Last Update: Monday September 27, 2021

# A Little Background About The Unfolding Of This Pentakis Dodecahedron

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. In 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 below 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.

The next step:   It was quite clear that there were many possible objects within this hexacontaqon.   On the weekend following the class, we filled the object
with Play Doh and waited a week for it to harden a little.

It was relatively straightforward to begin to discern an abundance of imperfect tetrahedrons and, in the middle, an icosahedron.