A Little Background About The Unfolding
Definitions: A sixty-sided pentagonal figure of the kind created below is known as the Pentakis Dodecahedron. A sixty-sided pentagonal figure is also known as a cumulated dodecahedron or hexacontagon.
Earlier Work: We were learning about the tetrahedral-octahedral clusters. There are many possible combinations of tetrahedrons and octahedrons. The two can be used to tile and tessellate the universe in three dimensions, perfectly filling the container.
April 2011: This structure below was created with sixty tetrahedrons that create the outer layer. Of the five platonic solids (in discussions with high school geometry classes), it seemed that the flat pentagonal face of the simple dodecahedron was not readily evident throughout nature. It seemed to beg questions about “really-real” structure of the dodecahedron and its place in nature.
Why not make the dodecahedron with the rather odd three-dimensional pentagonal of five tetrahedrons? It is odd simply because there is a very small gap
between the tetrahedrons.
This dodecahedron was created with twelve of those pentagonal objects, each made of five tetrahedrons.
The next step: Name this object. Smaller than a soccer ball, this 60-sided object had no name for many months. Finally, with a little help from Wolfram Research, it became apparent that it was in a class of objects called cumulated dodecahedrons. Hexacontagon is another class of 60-sided objects. Yet, the Pentakis dodecahedron seems to be most commonly accepted name.