Spheres with an internal Cavity and a weight
There are spherical dice with an internal cavity and a weight. When the die is rolled, the weight moves, which causes the die to settle in one of n orientations. For fair dice, the weight is a sphere itself, and the internal cavity of a spherical Dn is a often a platonic solid with n vertices (not faces!), which is also a platonic solid.
D6

Plastic
Wood 
Internal cavity ressembles an octahedron, the dual of the cube 

D8

Clsn / Shapeways 

Internal cavity is a cube, the dual of the octahedron. 
D12

Clsn / Shapeways 

Internal cavity is an icosahedron, the dual of the dodecahedron. 
Truncated or flattened spherical dice are made of spheres where n spherical caps of equal size are removed. This results in a sphere with n equal nonoverlapping circles. This can also be seen as the intersection of a special nsided polyhedron and a sphere.
When the die is rolled, it most likely stops on one of the circles. Because it might also stop on a curved part of the surface, such dice are not exactly fair in the mathematical sense, but if the circles are placed in a special manner (see below), these dice can be considered fair in a practical sense.
There are different approaches to design the intersecting polyhedron, as described by the following (not necessarily mutually exclusive) criteria:
The intersecting polyhedron is a platonic solid. This results in “fair” dice.
The intersecting polyhedron is an isohedron (note that platonic solids are a subset of isohedra). This results in “fair” dice.
The intersecting polyhedron is a symmetrical construction such that the centers of the faces are located at the poles of the sphere and on circles parallel to the equator. Then, the resulting polyhedron has some rotational symmetry around the axis between the poles.
An alternative view is to place the centers of n equal nonoverlapping circles according to the following (not necessarily mutually exclusive) criteria of distributing points on a sphere or packing circles on a sphere:
Packing: maximize
the minimum distance between any pair of points: This maximum distance is
called the covering radius, and the configuration is called a spherical code
(or spherical packing). Packings are the same if some orthogonal matrix
transforms one to the other.
The resulting circles that are locked into position by their neighbours are
called rigid, those are free to “move” in the small space between them and
their nearest neighbours are called rattlers.
The problem of spherical packing is sometimes known as the or Tammes problem. There
are exact solutions for 4, 6, and 12, but the general problem has not been
solved. Best known solutions are available here.
Antipodal Packing: maximize the minimum distance between any pair of points with the constraint that there are antipodal pairs of points. This constraint leads to parallel faces of the dice, making it possible to print the number on the faces. Except for n=6 and n=12, the optimum solutions of the Tammes problem (see above) are not antipodal, i.e., the resulting dice do not have parallel faces. No systematic approach was found in the literature for this problem. Here we try to construct optimum dice and back the results up by brute force search methods.
Potential: minimize the potential energy of the points: Sum 1 / dist(P_i, P_j ), minimal 1/r potential. This is also known as Thomson problem. Best known solutions are available here.
Check this page to display optimum truncated spheres.
The table below shows a collection of truncated spherical dice. Full size pictures are from my own collection. Halfsize pictures are from dice not (yet) in collection, picture sources are indicated in [Ref].
Symmetry is categorized according Wikipedia pages Point groups in three dimensions, Schoenflies notation and List of spherical symmetry groups.
The description was obtained from this blog or through private communication with the designers.
# circles, symmetry type 
Picture and Designer 
Polyhedron 
Description 
D2 / D∞h“Coin” Optimum 2fold 
Sylook / Shapeways
McTrivia / Shapeways Missing 
Equator 2
Equator 1
Pole (2fold) 
Intersection of two parallel planes and a sphere. “Fair”.
Optimum in the sense of “antipodal packing”. This can be seen by construction: the maximum angle between any two points on a sphere is 180°.
Parallel faces.
Phi=180°=360°/2
Symmetry type D∞h 
D3 / D3h“Elongated 3Prism” Optimum 3fold 
Magic / Shapeways
McTrivia / Shapeways Missing 
Intersecting polyhedron Constellation
Pole (3fold)
Equator

Intersection of an elongated 3prism (“Toblerone”) and a sphere. The 3 points form an equilateral triangle. “Fair”.
Phi=120°=360°/3
Symmetry type D3h (3fold prismatic symmetry)
No parallel faces, numbers are on the spherical part.

