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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. It would be the best gambling game since King Solomons slots. Imagine the odds.
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.)
It is important to fully understand the details behind the games we play and the odds and probabilities. Now that you can play dice game online understanding the odds becomes a software issue. There is a big difference between playing online roulette and roulette at a table. Either way by studying the game you will always get the best results.
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!