age, the philosophers of the Renaissance were perhaps more remarkable for the revolutionary nature of the questions they asked than for the answers they proposed. Paracelsus and Bruno, Gilbert and Tycho, Kepler and Galileo formulated some answers that are still valid; but first and foremost, they were giant question masters. Post factum, however, it is always difficult to appreciate the originality and imagination it required to ask a question which had not been asked before. In this respect, too, Kepler holds the record."

References

[1] Holton, G. (1973). Thematic Origins of Scientific Thoughts: Kepler to Einstein, p. 89. Cambridge, MA: Harvard University Press.
[2] Small, R. (1963). An Account of the Astronomical Discoveries of Kepler, a reprint of the 1804 text, with a foreword by W. D. Stahlman. Madison: The University of Wisconsin Press.
[3] Brewster, D. (1847). "Life of John Kepler". In The Martyrs of Science Or, The lives of Galileo, Tycho Brahe, and Kepler, pp. 187−240. New York: Harper & Brothers.
[4] Caspar, M. (1959). Kepler, translated and edited by C. D. Hellman. New York: Abelard-Schuman Ltd.
[5] Koestler, A. (1959). The Sleepwalkers. London: Hutchinson and Co.
[6] Koestler, A. (1960). The Watershed: A Biography of Johannes Kepler. Garden City, NY: Doubleday and Co.
[7] Lima-de-Faria, J. (1990). Editor. Historical Atlas of Crystallography. Dordrecht: Kluwer Academic Publishers.
[8] Hargittai, I. & Hargittai, M. (2000). In Our Own Image: Personal Symmetry in Discovery, ch. 3, pp. 26−51. New York: Kluwer Academic/Plenum.
[9] Kepler, J. (1966). De Nive Sexangula. English translation: The Six-Cornered Snowflake. Oxford: Clarendon Press.
[10] Descartes, R. (1637). Discours VI, Les Meteores, Leiden; cited in J. V. Field (1988). Kepler's Geometrical Cosmology, p. 173. University of Chicago Press.
[11] Vatsyayana (1963). Kama Sutra: The Hindu Ritual of Love, p. 14. New York: Castle Books.
[12] Hales, T. C. (2005). "A proof of the Kepler conjecture", Ann. Math. 162, 1065–1185.
[13] Hargittai, I. & Hargittai, M. (2021). Science in London: A Guide to Memorials, p. 34. Springer Nature.
[14] Mackay, A. L. (1981). "De Nive Quinquangula: On the Pentagonal Snowflake", Sov. Phys. Crystallogr. 26, 517–522.
[15] Hargittai, I. & Hargittai, B. (2020). "Penrose–Mackay–Shechtman and bringing down the dogma about fivefold symmetry", IUCr Newsletter, Vol. 28, No. 4.
[16] Ball, P. (2011). "On the Six-Cornered Snowflake", Nature, 480, 455.
[17] Hargittai, M. & Hargittai, I. (2009; 2010). Symmetry through the Eyes of a Chemist, 3rd ed., pp. 40−53. Springer.
[18] Danzer, L., Grünbaum, B. & Shephard, G. C. (1982). "Can All Tiles of a Tiling Have Five-fold Symmetry?", Am. Math. Mon. 89, 568–585.
[19] Shafranovskii, I. I. (1975). "Kepler's Crystallographic Ideas and His Tract The Six-Cornered Snowflake". In Beer, A. & Beer, P. (Editors), Kepler, Four Hundred Years, Vistas in Astronomy, Vol. 18, pp. 861−876. Oxford; New York: Pergamon Press.
[20] Kepler, J. (1981). Mysterium Cosmographicum (The Secret of the Universe), translated by A. M. Duncan. New York: Abaris Books.
[21] Kepler, J. (1997). The Harmony of the World, translated into English with an introduction and notes by E. J. Aiton, A. M. Duncan and J. V. Field, Memoirs of the American Philosophical Society, Vol. 209, p. 490. Philadelphia, PA: American Philosophical Society.
[22] Hargittai, I. (2002). Candid Science II: Conversations with Famous Biomedical Scientists, edited by M. Hargittai, ch. 20, pp. 306−329 (the actual reference is made to the top of p. 313). London: Imperial College Press.
[23] Kepler, J. (1609). Astronomia Nova, summary of ch. 45, cited in The Watershed, p. 59 (which served as our source for this quote).