D4 / Td“Tetrahedron” Optimum

Magic / Shapeways
McTrivia / Shapeways Missing 
Intersecting polyhedron Constellation (dual) 
Intersection of a
tetrahedron and a sphere. The four points form
a tetrahedron (dual). “Fair”. Optimum in the sense of “packing” and “potential”.
Phi= 109.4712206°. tan(phi/2)=sqrt(2).
Symmetry type Td (full tetrahedral symmetry).
No parallel faces, numbers are on the spherical part.
Upper die is numbered 1..4, lower 2,3,3,4 (same as 2 D2) 
D4 / C4“Elongated 4Prism” 4fold 
Sylook / Shapeways 
Intersecting polyhedron Constellation
Pole (4fold)
Equator

Intersection of an elongated 4prism and a sphere. The four points form a square “Fair”.
Phi=90°=360°/4
Symmetry type C4 (4fold rotational symmetry)
Optimum in the sense of “antipodal packing”. This can be seen by construction: two pairs of antipodal points on a sphere lie in a plane. The angle is maximized if they lie on a great circle, leading to a maximum angle of 360°/4=90°. 
D5 / D3h“Prism” Optimum 3fold 
Magic / Shapeways
McTrivia / Shapeways Missing 
Intersecting polyhedron Constellation (dual)
Pole (3fold)
Equator 
Intersection of a 3prism and a sphere. The 5 points form a 3bipyramid. “Fair”.
Phi= 90° (same as for D6).
3fold rotational symmetry, 1+3+1.
Symmetry type D3h (3fold prismatic symmetry)
Only 2 parallel faces, 3 numbers are on the spherical part
There is an infinite number of optima in the sense of “packing”. Starting with a cube with one of the equatorial circles removed, the 3 circles on the equator can be moved freely. Placing these 3 circles such that their centers form an equilateral triangle yields the 3prism. 
D6 / Oh“Cube” Optimum

Magic / Shapeways
BoJo dice
Sylook / Shapeways Missing 
Intersecting polyhedron Constellation (dual)
Pole (4fold)
Equator 
Intersection of a sphere and a cube. The 6 points form a regular octahedron. “Fair”.
Optimum in the sense of “packing”, “potential”, and “antipodal packing”.
Parallel faces.
Phi= 90°=360°/4.
Symmetry type Oh (full octahedral symmetry).
The circles of the BoJo die do not touch.

D7 / D5h“Prism” 5fold 
McTrivia / Shapeways 
Intersecting polyhedron Constellation (dual)
Pole (5fold)
Equator 
Intersection of a 5prism and a sphere. The 7 points form a 5bipyramid. “Fair”.
Phi= 72°=360°/5
Symmetry type D5h (5fold prismatic symmetry)

D7 / C2v2fold 
Magic / Shapeways 
Intersecting polyhedron
North Pole (2fold)
Equator
South Pole 
Symmetrical construction with 2fold rotational symmetry, 1+4+2 (1 rhombus, 4 quads, 2 triangles).
Phi=77.25°.
Symmetry type C2v=D1h (reflection in a plane and a 180° rotation through a line in that plane)
Tranformation of a cube, where one face is cut into two in diagonal. Then, the vertices were moved to maximize the radii of the circles. Nevertheless, this is not optimum in the sense of packing. 
D7 / C3vOptimum 3fold 
Magic /
Magic / Shapeways 
Intersecting polyhedron
North Pole (3fold)
Equator
South Pole 
Optimum in the sense of “packing”.
Phi=77.869542°. No parallel faces.
3fold rotational symmetry, 1+3+3 (1 equilateral triangle + 3 irregular pentagons + 3 deltoids).
Symmetry type C3v (3fold pyramidal symmetry)

D8 / Oh“Octahedron” 4fold 
Magic / Shapeways
Sylook / Shapeways Missing
LOGICAL LEARNING LTD
McTrivia / Shapeways Missing 
Intersecting polyhedron Constellation (dual)
Pole (4fold) Equator

