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**The use of mathematics to read the book of nature.
About Kepler and snowflakes**

**Capi Corrales Rodrigáñez**

*Department of Algebra, Faculty of Mathematics, Complutense University of Madrid, Madrid, Spain*

**Abstract.** “Philosophy is written in that great book which ever lies before our eyes – I mean the universe – but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometric figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth,” wrote Galileo (Il Saggiatore, chapter 6, p. 4). In 1611, the mathematician Johannes Kepler, a contemporary of Galileo and voracious reader of the book of the world, wrote his shortest and most surprising book, The Six-Cornered Snowflake: A New Year’s Gift. “Even as I write these things, it has begun to snow again, and more thickly than before. I have been attentively observing the tiny particles of snow, and although they were all falling with pointed radii, they were of two kinds. Some were exceedingly small, with varying numbers of radii that spread in every direction and were plain, without tufts or striations. These were most delicate, but at the same time joined together at the center in a somewhat larger droplet; and they were the majority. Sprinkled among them were the rarer, six-cornered snowflakes” (Kepler, 1611). This text by Kepler, little known outside the physics and mathematics community, marked a milestone in the use of mathematics to understand a part of the physical world that surrounds us. With this text as a map, this article covers part of the terrain explored by geometry, from the 3rd century AD until today.

**About the author**

Capi Corrales Rodrigáñez is professor of the Department of Algebra of the Faculty of Mathematics of the Complutense University of Madrid. She obtained her Ph.D. at the University of Michigan (USA), where she specialized in the theory of algebraic numbers. She combines research with the scientific popularization of contemporary mathematics.

Fancy some Christmas ornaments that defy conventional Euclidean geometry? In these animations, created by mathematician and artist Jos Leys, you can see what Christmas ball patterns would look like in hyperbolic space, a “bizarro world” where parallel lines can intersect and the three angles of a triangle add up to less than 180 degrees.

The different ornament designs show various ways that 2D hyperbolic space can be tiled with polyhedron shapes by varying the Poincaré disc model. Developed by Henri Poincaré of Poincaré conjecture fame these tesselations have also been explored by artist MC Escher in a series of prints.

For more mathematical imagery, check out a gallery of art in the hyperbolic realm, hyperbolic home decoration or visit Leys’ web site. You may also like to watch our series of One-Minute Math animations.

“Why aren’t we giving our students a chance to even hear about these things, let alone giving them an opportunity to actually do some mathematics, and to come up with their own ideas, opinions, and reactions? What other subject is routinely taught without any mention of its history, philosophy, thematic development, aesthetic criteria, and current status? What other subject shuns its primary sources— beautiful works of art by some of the most creative minds in history— in favor of third-rate textbook bastardizations?”