Snow crystals always appear to be of hexagonal symmetry, and Kepler seems to have been the first western scientist to point this out.187 In 1555 in Rome, the Swede Olaus Magnus (1490–1557), the Roman Catholic Archbishop188 of Uppsala, published his histories of the Northern people189 containing images of frost and snow190, although the Chinese philosopher Han Ying had already written of the six-pointed nature of the snow flakes around 135 BC. 191 Unlike the later works of Kepler and Descartes, the woodblock prints used in Olaus Magnus’ book did not acknowledge the typical six-fold nature of snow crystals, notwithstanding one single hexangular snowflake in the image in question.192
187 Philip Ball, Nature’s Patterns: a Tapestry in Three Parts; Branches, 2009, p. 2, and Ukichiro Nakaya, Snow Crystals: Natural and Artificial, 1954, p. 1. Ukichiro Nakaya (1900–1962) was the first person to succeed in producing artificial snowflakes in the laboratory. His book is a classic in the scientific study of snow crystals. His daughter Fujiko Nakaya (b. 1933) is an artist famous for her fog sculptures, creating the world’s first in 1970.
188 As Sweden was no longer Roman Catholic, this title was only nominal.
189 In Finnish, there is a great book available about snow crystals and their study throughout western history by Raimo Lehti (1931–2008), Lumihiutaleet ja maailmankuvat, [snowflakes and worldviews] 2000 (1998).
190 Olaus Magnus, Historia de Gentibus Septentrionalibus, Book 1, chapter 22. The image of snow and frost is on page 37 in the first 1555 edition, available online, for example, at https://archive.org/details/bub_gb_O9lEAAAAcAAJ (accessed 2015-08-19).
191 Ball, Branches (2009), p. 2.
192 Ibid. On the next page (p. 38), Olaus shows one classic way of enjoying fresh snow: boys having a snowball fight with a mighty snow castle. Sunt pueri pueri et pueri…
Figure 3.5 A quadrangular (A) and a triangular (B) packing of uniform spheres depicted in Johannes Kepler’s Strena Seu De Nive Sexangula (1611).
In the beginning of 1611, in between his larger scientific works, Kepler published the short, 24-page study Strena Seu De Nive Sexangula as a present dedicated to his friend and benefactor baron Wackher von Wackenfels.193 The title of the book means “A New Year’s Gift or On Hexagonal Snow”.194 Despite the name of Kepler’s book and the fact that it contains three images, none of these images actually depicts a snow crystal.195 What Kepler was looking for was not so much about describing real physical snowflakes196 but to ponder the processes and principles
193 The original 1611 edition is very rare, but it is available scanned, for example, at https://archive.org/details/den-kbd-pil-21055000404F-001 (accessed 2015-08-19). A critical edition of it is Johannes Kepler, The Six-Cornered Snowflake, Oxford University Press 1966, which is a bilingual Latin-English edition. A new printing was completed in 2014. This reprinting was clearly a response to another Latin-English bilingual edition published by Paul Dry Books, Philadelphia in 2010, which is based on the Latin text edited by M. Caspar and F. Hammer in Johannes Kepler Gesammelte Werke, München 1941, Vol. IV, pp. 259–280.
From other German translations of this snowflake-book, I will mention Neujahrsgabe oder vom sechseckigen Schnee, M. Caspar (ed.), Berlin 1943, Vom sechseckigen Schnee, Dorothea Goetz (ed.), Geest & Portig, Leipzig, (DDR) 1987, and Vom sechseckigen Schnee, Lothar Dunsch (ed.), Hellerau-Verlag 2005. See also Cecil J. Schneer [not Schnee], “Kepler’s New Year’s Gift of a Snowflake”, Isis, Vol. 51, Part 4 (December 1960), pp. 531–545.
