What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
There are two interpretations of this problem,
the shortest route between opposite faces of the dodecahedron going
across the outside of the solid and the shortest route boring a
hole through the middle.
While few tackled this challenging
problem, Luke, of Madras College found the shortest distance
through the centre. In his solution he used the centres of three
spheres: the circumsphere that passes through all the vertices of
the dodecahedron, the midsphere touching the midpoints of the
edges, and the insphere which touches the opposite faces of the
dodecahedron at their centres. Luke's answer is 2.23 approximately.
Before you look at Luke's solution here you might like to try to
try it for yourself.
On the surface this must be the distance between A and B on the
net of the solid.