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Odds of a dimpled die, out of 10,000 tosses, using a Markov Chain model

This page contains a complete list of all possible Fair Dice.  First off, objects which are ISOHEDRAL are fair.  To be isohedral, each face must have the same relationship with all other faces, and each face must have the same relationship with the center of gravity.  Thus, each object has a Group structure (a Group Theory topic).  Using Group Theory and Euler's formula (Vertices + Faces - Edges = 2), it can be proven that the following list contains all possible isohedra.  An interactive version of this page is at the CRC Concise Encyclopedia of Mathematics.  A good explanation of the math behind all this can be found at Klaus Æ. Mogensen's website.
 

Regular Tetrahedron

Isosceles Tetrahedron

Scalene Tetrahedron

Cube

Octahedron

Regular Dodecahedron

Octahedral Pentagonal Dodecahedron

Tetragonal Pentagonal Dodecahedron

Rhombic Dodecahedron

Trapezoidal Dodecahedron

Triakis Tetrahedron

Regular Icosahedron

Hexakis Tetrahedron

Tetrakis Hexahedron

Triakis Octahedron

Trapezoidal Icositetrahedron

Pentagonal Icositetrahedron

Dyakis Dodecahedron

Rhombic Triacontahedron

Hexakis Octahedron

Triakis Icosahedron

Pentakis Dodecahedron

Trapezoidal Hexecontahedron

Pentagonal Hexecontahedron

Hexakis Icosahedron
120 sides

Triangular Dihedron
Move points up/down - 4N sides

Basic Triangular Dihedron
2N sides

Trigonal Trapezohedron
Asymmetrical sides -- 2N Sides

Basic Trigonal Trapezohedron
Sides have symmetry -- 2N Sides

Triangular Dihedron
Move points in/out - 4N sides

Note to gaming companies:  I'd love to see someone make the exotic dice in the list above.

   Can a non-isohedral fair die exist?  Consider a pyramid made from 4 isosceles triangles and a square.  If the pyramid is short and fat, the square face will be landed upon more than a fifth of the time.  If the pyramid is tall and thin the square face will be landed upon less than a fifth of the time.  Is there a height where the square face will be landed upon exactly one fifth of the time???
   Yes, for a given set of conditions.  If you knew the height, force, elasticity, and throwing method, you could find the right height.  However, once the conditions changed, the die would no longer be fair.  (NOTE:  I have a strong argument for this, but no proof.)

If you find the polyhedrons above interesting, you're bound to enjoy the Pavilion of Polyhedreality by George Hart.

There are 6 regular 4-dimensional object.  Peek is a fascinating and very pretty piece of software that allows looking at 3-D cross-sections of complex 4-D objects.  The pictures are beautiful!