User scott sherman - MathOverflow

Answer by Scott Sherman for polyhedra with equilateral pentagons faces.

2011-12-11T22:57:03Z

The pentagonal isohedra with sides of equal length listed on http://loki3.com/poly/isohedra.html are the regular dodecahedron, non-convex equilateral pyritohedron, equilateral pentagonal icositetrahedron, non-convex equilateral pentagonal icositetrahedron, non-convex equilateral pentagonal hexecontahedron and non-convex equilateral pentagonal hexecontahedron.

(I should note that the page jolumij references, is a page I built to summarize my findings. I had tried to learn about the isohedra from pages such as http://mathworld.wolfram.com/Isohedron.html, but all sources I could find were very incomplete when it came to describing the pentagonal isohedra. They offer names such as "octahedral pentagonal dodecahedron" without describing how to construct them or mentioning they may represent an infinite family of shapes.)

I assume by "this kind of polyhedra," you're referring to isohedra with pentagonal faces. Mathworld offers this definition of an isohedron:

An isohedron is a convex polyhedron with symmetries acting transitively on its faces with respect to the center of gravity.

For my list of isohedra, I relax the definition to include non-convex polyhedra. The isohedral transforms can also be used to create polyhedra with intersecting faces "with symmetries acting transitively on [their] faces with respect to the center of gravity."

As to whether those 6 are the only isohedra with equilateral pentagonal faces, I believe the list is complete, but I haven't rigorously proven it. What I have done is start from the tetrahedral, octahedral and icosahedral symmetry groups and applied the isohedral pentagonal transform to them. This transform has two degrees of freedom. Then I explored the space for equilateral pentagons as well as other interesting symmetries or patterns. I haven't seen references to many of the shapes I found (including 5 of the 6 shapes listed here), so I'd be interested if other people know of any other references.

As to how to compute the angles of those pentagons, http://loki3.com/poly/transforms.html#penta gives a description of the transform and what my notation means. You can use the parameters and transform to derive the angles. For example, the non-convex equilateral pyritohedron is 4p(0.09549150, 0.6605596), which means you apply the isohedral pentagonal transform (p) to a tetrahedron (4) using the parameters 0.09549150 and 0.6605596. In this case, you get two 36 degree angles and three 108 degree angles.

Answer by Scott Sherman for Fair but irregular polyhedral dice

2011-12-09T06:45:29Z

Yes, there are conditions broader than isohedral symmetry that guarantee a convex polyhedron represents a fair die. Consider a teetotum or dreidl. These shapes aren't isohedra, but they are mathematically fair dice in the sense that each face has an equal chance of being rolled. True, they're not convex, but you can only land on the convex hull, so you could either consider them effectively convex or easily modify them so they're convex while preserving the property of being non-isohedral fair dice.

Those shapes are examples of a class of object I call polyisohedra ( http://loki3.com/poly/polyisohedra.html ), where sets of faces are equivalent. A polyisohedron modified with the proper polyisohedral symmetry preserves the property of being a fair die, which is how I derive teetotums and dreidls from this more general category of fair dice.

A simple example of a polyisohedron is a cube where you combine adjacent pairs of faces to essentially make a fair 3 sided die. More interesting, however, are cases where you have to combine multiple faces to make the shape into a fair die. One example is a gyrobifastigium, a Johnson solid with 4 square faces and 4 equilateral triangle faces. Obviously it's not a fair 8 sided die, but if you give the same label to square-triangle face pairs, you end up with a fair 4 sided die. For this shape, the triangular sides aren't stable, so you can only land on the square faces, but this isn't a required feature. However, unstable faces are used in other non-isohedral shapes commonly used as fair dice, such as barrels.

I have more examples and details posted at http://loki3.com/poly/fair-dice.html.

Comment by Scott Sherman

2011-12-14T04:48:39Z

I should point out that there are other isohedra with rhombic faces besides the three mentioned above: the rhombic hexecontahedron (non-convex, 60 faces) and the infinite family of trigonal trapezohedra with 6 sides. Is that all?