Shapes
Dice come in a variety of shapes. Shapes of mathematically fair dice are called isohedra, which are convex polyhedra with special properties. Other dice are based on barrels (prisms and antiprisms) and spheres.
My collection
From left to right and top to bottom:
Platonic Solids: Tetrahedron (D4), Hexahedron
(D6), Octahedron (D8), (Pentagonal) Dodecahedron (D12), Icosahedron (D20)
Catalan Solids: Rhombic Dodecahedron (D12),
Tetrakis Hexahedron (D24), Deltoidal Icositetrahedron (D24), Rhombic Triacontahedron
(D30)
Dipyramids: Square Dipyramid (D8, two versions),
Octagonal Dipyramid (D16). Special Isohedra: Isosceles Tetrahedron (D4)
Trapezohedra: Triagonal Trapezohedron (Hexahedron,
D6), Pentagonal Trapezohedron (D10, three versions), Heptagonal Trapezohedron
(D14), Tridecagongonal Trapezohedron (D26), Heptadecagonal Trapezohedron (D34),
Icosikaipentagonal Trapezohedron (D50)
Prisms: Triangular Prism (D5), Square Prism (Hexahedron,
D6), Pentagonal Prism (D7).
Modified Prisms (odd number of faces): Elongated
Prism (D3), Rounded-off Triangular Prism (D3, 2 versions), Rounded-off
Elongated Pentagonal Prism (D5), Rounded-off Elongated Heptagonal Prism (D7), Rounded-off
Elongated Nonagonal Prism (D9), Rounded-off Elongated Hendecagonal Prism (D11),
Rounded-off Elongated Tridecagonal Prism (D13), Rounded-off Elongated Pentadecagonal
Prism (D15)
Modified Prisms (even number of faces): Modified
Square Prism (D4, two versions), Rounded-off Hexagonal Prism (D6), Modified Decagonal
Prism (D10)
Antiprisms: (Degenerate) 2-Antiprism (Tetrahedron,
D4), 3-Antiprism (Octahedron, D8)
Modified Antiprisms: Modified 3-Antiprism, pipped
and numbered (D6), Modified 4-Antiprism (D8), Modified 5-Antiprism (D10), Modified
6-Antiprism (D12), Modified 10-Antiprism (D20),
Spheres: Sphere (D1), Spherical D4, Spherical D6,
Spherical Flattened D6, Spherical Flattened D8, Spherical Flattened D32, Spherical
Flattened D50, Spherical Flattened D100 (Zocchihedron)
Others:, Coin D2, Bone D6, Crooked D6, Jumping
D6, Elliptical D6, Tactile D6, Truncated Octahedron (D10), based on D14
Cuboctahedron
Fair Dice: Isohedra
An isohedron is a convex polyhedron with symmetries acting transitively on its faces. Every isohedron has an even number of faces. The isohedra make fair dice, and there are 30 of them (25 finite solids and 5 classes of infinite solids):
Platonic solids
Catalan solids
Dipyramids
Trapezohedra
Special Isohedra
Platonic Solids
Platonic Solids are congruent convex regular polygons. There are exactly five such solids: the cube (6 faces), dodecahedron (12), icosahedron (20), octahedron (8), and tetrahedron (4).
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Regular Tetrahedron (D4) |
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Composed of four equivalent equilateral triangular faces with four vertices and six edges. The tetrahedron is also a degenerate antiprism of 2 sides(see below). |
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Hexahedron, Cube (D6) |
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Composed of six square faces that meet each other at right angles with eight vertices and 12 edges. The cube is also a square prism (see below). |
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Regular Octahedron (D8) |
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Composed of eight equivalent equilateral triangular faces with six vertices and 12 edges. The octahedron of unit side length is the antiprism of three sides. The octahedron is also a square dipyramid with equal edge lengths. |
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Regular Dodecahedron (D12) |
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Composed of 12 regular pentagonal faces with 20 vertices and 30 edges.
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Icosahedron (D20) |
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Composed of 20 equivalent equilateral triangular faces with 12 vertices and 30 edges.
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Catalan Solids
Catalan Solids, also known as dual polyhedra of the Archimedean solids, are named in honor of the Belgian mathematician Eugène Charles Catalan who first published them in 1862. There are 13 Catalan solids with 12, 24, 30, 48, 60, and 120 faces.
