Kepler and the Rhombic Dodecahedron: Article Objectives

Roberto Cardil (matematicasVisuales)


The principal goal of this article is to share materials and activities designed to engage students in the study of mathematics (namely, of polyhedra) within an historical context. It focuses on the rhombic dodecahedron—a simple, beautiful and interesting polyhedron that can delight young and old alike. In our journey of exploring its properties, we follow paths traced by the wonderful mind of its discoverer: Johannes Kepler (1571–1630).

Although we do not know how Kepler first conceived of the rhombic dodecahedron, he wrote about it in at least three books: Strena Seu De Nive Sexangula [The Six-Cornered Snowflake. A New Year Gift, 1611], Epitome Astronomiae Copernicanae [Epitome of Copernican Astronomy, 1618–1621], and Harmonices Mundi [The Harmony of the World, 1619]. The activities that I share in this article relate Kepler’s thinking in these works with three interdisciplinary points of view that connect mathematics with other aspects of human activity:

  • Nature, based on Kepler’s fascinating look into how bees build the cells of their honeycombs.
  • Technology, based on Kepler’s exploration of the problem of finding the optimal way to pack cannonballs, in which he astonishingly imagined squeezing them, a thought experiment that led him to the proposal that is now called the ‘Kepler Conjecture’.
  • Art, based on Kepler’s view of the rhombic dodecahedron as a cube topped with pyramids, a view that Renaissance mathematicians and artists like Leonardo da Vinci were very close to discovering before Kepler did.

I have used these materials with students several times, and the experience is always a success. Both the beauty of polyhedra and the interdisciplinary approach of the activities help to engage students in the study of mathematics.

It is also enriching for our students to become aware that mathematics is not a finished science. On the contrary, it is a subject in continual development. The tale of Kepler’s discovery of the rhombic dodecahedron places us in a wonderful position to see this. Indeed, after his remarkable achievements with the rhombic dodecahedron, Kepler’s mind did not rest; he discovered a second rhombic polyhedron (now called the triacontahedron), and he was aware that the cube was a third rhombic polyhedron.

To follow these ideas is not easy without concrete mathematical models. A second objective of this article is thus to encourage readers and their students to build and play with their own polyhedra. As previously noted, modeling polyhedra is not only an enjoyable activity in itself with results that are appealing for their beauty; we also learn about the mathematical properties of these bodies when we make, examine, and touch them.

On the following pages, teachers and their students will find background historical information and classroom-tested resources (including several video animations) that offer guidance in building these models, exploring the mathematics that they embody, and placing the ideas that emerge within a historical context that unveils sociocultural settings for mathematics.