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The Six-Cornered Snowflake
"In 1611, Kepler wrote an essay wondering why snowflakes always had perfect, sixfold symmetry. It's a simple enough question, but one that no one had ever asked before and one that couldn't actually be answered for another three centuries. Still, in trying to work out an answer, Kepler raised some fascinating questions about physics, math, and biology, and now you can watch in wonder as a great scientific genius unleashes the full force of his intellect on a seemingly trivial question, complete with new illustrations and essays to put it all in perspective."—io9, from their list "10 Amazing Science Books That Reveal The Wonders Of The Universe"
When snow began to fall while he was walking across the Charles Bridge in Prague late in 1610, the eminent astronomer Johannes Kepler asked himself the following question: Why do snowflakes, when they first fall, and before they are entangled into larger clumps, always come down with six corners and with six radii tufted like feathers?
In his effort to answer this charming and never-before-asked question about snowflakes, Kepler delves into the nature of beehives, peapods, pomegranates, five-petaled flowers, the spiral shape of the snail's shell, and the formative power of nature itself. While he did not answer his original question—it remained a mystery for another three hundred years—he did find an occasion for deep and playful thought.
"A most suitable book for any and all during the winter and holiday seasons is a reissue of a holiday present by the great mathematician and astronomer Johannes Kepler…Even the endnotes in this wonderful little book are interesting and educationally fun to read."—Jay Pasachoff, The Key Reporter
New English translation by Jacques Bromberg
Latin text on facing pages
An essay, "The Delights of a Roving Mind" by Owen Gingerich
An essay, "On The Six-Cornered Snowflake" by Guillermo Bleichmar
Snowflake illustrations by Capi Corrales Rodriganez
John Frederick Nims' poem "The Six-Cornered Snowflake"
Notes by Jacques Bromberg and Guillermo Bleichmar
Johannes Kepler (1571-1631) was an important figure in the seventeenth century astronomical revolution. He is best known for his eponymous laws of planetary motion. Kepler wrote: "If there is anything that can bind the heavenly mind of man to this dusty exile of our earthly home…then it is verily the enjoyment of the mathematical sciences and astronomy."
When snow began to fall while he was walking across the Charles Bridge in Prague late in 1610, the eminent astronomer Johannes Kepler asked himself the following question: Why do snowflakes, when they first fall, and before they are entangled into larger clumps, always come down with six corners and with six radii tufted like feathers?
In his effort to answer this charming and never-before-asked question about snowflakes, Kepler delves into the nature of beehives, peapods, pomegranates, five-petaled flowers, the spiral shape of the snail's shell, and the formative power of nature itself. While he did not answer his original question—it remained a mystery for another three hundred years—he did find an occasion for deep and playful thought.
"A most suitable book for any and all during the winter and holiday seasons is a reissue of a holiday present by the great mathematician and astronomer Johannes Kepler…Even the endnotes in this wonderful little book are interesting and educationally fun to read."—Jay Pasachoff, The Key Reporter
New English translation by Jacques Bromberg
Latin text on facing pages
An essay, "The Delights of a Roving Mind" by Owen Gingerich
An essay, "On The Six-Cornered Snowflake" by Guillermo Bleichmar
Snowflake illustrations by Capi Corrales Rodriganez
John Frederick Nims' poem "The Six-Cornered Snowflake"
Notes by Jacques Bromberg and Guillermo Bleichmar
Johannes Kepler (1571-1631) was an important figure in the seventeenth century astronomical revolution. He is best known for his eponymous laws of planetary motion. Kepler wrote: "If there is anything that can bind the heavenly mind of man to this dusty exile of our earthly home…then it is verily the enjoyment of the mathematical sciences and astronomy."
158 pages, Paperback
First published January 1, 1966
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563 people want to read
About the author
Johannes Kepler
254 books165 followersJohannes Kepler (German pronunciation: [ˈkɛplɐ]) was a German mathematician, astronomer and astrologer, and key figure in the 17th century scientific revolution. He is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astronomy. These works also provided one of the foundations for Isaac Newton's theory of universal gravitation.
During his career, Kepler was a mathematics teacher at a seminary school in Graz, Austria, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to astronomer Tycho Brahe, the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II. He was also a mathematics teacher in Linz, Austria, and an adviser to General Wallenstein. Additionally, he did fundamental work in the field of optics, invented an improved version of the refracting telescope (the Keplerian Telescope), and mentioned the telescopic discoveries of his contemporary Galileo Galilei.