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Triakis Tetrahedron (D12) |
Paper model, Missing in my collection [7] |
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Non-regular dodecahedron composed of 12 isosceles triangular faces with eight vertices and 18 edges. Can be constructed by cumulation of a unit edge-length tetrahedron by a pyramid with height sqrt(6)/15. No parallel faces. Dual polyhedron of the truncated tetrahedron. |
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Rhombic Dodecahedron (D12) |
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Composed of 12 rhombic faces with 14 vertices and 24 edges. Can be constructed by affixing a square pyramid of height 1/2 on each face of a unit edge-length cube. Dual polyhedron of the cuboctahedron. |
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Small Triakis Octahedron (D24) |
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Composed of 24 isosceles triangular faces with 14 vertices and 36 edges. Can be constructed by cumulation of a unit edge-length octahedron by a pyramid with height sqrt(3)-2/3*sqrt(6). Dual polyhedron of the truncated cube. |
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Tetrakis Hexahedron (D24)
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Composed of 24 isosceles triangular faces with 14 vertices and 36 edges. Can be constructed by cumulation of a unit cube by a pyramid with height 1/4. Dual polyhedron of the truncated octahedron. |
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Deltoidal Icositetrahedron (D24) |
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Composed of 24 deltoidal faces with 26 vertices and 48 edges. Also called trapezoidal icositetrahedron. Dual polyhedron of the small rhombicuboctahedron. Franck Dutrain (top), D-Total / Alexander Simkin (bottom)
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Pentagonal Icositetrahedron (D24) |
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Composed of 24 irregular pentagonal faces with 38 vertices and 60 edges. Comes in two enantiomorphous forms, known as laevo (left) and dextro (right). Dual polyhedron of the snub cube.
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Rhombic Triacontahedron (D30) |
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Composed of 30 rhombic faces with 32 vertices and 60 edges. Dual polyhedron of the icosidodecahedron |
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Disdyakis Dodecahedron (D48) |
Prototype, missing in my collection [1] |
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Composed of 48 triangular faces with 26 vertices and 72 edges Also called Hexakis Octahedron. Dual polyhedron of the Archimedean great rhombicuboctahedron. |
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Deltoidal Hexecontahedron (D60) |
Paper model, missing in my collection [1] |
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Composed of 60 deltoidal faces with 62 vertices and 120 edges. Also called trapezoidal hexecontahedron or strombic hexecontahedron. Dual polyhedron of the small rhombicosidodecahedron. |
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Pentagonal Hexecontahedron (D60) |
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Composed of 60 irregular pentagonal faces with 92 vertices and 150 edges. Comes in two enantiomorphous forms, known as laevo (left) and dextro (right). Dual polyhedron of the snub dodecahedron
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Triakis Icosahedron (D60) |
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Composed of 60 isosceles triangular faces with 32 vertices and 90 edges. Can be constructed by cumulation of a regular icosahedron by a pyramid. Dual polyhedron of the truncated dodecahedron
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Pentakis Dodecahedron (D60) |
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Composed of 60 isosceles triangular faces with 32 vertices and 90 edges. Can be constructed by cumulation of a regular dodecahedron by a pentagonal pyramid. Dual polyhedron of the truncated icosahedron. |
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Disdyakis Triacontahedron (D120) |
Paper model, missing in my collection [1] |
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Composed of 120 irregular triangular faces with 62 vertices and 180 edges. Also known as the hexakis icosahedron. Dual polyhedron of the Archimedean great rhombicosidodecahedron. Largest non-dipyramidal fair die. |
Dipyramids
Dipyramids are also called bipyramids or basic triangular dihedral. An infinite number of solids can be created by placing two pyramids symmetrically base-to-base. The height is arbitrary. A dipyramid whose base is a regular n-sided polygon is composed of 2n identical isosceles triangular faces with n+2 vertices and 3n edges.
The dipyramids are duals of the regular prisms.
In order to end up with opposite pairs of faces for convenient numbering, there need to be an even number of faces on each side of the "equator", such that n has to be even (n=2m). Then, the total number of faces is 2n=4m, m>1. (If dipyramids with odd n are chosen, the numbers have to be printed at the edges).
