The Kepler Solids


One of the first people in modern times to study polygons, polyhedra, and crystals was the astronomer Johannes Kepler, who discovered the two solids above. In each, faces of a dodecahedron are extended outward into a pentagonal star (pentalpha). Both are stellated dodecahedra; they are called "small" and "great" to distinguish them.

The small stellated dodecahedron (on the left above) has twelve pentagonal pyramids built on the faces of a dodecahedron. All the triangles are isosceles 36°-72°-72° (golden) triangles. Each star face has a pentagonal pyramid sticking out of its center. The solid appears in M. C. Escher's Order and Chaos. In the diagram above, the faces have been colored to show the pentagonal stars.

The great stellated dodecahedron (on the right above) also has twelve interpenetrating pentagonal star faces. It can be thought of as an icosahedron with a triangular pyramid on each face. For this reason, it is sometimes called a stellated icosahedron. Again, the triangles are isosceles 36°-72°-72° triangles. The pentagonal star faces are harder to see because each star has a rosette of five triangular pyramids in its center. This solid makes a lovely Christmas decoration, and is often seen as such in store windows and commercial displays during the holiday season. The faces in the diagram above has been colored to show the pentagonal stars.

In their colored versions, each solid consists of ten golden triangles of each of six different colors - sixty triangles in all. Each in turn can be made by adding pentagonal or triangular pyramids to an dodecahedron or icosahedron base respectively. An alternative approach is to make the solids completely hollow. Instructions for the hollow approach follow.

In the second of the solids, the great stellated dodecahedron, the twenty  triangular pyramids are given the color scheme below. Distinct colors have been represented by distinct numbers.
 

1 2 3 xxxxxxxxxxxxxxx 3 2 1 
3 1 4 4 1 3
4 3 5 5 3 4
5 4 2 2 4 5
2 5 1
mirror
1 5 2
6 2 3  images 3 2 6
6 1 4 4 1 6
6 3 5 5 3 6
6 4 2 2 4 6
6 5 1 1 5 6

The first five triangular pyramids (specifically 1 2 3, 3 1 4, 4 3 5, 5 4 2 and 2 5 1) are joined in a ring with the first color in each color trio on the outer edge, which becomes pentagonal. Join parts with all pyramids pointing away from you and working from left to right. Then color 6 of the next five triangular pyramids (specifically 6 2 3, 6 1 4, 6 3 5, 6 4 2 and 6 5 1) is attached to the first or outer color of the first five triangular pyramids (first pyramid to first pyramid, second pyramid to second pyramid, and so forth). If you work in a systematic fashion, you will not find this difficult. In fact, the colors will help you if you remember that each triangle is an arm of a pentagonal star and you want each star to have five arms of the same color. The remaining ten triangular pyramids have colors schemes which are mirror images of the first ten and are placed diametrically opposite their counterparts. Planes that are parallel to one another will be the same color. You will see an icosahedron (five equilateral triangles per vertex) forming inside the model as you put it together. Visit Math World and Virtual Flower for interactive versions you can rotate with your mouse.

In the small stellated dodecahedron, the twelve stellating pentagonal pyramids have the following color scheme.

1 2 3 4 5 xxxxxxxxxxxxxxx 5 4 3 2 1
6 5 3 4 2 2 4 3 5 6 
6 1 4 5 3 mirror 3 5 4 1 6 
6 2 5 1 4 images 4 1 5 2 6 
6 3 1 2 5   5 2 1 3 6 
6 4 2 3 1 1 3 2 4 6

Its construction is similar to that of the great stellated dodecahedron. We leave it to you as a challenge. Visit Math World and Virtual Flower for interactive versions you can rotate with your mouse.

All interactive links require a Java-capable browser. Those by Virtual Flower need Internet Explorer.

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Jill Britton Home Page
10-August-2008
Copyright Jill Britton