Kepler lived in an era when there was no clear distinction between astronomy and astrology, but there was a strong division between astronomy (a branch of mathematics within the liberal arts) and physics (a branch of natural philosophy).
During his career, Kepler was a mathematics teacher at a seminary school in Graz, Austria, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to astronomer Tycho Brahe, the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II. He was also a mathematics teacher in Linz, Austria, and an adviser to General Wallenstein. Additionally, he did fundamental work in the field of optics, invented an improved version of the refracting telescope (the Keplerian Telescope), and mentioned the telescopic discoveries of his contemporary Galileo Galilei.
Kepler lived in an era when there was no clear distinction between astronomy and astrology, but there was a strong division between astronomy (a branch of mathematics within the liberal arts) and physics (a branch of natural philosophy).
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Displaying 1 - 16 of 16 reviews
October 8, 2023
Let us continue on with another of the astronomer Johannes Kepler’s minor works, The Six-Cornered Snowflake, a long letter presented as a New Year’s gift to his patron John Matthew Wacker, counsellor to the imperial court in Prague in the year 1611. Just as with the Mysterium Cosmographicum of 1596, it offers a precious portrait of the mind of a contributor to the nascent science of the seventeenth century, caught between old and new. But Kepler is now fifteen years older and we can observe his progress towards a scientific maturity.
What problem does Kepler take up, what were his conceptual resources and what methods does he avail himself of? The editor Lancelot Law Whyte writes in his foreword:
1) To the historically oriented physicist and mathematician Kepler’s essay provides the first published evidence, in diagrams as well as text, of the ideas of regular arrangements and close packing. [p. v]
2) This would be enough to justify attention to this gem of disciplined speculation on the part of one of the greatest figures of exact science. But these retrospective features of the New Year’s Gift are supplemented by another, which points ahead. This essay is the first recorded step towards a mathematical theory of the genesis of inorganic or organic forms, a theory which still lies in the future. In 1610 Kepler recognized that the apparently hexagonal form of a few snowflakes that fell on his coat one day in Prague presented a challenge to the new mathematical science that was struggling to birth in his own mind: Why six? What was the physical cause of the six? What principle selected six from the other possible numbers? Moreover he regarded this as a special case of the general problem of the genesis of forms. In effect Kepler challenged those who followed him to discover the mathematics of the emergence of visible forms in crystals, plants and animals. [pp. v-vi]
How does modern theory resolve Kepler’s problems? The crystallographers Hessel (1830) and Bravais (1849) show that if the microphysical structure is that of a lattice, it can display only 2-, 3-, 4- or 6-fold rotational symmetry [p. 60]. By the end of the nineteenth century, it was clear that matter is atomistic on microscopic enough scales and that its lattice structure is in principle predictable from physical theory. But presumably Kepler himself would have been aghast at this development:
Here Kepler would gain a point. For the six cannot be predicted from fundamental theory alone. It cannot predict under what conditions particular complex groupings of water molecules will be stable in the solid state. Snowflakes are known to be much more complex than Kepler, who repudiated atomism, could have imagined, and no theory yet exists which covers all the observable properties of such complex systems. Visually simple facts are often too complex to be treated theoretically. [p. 62]
For six cannot be characteristic of the snowflake as we know at low enough temperatures they become cubic. Even so, Kepler’s question sets the agenda for a condensed-matter physics, as yet incomplete. A few comments on the status of Kepler’s research program. Even yet today with a mature condensed-matter physics grounded in quantum mechanics, the general problem of predicting thermodynamic phases is hard to solve: the first principles are known but, aside from a handful of exactly solvable cases such as the Ising model, the determination of precisely what atomic configurations optimize the free energy is computationally involved. From the appendix by Whyte:
1) We certainly have not yet a comprehensive scientific substitute for Kepler’s facultas formatrix. [p. 62]
2) We must not ask ‘What is the characteristic form of structural order in a snowflake?’ but ‘Under the given conditions what particular blend of order and disorder is most stable and therefore dominant?’ For disorder is now known to play a basic role in stabilizing particular kinds of visual forms at normal temperatures. It would be impossible for Kepler to like this interpretation, which is not only atomic, but introduces disorder into the divine regulation of the universe. The visually perfect hexagon starlet is the progeny of a union of order and disorder. [p. 61]
What can we learn about how Kepler’s mind works? Compare with the Mysterium Cosmographicum (just reviewed by us here): in the present work, he tones down the mystical and astrological elements, stays more sober in sticking to knowable physics and in eschewing purely formal speculations, and enumerates five causes, involving a so-called facultas formatrix which sets to work under given conditions to produce a regular structure, here characterized by six-fold symmetry [p. 41]. Is he still overly inclined to analogy? This review would judge not so. Let us rather label him pre-mechanistic in that he is groping towards a concept of a stable configuration of the system under the play of interparticle forces. Moreover, although he fails to arrive at a comprehensive theory of the phenomenon of six-fold symmetry in snowflakes, his inferences (deductions would be too strong a term) such as they are remain valid. Kepler has come a long way from the youthful enthusiast he was and disciplined himself to publish only speculations that meet the test of provable rationality.
