# 5 Pythagorean Solids

This OER was designed by the OU Academy of the Lynx (oulynx.org) in conjunction with the "Galileo's World" (galileo.ou.edu) exhibition at the University of Oklahoma.

This activity is designed to be completed in 5 minutes by a typical visitor to the exhibition. For adaptations to other age levels and pedagogical settings, visit the "Pythagorean Solids Educational Cluster" below.

## Introductory Pythagorean Solids Activity

We can define a solid as regular when every face, edge and corner angle is identical, whether a square on every side of a cube, or a triangle on every side of a tetrahedron. The Pythagoreans proved that there are only five regular solids: The octahedron has 8 sides; the dodecahedron has 12 sides; and the icosahedron has 20 sides. There are no others.

Which is which?  Given a set of three-dimensional regular solids, identify them using the information in the table above.

Which figure below is NOT a regular solid?  Why not?

Create your own. Given the interlocking colored plastic triangles, squares and pentagons, assemble each of the five regular solids.

A “net” is a flat, two-dimensional pattern that could be used as a template to cut out models of the solids, printed on card stock.  Use the plastic pieces to create your own “net.”  Is there more than one way to make a net for each solid?

## Historical Background for the Pythagorean Solids

Because Plato used the 5 regular solids to explain the structure of the Universe in his dialog, The Timaeus, they are also called the Platonic Solids.  After Plato, astronomers supposed that the geometry of these five solids would hold an essential clue to the true structure of the universe. See Plato's book, https://galileo.ou.edu/exhibits/divine-plato.

Euclid analyzed the properties of the regular solids in Book 13 of his Elements of Geometry. See a 1570 English edition of Euclid, https://galileo.ou.edu/exhibits/elements-geometry-1570.

In his earliest book Kepler used the regular solids to prove Copernicanism.  The mystery of the universe was now revealed, he argued, because the Divine Architect knew Pythagorean geometry and used it to construct a Copernican universe! See Kepler's book, https://galileo.ou.edu/exhibits/sacred-mystery-structure-cosmos.

The works of Leonardo da Vinci, Luca Pacioli, Albrecht Dürer, and Lorenzo Sirigatti show that artists, in addition to astronomers, mathematicians and philosophers, were also deeply familiar with the properties of regular solids. See the work by Leonardo da Vinci, https://galileo.ou.edu/exhibits/treatise-painting, Luca Pacioli, https://galileo.ou.edu/exhibits/divine-proportion, Albrecht Dürer, https://galileo.ou.edu/exhibits/principles-geometry, and Lorenzo Sirigatti, https://galileo.ou.edu/exhibits/practice-perspective.

## Further OER's on the Pythagorean Solids

Use the following OER's to further explore the Galileo's World exhibition.

## Pythagorean Educational Cluster

We want to create variations on this activity that connect the Duochord to a variety of ages. Use the following chart and hyperlinks to find the one to best fit your group.