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Kepler and the Rhombic Dodecahedron: Kepler and His Time

Roberto Cardil (matematicasVisuales)


Johannes Kepler (1571–1630) was a German scientist, mathematician, astronomer, and astrologer. He lived in a time full of new ideas and discoveries.

The timeline below gives a sense of the historical context of his time in relation to the cultural Renaissance (or “Rebirth”) that came just before it and the important mathematical period (in which calculus was discovered by Newton and Leibniz) that followed.

Basic timeline of the historical context of Kepler’s time
Figure 4. Basic timeline of the historical context of Kepler’s time, showing Kepler was a man who lived and worked between the Renaissance and the great mathematical discoveries of Newton and Leibniz. The names shown in red are those of important mathematicians and scientists, and those in green are famous artists (although Dürer is also remembered as a mathematician).

The Renaissance that preceded Kepler’s time was a period in European history that is often seen as a transition between the medieval era and modernity. More people had access to old and new ideas, thanks to the invention of the printing press by Johannes Gutenberg (ca 1400–1468) and the slow expansion of literacy that followed. Artists developed new techniques, such as the use of linear perspective, that in part encouraged the creation of more realistic representations of the natural world. It was also a time of significant religious changes, most notably the Protestant Reformation instigated by the actions of Martin Luther (1483–1546), and the European exploration of the world was underway.

One of the most important new scientific ideas of the Renaissance was the sun-centered (or heliocentric) cosmology, which posited that the Earth and all planets revolve around the Sun, posited by Nicolaus Copernicus (1473–1543). The 1543 publication of Copernicus’ book De Revolutionibus orbium coelestium (On the revolution of the heavenly spheres) started what has been called the Copernican Revolution, as the astronomical theory that it described challenged the long-held assertion that the Earth was the stationary center of the universe. Copernicus’ ideas influenced both the work of Kepler, who from his youth was an early supporter of the heliocentric model, and the thinking of Kepler’s famous scientific contemporary, Galileo Galilei (1564–1642).

While we might assume that this “revolution” led to the abandonment of earlier understandings of mathematics and science, in fact the late 16th and early 17th centuries were an age in which the Greek mathematical masterworks—such as the Elements of Euclid (ca 325–ca 265 BCE), the Conics of Apollonius (ca 240–ca 190 BCE), and the works of Archimedes (ca 287–ca 212 BCE)—were also studied seriously. Geometry interested not only mathematicians and scientists, but also artists and craftsmen.

It was into this fertile period for mathematics that Johannes Kepler was born in Weil der Stadt (now in southwest Germany) in 1571. His parents were poor. He was an intelligent boy whose health was delicate. Throughout his life, his place of residence changed several times. Before he was 30, he studied at the University of Tübingen; was appointed as a mathematics teacher in Graz; and briefly worked with the famous astronomer Tycho Brahe (1546–1601) in Prague. Kepler remained in Prague, serving as Imperial Mathematician, for 11 years following Brahe’s death. Due in part to growing political-religious tension, Kepler then moved first to Linz (where he lived from 1612 to 1626) and later to Ulm and the Silesian town of Sagan (Żagań). He died after a short illness while visiting Regensburg in 1630.

Basic timeline of the historical context of Kepler’s time
Figure 5. Portrait of Kepler (MAA Convergence Portrait Gallery).

Kepler’s life was not easy, and he lived during troubled political times. He struggled with bad health, financial troubles, and family difficulties (for example, his mother was imprisoned for a time on charges of witchcraft). He was also affected by religious conflicts (a deeply religious man, Kepler left his teaching post at Graz due to the Catholic Counter-Reformation) and even war (the religious strife of the Thirty Years’ War that devastated Germany broke out in 1618).

Despite these circumstances, Kepler was able to make important contributions to science, in particular to astronomy, optics, and mathematics.

As an astronomer, Kepler is famous for his discovery of the three laws of planetary motion now known as Kepler’s Laws. Newton later showed that these could be deduced from Newton's laws of motion and the universal law of gravitation. His work on optics included conclusions about the intensity of light based on Tycho’s observations of eclipses as well as improvements to the refracting telescope.

As a mathematician, Kepler discovered two new regular polyhedra (today called the great and small stellated dodecahedron) and two new rhombic polyhedra (the rhombic dodecahedron and the triacontahedron), worked on the problem of close packing of equal spheres, was interested in logarithms, and found volumes of solids of revolution using methods that were precursors to the important calculus discoveries made by Newton and Leibniz.

Kepler enjoyed a fertile imagination. As an example, we can consider how a life event led him to think about the volume of solids. When he married for the second time, Kepler was surprised by the wine merchant’s technique for measuring the volume of wine barrels. As a result, Kepler studied how to calculate the volume of solids of revolution.

Formation of cuboctahedron by cutting corners off a cube.

TO INVESTIGATE    Kepler: The Volume of a Wine Barrel.

To learn more about Kepler’s contribution to the development of calculus, read this article published in Convergence in 2011.






A similar anecdote relates to the origin of the book that is our main source for Kepler’s discovery of the rhombic dodecahedron.

The title of this book is Strena Seu De Nive Sexangula [The Six-Cornered Snowflake. A New Year Gift, 1611].[1] It was written as a 1611 New Year’s gift to Kepler’s friend and benefactor Matthew Wacker von Wackenfels (1550–1619). As Kepler described how he came up with the idea behind this gift, he was thinking and walking along Prague’s Charles Bridge one winter day when

Just then, by a happy occurrence, some of the vapor in the air was gathered into snow by the force of the cold, and a few scattered flakes fell on my coat, all six-cornered, with tufted radii. By Hercules! Here was something smaller than a drop, yet endowed with a shape. Here indeed, was a most desirable New Year’s gift for the lover of Nothing, and one worthy as well of a mathematician (who has Nothing, and receives Nothing) since it descends from the sky and bears a likeness to the stars [Kepler 1611, 33].

As Kepler went on to try to explain the reason for the six-angled shape of snow crystals, his rich imagination flew from honeycombs to peas, and from pomegranates to cannonballs. In the midst of these thoughts, he described a new polyhedron: the rhombic dodecahedron.

In the next section of this article, we will explore how Kepler thought about this polyhedron, how he identified it by looking into the bottom of bee cells, and how we can understand its main properties intuitively.

[1] This beautiful book was written in Prague and published in Frankfurt am Main in 1611. Copies of the first edition are very rare. We can enjoy some images from a first edition in Frank Swetz’s Convergence article, “Mathematical Treasure: Kepler and Sphere-packing,” and a link to a fully-digitized version of the copy owned by the University of Toronto appears above.


Roberto Cardil (matematicasVisuales), "Kepler and the Rhombic Dodecahedron: Kepler and His Time," Convergence (March 2022)