To close on a speculative note: could Kepler’s facultas formatrix be given a meaning acceptable under the current state of the art? A reductionistic atomistic description by itself is insufficient, one also has to specify the conditions under which natural forces operate. The snowflake represents a many-body system intermediate between few-body systems and the organic bodies of biology: in the case of the former we would reject the concept of a soul, in that of the latter we might retain it, yet where to draw the line? It is perhaps aptest to understand by soul the confluence of natural forces that act under given conditions to shape and inform matter. As such it ought to be unobjectionable to anyone. This statement has to do with Sara Imari Walker and Paul Davies’ notion of causal information, i.e., a process in which information gives rise to a physical state obeying its own dynamical law, which does not exist antecedently (see for instance our review of Sara Imari Walker, Paul Davies and George Ellis, From Matter to Life: Information and Causality). Thus, as in the case of DNA in molecular biology, information can impart a generative faculty, leading to a different regime of behavior. What is amazing, however, is the sheer spontaneity nature displays in producing organized forms, even if these are in principle explicable in weak emergentist terms – to borrow a phrase from Eugene Wigner speaking about the unreasonable effectiveness of mathematics in physics, something we neither understand nor deserve!
What problem does Kepler take up, what were his conceptual resources and what methods does he avail himself of? The editor Lancelot Law Whyte writes in his foreword:
1) To the historically oriented physicist and mathematician Kepler’s essay provides the first published evidence, in diagrams as well as text, of the ideas of regular arrangements and close packing. [p. v]
2) This would be enough to justify attention to this gem of disciplined speculation on the part of one of the greatest figures of exact science. But these retrospective features of the New Year’s Gift are supplemented by another, which points ahead. This essay is the first recorded step towards a mathematical theory of the genesis of inorganic or organic forms, a theory which still lies in the future. In 1610 Kepler recognized that the apparently hexagonal form of a few snowflakes that fell on his coat one day in Prague presented a challenge to the new mathematical science that was struggling to birth in his own mind: Why six? What was the physical cause of the six? What principle selected six from the other possible numbers? Moreover he regarded this as a special case of the general problem of the genesis of forms. In effect Kepler challenged those who followed him to discover the mathematics of the emergence of visible forms in crystals, plants and animals. [pp. v-vi]
How does modern theory resolve Kepler’s problems? The crystallographers Hessel (1830) and Bravais (1849) show that if the microphysical structure is that of a lattice, it can display only 2-, 3-, 4- or 6-fold rotational symmetry [p. 60]. By the end of the nineteenth century, it was clear that matter is atomistic on microscopic enough scales and that its lattice structure is in principle predictable from physical theory. But presumably Kepler himself would have been aghast at this development:
Here Kepler would gain a point. For the six cannot be predicted from fundamental theory alone. It cannot predict under what conditions particular complex groupings of water molecules will be stable in the solid state. Snowflakes are known to be much more complex than Kepler, who repudiated atomism, could have imagined, and no theory yet exists which covers all the observable properties of such complex systems. Visually simple facts are often too complex to be treated theoretically. [p. 62]
For six cannot be characteristic of the snowflake as we know at low enough temperatures they become cubic. Even so, Kepler’s question sets the agenda for a condensed-matter physics, as yet incomplete. A few comments on the status of Kepler’s research program. Even yet today with a mature condensed-matter physics grounded in quantum mechanics, the general problem of predicting thermodynamic phases is hard to solve: the first principles are known but, aside from a handful of exactly solvable cases such as the Ising model, the determination of precisely what atomic configurations optimize the free energy is computationally involved. From the appendix by Whyte:
1) We certainly have not yet a comprehensive scientific substitute for Kepler’s facultas formatrix. [p. 62]
2) We must not ask ‘What is the characteristic form of structural order in a snowflake?’ but ‘Under the given conditions what particular blend of order and disorder is most stable and therefore dominant?’ For disorder is now known to play a basic role in stabilizing particular kinds of visual forms at normal temperatures. It would be impossible for Kepler to like this interpretation, which is not only atomic, but introduces disorder into the divine regulation of the universe. The visually perfect hexagon starlet is the progeny of a union of order and disorder. [p. 61]
What can we learn about how Kepler’s mind works? Compare with the Mysterium Cosmographicum (just reviewed by us here): in the present work, he tones down the mystical and astrological elements, stays more sober in sticking to knowable physics and in eschewing purely formal speculations, and enumerates five causes, involving a so-called facultas formatrix which sets to work under given conditions to produce a regular structure, here characterized by six-fold symmetry [p. 41]. Is he still overly inclined to analogy? This review would judge not so. Let us rather label him pre-mechanistic in that he is groping towards a concept of a stable configuration of the system under the play of interparticle forces. Moreover, although he fails to arrive at a comprehensive theory of the phenomenon of six-fold symmetry in snowflakes, his inferences (deductions would be too strong a term) such as they are remain valid. Kepler has come a long way from the youthful enthusiast he was and disciplined himself to publish only speculations that meet the test of provable rationality.