Intersection of an octahedron and a sphere. The 8 points form a cube. “Fair”.
Parallel faces.
Phi= 70.528779°. sin(phi/2)= 1/sqrt(3).
Symmetry type Oh (full octahedral symmetry)
Note that although the octahedron is a platonic solid, this shape is not optimum in the sense of “packing” (but the intersection of a tetragonal trapezohedron and sphere is).
Optimum in the sense of “antipodal packing”. This can be seen by construction: 2 poles, 6 points on the equator (1+6+1), leading to a mimumum angle of 360°/6=60° 2 poles, 3 points each on two planes (1+3+3+1), leading to the 3antiprism (which is the cube!) with phi= 70.528779° 2 poles, 2 points each on three planes (1+2+2+2+1). This is the same as 1+6+1 rotatated by 90°,as the blue points (1+2+2+2+1) lie in the same plane No poles, 2 planes of 4 points each, yielding a 4prism (cube) (Note: the 4antiprism is not antipodal) in some asymmetric way. I am not aware of an analytic proof that this does not lead to a better configuration, but extensive numeric search has not found a better result than the cube
From top: numbered (2x), pipped (29), moon phases 
D8 / D4d“4Trapezohedron” Optimum 4fold 
Magic / Shapeways 
Intersecting polyhedron Constellation (dual)
Pole (4fold)
Equator 
Intersection of a tetragonal trapezohedron and a sphere. The 8 points form a square antiprism. “Fair”.
Optimum in the sense of “packing”.
No parallel faces.
Phi= 74.8584922°.
Symmetry type D4d (4fold antiprismatic symmetry)

D9 / D3hOptimum 3fold 
Magic / Shapeways
Clsn / Shapeways
McTrivia / Shapeways 
Pole (3fold)
Equator 
Optimum in the sense of “packing”,
Phi=70.528779°= sin(phi/2)= 1/sqrt(3). Note that this is the same angle as for the octahedron. Explanation?
3fold rotational symmetry, 3+3+3 (3 rhombi, 6 irregular pentagons).
Symmetry type D3h (3fold prismatic symmetry)
Magic: 1,5,9 on equator Clsn: 3,5,6 on equator McTrivia: 2,3,5 on equator
Magic’s die is numbered from 1 to 9: 6 numbers are on vertices, 3 on edges, none on the faces. The sum of the numbers on the upper vertices is 15. Same thing for the sum of the 3 numbers on the lower vertices and the sum of the 3 numbers on the edges. This numbering is closely linked to the well known 3x3 magic square.
McTrivia’s die is numbered to get the same results as throwing 2 D3s (6,5,5,4,4,4,3,3,2) 
D10 / D5d“5Trapezohedron” 5fold 
Magic / Shapeways 
Intersecting polyhedron Constellation (dual)
Pole (5fold)
Equator 
Intersection of a 5trapezohedron and a sphere. The 10 faces are parallel to those of a dodecahedron, whose top and bottom faces are removed. The 10 points form a 5antiprism. “Fair”.
Parallel faces.
Phi 63.44° = acos(1/sqrt(5))
5fold rotational symmetry.
Symmetry type D5d (5fold antiprismatic symmetry)
Optimum in the sense of “antipodal packing”. This can be seen by construction: 2 poles, 8 points on the equator (1+8+1), leading to a mimumum angle of 360°/8=45° 2 poles, 4 points each on two planes (1+4+4+1), leading to an elongated square dipyramid with phi=60° (see below) 2 poles, 4 points on equator, 2 points each on two intermediate planes (1+2+4+2+1). The angle is less than 60°. No poles, 2 planes of 5 points each, yielding ta 5antiprism in some asymmetric way. I am not aware of an analytic proof that this does not lead to a better configuration, but extensive numeric search has not found a better result than the 5antiprism
Numbered 0..9, also available in 00..99. 
D10 / D4h“Square bifrustum” 4fold 
No design available yet 
Intersecting polyhedron Constellation (dual)

Intersection of a square bifrustum and a sphere. The 10 faces are parallel to those of a dodecahedron, whose top and bottom faces are removed. The 10 points form an elongated square bipyramid.
The design can be described as 1+4+4+1. The maximum angle can be achieved if the triangles are equilateral, yielding phi=60°.
4fold rotational symmetry.
Symmetry type D4h (4fold prismatic symmetry)

