The Dodecahedron Explained

What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?

The Dodecahedron

What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?

More Dicey Decisions

Why do this problem?

This problem offers students the opportunity to consider the symmetry of dodecahedra, and to develop insights from reasoning about dodecahedra to help them to analyse symmetry in icosahedra.

Possible approach

This problem follows on from Dicey Decisions.
The problem could be given to students to work on independently (perhaps as a homework task) but could also be used as a whole class activity as follows:
Start by making sure students are happy with the idea that opposite faces of a six-sided die sum to 7, and then challenge them to come up with a dodecahedral die where the opposite faces sum to 13. Some students may wish to visualise the die, some may be happy working with a two-dimensional representation, and some may prefer to create a net and actually make the die in three dimensions.
There is opportunity for rich discussion about the number of distinct dodecahedral dice it is possible to make (with the constraint that opposite faces sum to 13), and different groups of students could be encouraged to work with distinct dice.

Once students have created (or visualised) their die, ask them to create a frequency table showing the frequency with which the different edge scores occur.
"Does everyone get a symmetric distribution?"
"Are you surprised that everyone got a symmetric distribution even if they started with different dice?"
"Can you explain why it was symmetric?"
This could lead into rich discussion about the symmetry of the distribution and how it relates to the constraint that opposite faces sum to 13.

Finally, once students have explanations of where the symmetry in the distribution comes from, they could be challenged to do a similar analysis for corner totals of the dodecahedral die, or for edges or corner totals on an icosahedral die.

Key questions

If a face of the die is numbered $n$, what would the opposite face be numbered?
If adjacent faces were numbered $n$ and $m$ (with an edge total between them of $n+m$, what would the opposite edge total be?

Possible extension

Write up a really clear proof of the result concerning the symmetric distribution of the edge totals.

I have a 100-sided die. To the eye, it looks pretty regular, but it can't be exactly regular because there is no such thing as a 100 sided Platonic solid. Consider the issues which might arise in an analysis of the various totals.

Possible support

Spend time considering the edge totals of the six-sided die to make sense of the symmetry there before working on the dodecahedron.