To close on a speculative note: could Kepler’s facultas formatrix be given a meaning acceptable under the current state of the art? A reductionistic atomistic description by itself is insufficient, one also has to specify the conditions under which natural forces operate. The snowflake represents a many-body system intermediate between few-body systems and the organic bodies of biology: in the case of the former we would reject the concept of a soul, in that of the latter we might retain it, yet where to draw the line? It is perhaps aptest to understand by soul the confluence of natural forces that act under given conditions to shape and inform matter. As such it ought to be unobjectionable to anyone. This statement has to do with Sara Imari Walker and Paul Davies’ notion of causal information, i.e., a process in which information gives rise to a physical state obeying its own dynamical law, which does not exist antecedently (see for instance our review of Sara Imari Walker, Paul Davies and George Ellis, From Matter to Life: Information and Causality). Thus, as in the case of DNA in molecular biology, information can impart a generative faculty, leading to a different regime of behavior. What is amazing, however, is the sheer spontaneity nature displays in producing organized forms, even if these are in principle explicable in weak emergentist terms – to borrow a phrase from Eugene Wigner speaking about the unreasonable effectiveness of mathematics in physics, something we neither understand nor deserve!
October 30, 2018
In 1611 Johannes Kepler was apparently the first westerner to notice that snow crystals have six-fold symmetry -- Chinese people had written about this many centuries before -- and he tried to figure out why. Given what is known at that time, it is an impossible task. Kepler, to his great credit, realizes this and basically concludes with "I don't know".
¯\_(ツ)_/¯
Along the way, though he considers ideas related to close-packing of spheres in several different arrangements. The existence of atoms was not known, and barely suspected, so he was considering packing of tiny drops of water. He also considers the packing of arils in pomegranates and cells in honeycomb. Ultimately he concludes that this doesn't lead anywhere.
The text is a bit of a bore now, despite some attempts at jokes by Kepler. The surrounding discussion by C. Hardie, B. J. Mason, and L. L. Whyte in the 1966 edition are more interesting.
Possibly unrelated, one year later in 1612 Galileo also wrote a book about ice. Again, that original text is a bit of a bore now, but the surrounding essays in Cause, Experiment and Science are fascinating look at the early separation between Aristotelian physics and modern science. If you read only one book from the 1610's about ice, I suggest the Galileo one.
¯\_(ツ)_/¯
Along the way, though he considers ideas related to close-packing of spheres in several different arrangements. The existence of atoms was not known, and barely suspected, so he was considering packing of tiny drops of water. He also considers the packing of arils in pomegranates and cells in honeycomb. Ultimately he concludes that this doesn't lead anywhere.
The text is a bit of a bore now, despite some attempts at jokes by Kepler. The surrounding discussion by C. Hardie, B. J. Mason, and L. L. Whyte in the 1966 edition are more interesting.
Possibly unrelated, one year later in 1612 Galileo also wrote a book about ice. Again, that original text is a bit of a bore now, but the surrounding essays in Cause, Experiment and Science are fascinating look at the early separation between Aristotelian physics and modern science. If you read only one book from the 1610's about ice, I suggest the Galileo one.
January 20, 2026
Das Buch an sich gab Mir an sich nicht viel, aber vom Prinzip ist Kepler einfach genial.