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Triangular Dipyramid (D6) |
Paper model, missing in my collection [1] |
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n=3, base is a triangle, six faces, five vertices, and nine edges. Two tetrahedra placed symmetrically base-to-base. Numbers are printed on the edges |
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Square Dipyramid (D8) |
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n=4, base is a square, eight faces, six vertices, and twelve edges. Regular octahedron (all edges are of same length) and elongated octahedron |
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Pentagonal Dipyramid (D10) |
Paper model, missing in my collection [1] |
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n=5, base is a regular pentagon, 10 faces, seven vertices, and 15 edges. Numbers are printed on the edges |
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Hexagonal Dipyramid (D12) |
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n=6, base is a regular hexagon, 12 faces, eight vertices, and 18 edges. |
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Octagonal Dipyramid (D16) |
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n=8, base is a regular octagon, 16 faces, 10 vertcies, and 24 edges. |
Trapezohedra
Trapezohedra (also called antidipyramids or deltohedra) are the dual polyhedra of the Archimedean antiprisms. (The name for these solids is not particular well chosen since their faces are not trapezoids but deltoids).
The n-gonal trapezohedron is composed of 2n faces which are congruent deltoids (or kites) with 2n+2 vertices and 4n edges. The faces are symmetrically staggered. The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry.
In order to end up with opposite pairs of faces for convenient numbering, there need to be an odd number of faces on each side of the zig-zagged "equatorial" line, such that n has to be odd (n=2m-1). Then, the total number of faces is 2n=4m-2, m>1. (If trapezohedra with even n are chosen, the numbers have to be printed at the edges).
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Triagonal Trapezohedron (D6) |
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n=3, six faces, eight vertices, and 12 edges. A cube is a special case trigonal trapezohedron with square faces |
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Tetragonal Trapezohedron (D8) |
Paper model, missing in my collection [1] |
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n=4, eight faces, ten vertices, and 16 edges. Numbers are printed on edges |
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Pentagonal Trapezohedron (D10) |
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n=5, ten faces, 12 vertices, and 20 edges. Top: all vertices normal Center: top and bottom vertices blunt Bottom: center vertices blunt |
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Hexagonal Trapezohedron (D12) |
Paper model, missing in my collection [1] |
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n=6, twelve faces, 14 vertices, and 24 edges. Numbers are printed on edges |
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Heptagonal Trapezohedron (D14) |
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n=7, 14 faces, 16 vertices, and 28 edges |
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Enneagonal Trapezohedron (D18) |
Paper model, missing in my collection [1] |
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n=9, 18 faces, 20 vertices, and 36 edges |
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Tridecagongonal Trapezohedron (D26) |
Paper model, missing in my collection [1] |
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n=13, 26 faces, 28 vertices, and 52 edges |
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Heptadecagonal Trapezohedron (D34) |
Chessex |
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n=17, 34 faces, 36 vertices, and 68 edges
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Icosikaipentagonal Trapezohedron (D50) |
GameScience |
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n=25, 50 faces, 52 vertices, and 100 edges |
Special Isohedra
There are ten more solids which can be created by skewing platonic solids, catalan solids, dipyramids, or trapezohedra. They are convex with symmetries w.r.t an even number of faces.
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Isosceles Tetrahedron (D4) |
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Composed of four identical isosceles triangular faces with four vertices and six edges. Tetragonal Disphenoid.
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Scalene tetrahedron (D4) |
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Composed of four identical scalene triangular faces with four vertices and six edges.
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Octahedral Pentagonal Dodecahedron (D12)
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Composed of 12 irregular pentagonal faces with 20 vertices and 30 edges. Skewed dodecahedron which has a symmetry that mirrors the octahedron.
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Tetragonal Pentagonal Dodecahedron (D12)
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Composed of 12 irregular pentagonal faces with 20 vertices and 30 edges. Skewed dodecahedron which looks similar to the Tetrahedron. |
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Trapezoidal Dodecahedron (D12) |
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Composed of 12 identical trapezoidal faces with 14 vertices and 24 edges. Skewed version of the rhombic dodecahedron. |
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Hexakis Tetrahedron (D24) |
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Composed of 24 scalene triangular faces with 14 vertices and 36 edges. |
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Dyakis Dodecahedron (D24) |
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Composed of 24 irregular tetragonal faces with 26 vertices and 30 edges. Can be constructed by warping the dodecahedron in such a way that each face folds in half. |
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Triangular Dihedron skewed up/down (D4n) |
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Can be constructed by skewing the equatorial vertices of a dipyramid with 4n faces up and down
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Triangular Dihedron skewed in/out (D4n) |
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Can be constructed by squeezing the equatorial vertices of a dipyramid with 4n faces in and out from the center of the solid |
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Skewed Trapezoidal Dihedron (D2n) |
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Can be constructed by grasping a trapezohedron by two opposing vertices and stretching it, to form an elongated solid whose sides are all of equal length, and then twisting it a bit, so the sides become uneven. |
A two-dimensional lamina such as a coin can also be viewed as a degenerate case of a fair 2-sided solid.
