An Alternate Derivation of the Pentagonal Hexecontahedron
John Savard
http://www.maa.org/editorial/mathgames/mathgames_05_16_05.html
notes that there are exactly 25 isohedra not belonging to infinite
classes of isohedra, and in addition there are 5 infinite classes of
isohedra.
Excluded from this are spheres, lenses, and rolling logs.
Crystal Caste, a manufacturer of dice for role-playing games, came up
with a novel form of rolling log, where the faces are triangles of
alternating direction instead of rectangles.
Some of the Archimedian duals can be obtained from the Platonic solids
by simple processes easier to understand by the nonmathematician.
Thus, for example, a fair 24-sided die can be obtained by placing a
shallow square pyramid on each face. Note that the precise shape of that
pyramid is not critical, so one doesn't have to end up with the Tetrakis
Hexahedron as an Archimedian dual. This shape is currently used to make
commercial 24-sided dice; I think the Trapezoidal Icositetrahedron would
do a better job.
Starting with a dodecahedron, and puting shallow pentagonal pyramids on
each face, one obtains what is perhaps the simplest 60-sided fair die,
the Pentakis Dodecahedron.
If the height of the pyramids is increased just enough that the
triangles on adjoining pyramids form flat rhombic faces - which is as
far as they can be increased before the solid stops being convex - then
one gets the famous Rhombic Triacontahedron, in which shape 30-sided
dice are currently being manufactured for role-playing games
enthusiasts.
A snub dodecahedron has 12 pentagonal faces, 60 triangular faces which
adjoin one pentagonal face and two triangular faces, and an additional
20 triangular faces which adjoin only triangular faces.
Its dual is the Pentagonal Hexecontahedron.
Putting five-sided pyramids on the 12 pentagonal faces of the snub
dodecahedron, and three-sided pyramids on the 20 triangular faces which
are surrounded by triangular faces, and then raising their height
exactly so that the faces of the pyramids extend the 60 triangular faces
on which no pyramid is placed... leads to a Pentagonal Hexecontahedron
as well, without taking the dual.
Doubtless, this was well known already; having forgotten about the 20
extra triangular faces, I was hoping to have found a new 60-sided
isohedron.
John Savard
http://www.quadibloc.com/index.html