November 27, 2020
Check back in 3 years...
This entire review has been hidden because of spoilers.
November 22, 2025
The Six-Cornered Snowflake
One of my great heroes of astrophysics, was a true polymath as this intimate and ingenious discussion of snowflakes proves. A must read.
One of my great heroes of astrophysics, was a true polymath as this intimate and ingenious discussion of snowflakes proves. A must read.
January 10, 2026
I had this on my list but removed it after reading Somnium which wore me out. I'm so glad I decided to give Six-cornered Snowflake a try after all. I really enjoyed reading Kepler's (translated) original musings. I will never look at a pomegranate the same again.The diagrams throughout really helped with understanding. I am in awe at his brilliance, I don't know how he attempted to calculate all of his theories at a time when there were no special tools. Didn't the snowflakes melt too fast? Just amazing! Discussion at the end of the scientific review said that Kepler's New Year's Gift had little influence but that he presented a new inquiry -the mathematics of genesis of form. I say kudos to him. Very scientific and mathematical, much of the analysis was beyond my full understanding, but I really liked Kepler's original writing: A New Year's Gift! So much thought put into it and what a great idea. I wish we still did New Year's Gifts!
August 24, 2019
I'm not sure just how much the actual essay gave me, but this particular edition definitely made it worth my money.
Two forewords, notes, illustrations (the snowflake bonus is gorgeous), a poem (!), and even the original Latin version of the text at the end.
And as someone born, raised and living in Prague I also cherish all the hometown connections of this work. (And I rate that the poem used all the correct diacritic.)
I suppose the essay itself has more worth for me as a piece of history than as a...scientific expression. But I rate the nix/nichts play and I think as an expression of curiosity it's a beatiful piece of work.
Two forewords, notes, illustrations (the snowflake bonus is gorgeous), a poem (!), and even the original Latin version of the text at the end.
And as someone born, raised and living in Prague I also cherish all the hometown connections of this work. (And I rate that the poem used all the correct diacritic.)
I suppose the essay itself has more worth for me as a piece of history than as a...scientific expression. But I rate the nix/nichts play and I think as an expression of curiosity it's a beatiful piece of work.
August 17, 2017
Interesting inside into a method of inquiry back in Keplers' time. Also style of writing was quite interesting.
June 29, 2025
The book was short and sweet. Just left me wishing I were smarter so I could track some concepts he was explaining.
April 3, 2018
Excellent edition of Kepler’s impressive, albeit brief, foray into crystal form. Kepler was perhaps the first European to ask “why six?” He is also one the first to describe close packing of spheres and how that might explain a snowflake’s hexagonal symmetry. Most impressive perhaps is Kepler’s acknowledgement that he is unable to answer his own question, despite suggesting several hypotheses. An fine example of scientific reasoning. Turns out Kepler was a pretty funny guy to boot!
This edition has the English translation and Kepler’s Latin text (modernized) on facing pages, and includes detailed notes that provide context and help explain passages; and also compare three earlier German translations.
This edition has the English translation and Kepler’s Latin text (modernized) on facing pages, and includes detailed notes that provide context and help explain passages; and also compare three earlier German translations.
February 19, 2013
I am not the ideal audience for this book. It spends most of its time theorizing physical shapes (why snowflakes are the shape they are), and I am /terrible/ at this. It generally washed over me. The bits I was able to follow certainly showed some wit and even a little literary zest, and there were almost surprisingly long and detailed notes provided by the editor, but in the end (something like Kepler himself does in the last paragraphs of the work), I have to send this out to better heads for fair reaction.
August 11, 2010
I love this piece. It's been a few years since I read it, but I remember the apology—Kepler's self-admonition for not having a real gift. Instead he grasps at the world, following the geometry of a snowflake as far as the mind can follow it. There is a playful quality flavoring this meditation. It is a generous gift and a unique glimpse into the person of Kepler.
November 26, 2014
Enjoyed this. Kepler is one of my favorite scientists/mathematicians. It was fun to this lighter inquiry. This version includes a preface, introduction and poem (in the shape of snowflakes). The text is in both Latin on the left side page and English on the right side.
April 23, 2015
"There is then a formative faculty in the body of the Earth, and this carrier is vapour as the human soul is the carrier of spirit."
July 27, 2022
Gorgeous little book packed with wit and genius, as sparkling and mysterious as a snowflake.
August 3, 2015
Very informative. Tends to ramble often.
Displaying 1 - 16 of 16 reviews