194 Kepler’s small tractate is sometimes referred to in literature briefly as Strena, which is the
“New Year’s Gift” part of its title, or as De Nive Sexangula, which is the “On Hexagonal Snow”
part of its title. Between them, seu means “or” in Latin.
195 See, for example, https://archive.org/details/den-kbd-pil-21055000404F-001 (accessed 2015-08-19)
196 As was the American farmer and enthusiastic amateur photographer Wilson A. Bentley Figure 3.6 Wilson A. Bentley, photographs of snow crystals, 1931, detail (left), and Ukichiro Nakaya, the classification of regular planar snow crystals, 1954, detail (right).
behind the mysterious and so far unexplained six-fold symmetry-principle, which snowflakes so rigidly seem to obey. But where does this six-fold symmetry come from?
In his De Nive Sexangula, Kepler hypothesized that the six-cornered external structure of snowflakes reflected their internal structure. He speculated that the internal structure of the snow crystals could be depicted with uniform regular spheres representing atoms or other corresponding unobservable tiny particles.
If a fairly large number of uniform spheres or discs are placed on a flat plane, as close to each other as possible, the result is a regularly repeating structure. In such a triangular structure, the centres of the spheres, or discs, take positions in the vertices of equiangular triangles, and six spheres always surround one at the centre; see Fig. 3.5 (B). Two decades after Kepler, the French philosopher René Descartes (1596–1650) also made observations on snow crystals. His drawings of them were published in 1635 in Amsterdam, and they are nowadays thought to be the first actual realistic-looking depictions of the six-folded snowflakes.197 Descartes cherished the same idea as Kepler, as evidenced in his writing: “And the small little clusters of ice […] are obliged to arrange themselves in such a way that each has six others surrounding it; one cannot conceive of any reason that would prevent them from doing this, because all round and equal bodies that are moved in the same plane by the same kind of force naturally arrange themselves in this manner, as one can see by an experiment, in throwing a row or two of completely round unstrung pearls confusedly on a plate, and shaking them, or only blowing against them slightly, so that they approach one another.”198
The English chemist and father of modern atom theory, John Dalton (1766–
1844), almost two hundred years later published the same explanation as Kepler and Descartes for the shape of a snowflake in his A New System of Chemical Philosophy (1808). He explained the images in his book’s Plate 3, reproduced here as Fig. 3.7, in the following way: “Fig. 5 represents one of the small spiculæ of ice
(1865–1931), who captured more than 5000 microscopic images of snowflakes in his lifetime. Snow Crystals by W. A. Bentley and W. J. Humphreys, 1931, a classic of the subject, is still in print to this day by the Dover Publications from 1962 onwards. To achieve greater visual impact and sharper edges, Bentley used to render the backgrounds of his white snowflakes totally black by cutting them out carefully with scissors from the duplicates, but not from the original negatives, as it is sometimes stated. See Bentley (1931), p. 14 and Fig.
3.6 in this thesis.
197 Nakaya (1954), p. 1.
198 John G. Burke, Origins of the Science of Crystals, 1966, pp. 35–38. Burke refers in turn to Charles Adam and Paul Tannery (eds.), Œuvres de Descartes, 12 volumes, Paris, 1897–1913, Vol. VI, pp. 288.
formed upon the sudden congelation of small water cooled below the freezing point […]. Fig 6 represents the shoots or ramifications of ice at the commencement of congelation. The angles are 60° and 120°.”199
It should be noted that the small curvy areas in between the spheres, or discs, are always left uncovered as a consequence of the circular form. With regular uniform triangles, squares, or hexagons, 100% of the plane can be covered. 200 With uniform spheres, or discs, the densest possible packing covers only c. 90.69% of the plane.201 This optimal configuration was depicted by Kepler; Fig. 3.5 (B), and by John Dalton; Fig. 3.7 (2). The square-based packing of spheres, or discs, is also seen in Figs. 3.5 (A), and 4.7 (1). This quadrangular packing is even less dense, covering only c. 78,54% of the plane.