D10 / C3v“Cuboctahedron” 3fold 
Magic / Shapeways 
Pole(3fold)
Equator

Based on a D14 (nonregular Cuboctahedron), with 4 spherical caps added to "cancel" 4 out of the 14 faces.
Phi=54.74° = arccos(1/sqrt(3)).
3fold rotational symmetry, 1+3+3+3.
Symmetry type C3v (3fold pyramidal symmetry)
Numbered from 0 to 9. 
D10 / D4d4fold 
Magic / Shapeways 
Pole (4fold)
Equator 
Symmetrical construction with 4fold rotational symmetry. The dual polyhedron has 2 squares and 8 irregular pentagons, with the maximized radii.
Phi=65.53°.
4fold rotational symmetry, 1+4+4+1.
Symmetry type D4d (4fold antiprismatic symmetry)
Numbered from 0 to 9. 2 parallel faces. 
D10 / C2vOptimum 2fold 
No design available yet 

Optimum in the sense of “packing.
Phi= 66.146822°.
Symmetry type C2v=D1h (reflection in a plane and a 180° rotation through a line in that plane)
2fold rotational symmetry, 2+4+2+2. Polyhedron with 2+4 quads and 2+2 pentagons, 
D11 / C2v2fold 
McTrivia / Shapeways 
North Pole (2fold)
Equator
South Pole 
2fold rotational symmetry, 2+4+4+1.
Symmetry type C2v (2fold pyramidal symmetry)

D11 / unk3fold 

Pole (3fold)
Equator
Polyhedron? 
Symmetrical construction with 3fold rotational symmetry: 1+3+3+3+1
Phi=60°=360°/6.
Symmetry type ???
Numbered such that the sum of the numbers on each of the 3 horizontal planes is 18. 
D11 / C5vOptimum 5fold 
Magic / Shapeways

North Pole (5fold)
Equator
South Pole

Intersection of a dodecahedron and a sphere, with 1 spherical caps added to the dodecahedron to "cancel" 1 out of the 12 faces.
Optimum in the sense of “packing”. However, the center of gravity is not at the center of the circle à not fair.
Phi 63.44° = arccos(1/sqrt(5)), same as Pentagonal Dodecahedron.
5fold rotational symmetry, 1+5+5.
Symmetry type C5v (5fold pyramidal symmetry) 
D12 / Ih“Pentagonal Dodecahedron” Optimum

Magic / Shapeways

Intersecting polyhedron Constellation (dual)
Pole (5fold)
Equator 
Intersection of a dodecahedron and a sphere. The 12 points form a regular icosahedron. “Fair”.
Optimum in the sense of “packing” and “antipodal packing”
Parallel faces.
Phi 63.44° = arccos(1/sqrt(5))
Symmetry type Ih (full icosahedral symmetry) 
D13 / unk 
McTrivia / Shapeways 

Unknown design 
D13 / C4vOptimum 4fold 
Magic / Shapeways 
North Pole (4fold)
Equator
South Pole 
Optimum in the sense of “packing”, phi= 57.1367031°.
4fold rotational symmetry, 1+4+4+4.
Symmetry type C4v (4fold pyramidal symmetry)

D14 / OhCuboctahedron

Magic /
Magic / Shapeways
Card Dice

Pole (4fold)
Equator

Intersection of a cuboctahedron and a sphere.
Phi=54.74° = arccos(1/sqrt(3)).
4fold rotational symmetry, 1+4+4+4+1 and 3fold rotational symmetry, 1+3+3+3+3+1.
Symmetry types Oh (full octahedral symmetry) and Td (full tetrahedral symmetry)

D14 / D6h“Hexagonal bifrustum” 6fold 
No design available yet 
Intersecting polyhedron Constellation (dual)

Intersection of a hexagonal bifrustum and a sphere. The 14 points form an elongated hexagonal bipyramid.
The design can be described as 1+6+6+1. The maximum angle can be achieved if the quadrangles are squares, corresponding to an angle theta of the horitontal plane of 63.435°, yielding
Phi=53.13°
Parallel faces.
6fold rotational symmetry.
Symmetry type D6h (6fold prismatic symmetry) 
D14 / D2dOptimum 2fold

No design available yet 

Polyhedron consists of 4+8 pentagons and 2 hexagons
Optimum in the sense of “packing”.
Phi= 57.1367031°.
2fold rotational symmetry, 1+4+2+2+4+1
Symmetry type D2d (2fold antiprismatic symmetry) 
D15 / D5h5fold 
Magic / Shapeways