Barrels: (Modified) Prisms and Antiprisms
Two more infinite families of polyhedra which can be used to make dice are prisms and antiprisms. With three exceptions, these dice are not mathematically fair in the strong sense, but they can be transformed into shapes that allow rolling numbers with equal probability.
Prisms
An n-sided right prism is a polyhedron composed of two parallel copies of a regular n-sided polygon, connected by a band of 2n rectangles. The joining edges and faces are perpendicular to the base faces. An n-sided prism is composed of n+2 faces, 2n vertices and 3n edges.
The dual of a right prism is a dipyramid.
The only fair die that is a prism is the cube.
If two regular short n-pyramids (or some round shapes) are put on each of the base polygons, we get a die which will always “land” on one of the n rectangles. Since these rectangles are all equal, this die can be considered fair in the wide sense.
Because isohedra have an even number of faces, these modified prisms are the only way to make “almost” fair dice with an odd number of faces.
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Triangular Prism (D5) |
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Triangular right prism with numbers on the two base triangles and on the perpendicar edges. This die does not “land” on all faces with equel probabilities, i.e., it is not fair. |
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Square prism (D6) |
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The cube is an equilateral square prism. It is the only fair prism. |
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Pentagonal Prism (D7) |
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Pentagonal right prism with numbers on the two base pentagons and on the perpendicar edges. This die does not “land” on all faces with equal probabilities, i.e., it is not fair. |
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Heptagonal Prism (D9) |
Missing in my collection [1] |
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Heptagonal right prism. This die does not “land” on all faces with equal probabilities, i.e., it is not fair. |
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Elongated |
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Elongated right triangular prism. The die “lands” on each of the three faces with equal probability. Numbers are printed on the faces or on the edges |
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Rounded-off |
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Rounded-off elongated right triangular prism. The die “lands” on each of the three faces with equal probability. Numbers are printed on or near the edges. GameScience (top) Crystal Caste (center) Abraham Neddermann (bottom) |
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Modified |
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Elongated right square prism with two square pyramids, such that the die “lands” on each of the four rectangles with equal probability Crystal Caste (top) Bear Cub Machine (bottom) |
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Rounded-off Elongated Pentagonal Prism (D5) |
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Elongated right pentagonal prism, The die “lands” on each of the five rectangles with equal probability. Numbers are printed on the edges. Abraham Neddermann |
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Modified Hexagonal Prism (D6) |
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Rounded-off elongated right hexagonal prism. The die “lands” on each of the six faces with equal probability. Hasbro / Monopoly |
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Rounded-off Elongated Heptagonal Prism (D7) |
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Elongated right heptagonal prism. The die “lands” on each of the seven rectangles with equal probability. Numbers are printed on the edges. Abraham Neddermann |
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Rounded-off Elongated Nonagonal Prism (D9) |
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Elongated right nonagonal prism. The die “lands” on each of the nine rectangles with equal probability. Numbers are printed on the edges. Abraham Neddermann |
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Modified Decagonal Prism (D10) |
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Elongated right decagonal prism with two “cones”, such that the die “lands” on each of the nine rectangles with equal probability Bear Cub Machine |
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Rounded-off Elongated Hendecagonal Prism (D11) |
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Elongated right hendecagonal prism. The die “lands” on each of the eleven rectangles with equal probability. Numbers are printed on the edges. Abraham Neddermann |
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Rounded-off Elongated Tridecagonal Prism (D13) |
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Elongated right tridecagonal prism. The die “lands” on each of the thirteen rectangles with equal probability. Numbers are printed on the edges. Abraham Neddermann |
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Rounded-off Elongated Pentadecagonal Prism (D15) |
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Elongated right pentadecagonal prism. The die “lands” on each of the fifteen rectangles with equal probability. Numbers are printed on the edges. Abraham Neddermann |
Antiprisms
An n-sided right antiprism is a polyhedron composed of two parallel copies of a regular n-sided polygon (twisted by an angle 180°/n), connected by an
alternating band of 2n isosceles triangles. The line connecting the base centers is perpendicular to the base planes. An n-sided antiprism is composed of n+2 faces, n vertices and 2n edges.
Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterials: the vertices are symmetrically staggered.
The duals of the antiprisms are the trapezohedra.
The only fair dice that are antiprisms are the tetrahedron (a degenerate 2-antiprism) and the octrahedron (3-antiprism with equilateral triangles).
If two regular short n-pyramids (or some round shapes) are put on each of the base polygons, we get a die which will always “land” on one of the n triangles. Since these triangles are all equal, this die can be considered fair in the wide sense.
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2-Antiprism (D4) |
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The tetrahedron is a degenerate antiprism: the polygon at the top and the bottom are degenerated to lines
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3-Antiprism (D8) |
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The octahedron is a 3-antiprism with eight equilateral triangles |
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Modified |
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Right 3-Antiprism with two triangular pyramids, such that the die “lands” on each of the six larger isosceles triangles with equal probability |
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Modified |
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Right 4-Antiprism with two square pyramids, such that the die “lands” on each of the eight larger isosceles triangles with equal probability |
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Modified |
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Right 5-Antiprism with two pentagonal pyramids, such that the die “lands” on each of the ten larger isosceles triangles with equal probability. The arrangement of the faces corresponds to an “elongated icosahedron”. |
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Modified |
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Right 6-Antiprism with two hexagonal pyramids, such that the die “lands” on each of the twelve larger isosceles triangles with equal probability |
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Modified |
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Right 10-Antiprism with two decagonal pyramids, such that the die “lands” on each of the twenty larger isosceles triangles with equal probability |
Other Shapes
Spheres
Some dice are made of spherical shapes.
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Sphere (D1) |
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The sphere is a solid with a single face. |
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Spherical D4 |
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Colours instead of numbers. Easier to roll than a tetrahedron |
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Spherical D6 Plastic |
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Spherical die with internal cavity in which a weight moves which causes the die to settle in one of six orientations when rolled. |
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Spherical D6 Wood |
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Hollow, with internal cavity. |
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Flattened Spherical D6 |
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Flattened spherical D6. Hollow, with internal cavity. Bo-Jo dice. |
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Flattened Spherical D8 |
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Flattened spherical D8. Pipped, 2..9 |
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Flattened Spherical D32 |
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Flattened Glass sphere, numbered 00,0,1..30 |
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Flattended Spherical D50 |
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Flattened plastic sphere (Alan Davies) |
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Zocchihedron (D100) |
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Zocchihedron is the trademark of the most common 100-sided die, which was invented by Lou Zocchi, and debuted in 1985. It is not a polyhedron. Rather, it is more like a ball with 100 flattened planes. It is not fair. |
Various
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D2 |
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“Coin” |
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D6 |
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Concave , a.k.a “Bones”
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D6 |
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Crooked |
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D6 |
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Jumping
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D6 |
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Elliptical |
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D6 |
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Tactile |
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D10 |
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Truncated octahedron, Top: Blue Japanese “Trinity” die, numbered 0..9 Bottom: unknown white die, numbered 0..9 |
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D14 |
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Derived from a Cuboctahedron, an Archimedan solid with eight triangular faces and six square faces, i.e., 14 faces, 24 edges and 12 vertices. Pictures show die from different angles Card Dice |
References
[1] Huge collection of
shapes, including some paper models and prototypes not commercially available: http://www.dicecollector.com/diceinfo_how_many_shapes.html
[2] 3D pictures of
isohera: http://mathworld.wolfram.com/Isohedron.html
[3] Wikipedia http://wikipedia.org/
[4] Mathematical
derivation of all fair dice: http://hjem.get2net.dk/Klaudius/Dice.htm
[5] Another
mathematical derivation of all fair dice: http://www.geocities.com/dicephysics/thesis7.doc
[6] Detailed
description of all fair dice: http://www.mathpuzzle.com/Fairdice.htm
[7] International Bone
Rollers’ Guild: http://members.aol.com/dicetalk/polymore.htm
$
[8] Polygon Names: http://www.math.com/tables/geometry/polygons.htm