Pole (5fold)
Equator 
Symmetrical construction with 5fold rotational symmetry: 5+5+5.
Phi=52.501308°.
Symmetry type D5h (5fold prismatic symmetry)
Numbered from 1 to 15: all numbers are on edges. The sum of the numbers on each tropic is 40 and the same applies to the equator. Except for 8, all the numbers go by pair summing to 16: on the tropic two numbers symmetric relatively to the plane of the equator sum to16 and on the equator circle itself two numbers symmetric relatively to number 8 sum to 16. 
D15 / unk 
McTrivia / Shapeways 

Unknown design 
D15 / C1Optimum No symmetry 
No design available yet 

Polyhedron consists of 12 pentagons and 3 hexagons
Optimum in the sense of “packing”.
Phi= 53.6578501°.
Symmetry type C1 (no symmetry) 
D16 / D4dOptimum 4fold 
Magic / Shapeways

Pole (4fold)
Equator

Symmetrical construction with 4fold symmetry: 4+4+4+4, which is indeed optimum in the sense of “packing”.
Phi= 52.2443957°.
Symmetry type D4d (4fold antiprismatic symmetry)

D16 / unk 
McTrivia / Shapeways 

Unknown design
Numbered to get the same results as throwing 2 D4s (8,7,7,6,6,6,5,5,5,5,4,4,4,3,3,2) 
D17 / D5h5fold 
Magic / Shapeways

Pole (5fold)
Equator

Symmetrical construction with 5fold rotational symmetry: 1+5+5+5+1.
Phi=51.03° (almost optimum)
Symmetry type D5h (5fold prismatic symmetry)
Numbered such that the sum of the 5 numbers of the two tropics and of the equator is constant (and equals to 51) and that the sum of the 2 poles, of two numbers of the tropics that are symmetric relatively to the equator, of two numbers of the equator that are symmetric relatively to number 9 is constant (and equals to 18) 
D17 / unk 
McTrivia / Shapeways 

Unknown design 
D17 / C2vOptimum 2fold 
No design available yet 

Polyhedron consists of 1 rhombus, 10 pentagons and 6 hexagons
Optimum in the sense of “packing”.
Phi= 51.0903285°.
Symmetry type C2v=D1h (reflection in a plane and a 180° rotation through a line in that plane) 
D18 / Oh“Rhombicuboctahedron” 4fold 
Magic /
Magic / Shapeways

Pole (4fold)
Equator

Intersection of the squares of Rhombicuboctahedron and a sphere (there are 18 squares; the 8 triangles are not used)
Phi=45°=360°/8
Symmetry type Oh (full octahedral symmetry)
All faces are parallel.
Numbered such that opposite faces sum to 19 and that the 8 numbers around any of the 3 diameters sum to 76 (8 times the average value, which is 9.5) and that the 6 numbers surrounding a spherical zone to sum to 57 (six times 9.5).

D18 / D2h2fold Optimum 
Shapeways

North Pole (2fold)
Equator
South Pole (2fold) 
Optimum in the sense of “antipodal packing”.
Phi= 47.9821°
Symmetry type D2h (prismatic symmetry)
All faces are parallel.
See this thread for more information. 
D18 / C2Optimum 2fold 
No design available yet 

Polyhedron consists of 1 rhombus, 10 pentagons and 6 hexagons
Optimum in the sense of “packing”.
Phi= 49.5566548°.
Symmetry type C2 (2fold rotational symmetry), no reflectional symmetry! 
D19 / unk2fold 
Magic / Shapeways 

Designed based on minimum energy, but then maximized the radii. The repartition of the faces by layers is 1+4+2+4+2+2+4.
2fold rotational symmetry.
Phi=?°.
Symmetry type unknown (C2?) 
D19/ unk 
McTrivia / Shapeways Missing 

Unknown design 
D19 / CsOptimum

No design available yet 

Polyhedron consists of 14 pentagons and 6 hexagons
Optimum in the sense of “packing”.
Phi= 47.6919141°.
Symmetry type Cs (reflection symmetry), no rotational symmetry!
This design has one rattler. 
D20 / Ih“Icosahedron”

Magic / Shapeways

Intersecting polyhedron
Constellation (dual)
Pole (3fold)

Intersection of a regular icosahedron and a sphere. The 20 points form a pentagonal dodecahedron.
“Fair”, but not optimum.
Phi=41.810315°=2*arcsin(1/sqrt(3)/psi) where psi=(1+sqrt(5))/2, the golden ratio
Symmetry type Ih (full icosahedral symmetry)
All faces are parallel.
Due to the fact that a vertex is surrounded by 5 triangles, the rounded area is larger than in other truncated sphere dice (34.2%, about twice as much as average), yielding to smaller faces. 
D20 / D3hOptimum 3fold

No design available yet 

Polyhedron consists of 14 pentagons and 6 hexagons
Optimum in the sense of “packing”.
Phi= 47.4310362°.
Symmetry type D3h (3fold prismatic symmetry)
This design has two rattlers at the poles. It was found by van der Waerden in 1952. 
D21 / C1Optimum No symmetry 
No design available yet 

Polyhedron consists of 12 pentagons and 9 hexagons
Optimum in the sense of “packing”.
Phi= 45.6132231°.
Symmetry type C1 (no symmetry) 
D21 / unk 
Magic / Shapeways Not yet available


4fold rotational symmetry, 4+4+4+4+4+1
these two arrangements of 21 circles around the sphere seem as efficient (the diameter of the circles is the same for a given sphere). 
D21 / unk 
Magic / Shapeways Not yet available


2fold rotational symmetry 2+4+2+4+2+2+4+1 
D22 / D5d 
Magic /
Magic / Shapeways 
Pole (5fold)
Equator 
5fold rotational symmetry, 1+5+5+5+5+1
Symmetry type D5d (5fold antiprismatic symmetry)
Optimum in the sense of “antipodal packing”.

D22 / Td 
Magic / Shapeways 
Intersecting polyhedron
North Pole
South Pole
Equator 
Polyhedron consists of 12 pentagons and 10 hexagons
Optimum in the sense of “energy”.
Symmetry type Td (full tetrahedral symmetry).
Phi= 43.302°. 
D22 / unk 
McTrivia / Shapeways 
Top
Bottom 
Unknown design
Parallel faces 
D22 / C1Optimum No symmetry 
No design available yet 

Polyhedron consists of 12 pentagons and 10 hexagons
Optimum in the sense of “packing”.
Phi= 44.7401612°.
Symmetry type C1 (no symmetry) 
D24 / Oh“Deltoidal Icositetrahedron”

Magic / Shapeways

Intersection of deltoidal icositetrahedron and a sphere.
“Fair”, but not optimum.
Phi= 41.89055556°
Symmetry type Oh (full octahedral symmetry)
All faces are parallel.
Parallel faces, opposite faces sum to 25. 

D24 / O“Pentagonal Icositetrahedron” Optimum

Magic / Shapeways 

Intersection of pentagonal icositetrahedron and a sphere.
“Fair” and optimum in the sense of “packing”.
Phi= 43.6907671°
Symmetry type O (chiral octahedral symmetry)
No parallel faces. 
D25 / unk 
McTrivia / Shapeways 

Unknown design
Numbered to get the same results as
throwing 2 D5 (10,9,9,8,8,8,7,7,7,7,6,6,6,6,6, 
D28 / unk 
McTrivia / Shapeways 
Top
Bottom 
Unknown design
Parallel faces 
D30 / unk 
McTrivia / Shapeways Missing 

Unknown design
Alphabet, including spanish letters 
D32 / D3d3fold 
Glass, made in Czechoslovakia 

Symmetrical construction with 3fold rotational symmetry: 1+6+9+9+6+1.
Symmetry type D3d (3fold antiprismatic symmetry)
Numbered 00,0,1..30 
D32 / Ih“Nonuniform

Magic / Shapeways

Pole (29)
Equator (between 11 and 18) 
Intersection of nonuniform truncated icosahedron (convex hull of the rhombidodecahedron), a kind of polyhedral soccer ball, and a sphere.
Optimum in the sense of “energy”, “covering”, and “volume”, but not “packing”.
Phi= 37.377°.
Symmetry type Ih (full icosahedral symmetry).
Numbered from 1 to 32 with the difference between two numbers on opposite faces being always 16. Taking the sum of a face surrounded by 5 neighbours (there are twelve of them, that orrespond to pentagons on the underlying polyhedron) and its 5 surrounding faces equas 99.

D32 / C3Optimum 3fold 
No design available yet 

Polyhedron consists of 12 pentagons and 20 hexagons
Optimum in the sense of “packing”.
Phi= 37.475214°.
C3 (3fold rotational symmetry) 
D33 / D3h 
Magic / Shapeways

Pole (center point between 6,12,33)
equator 
Symmetrical construction with 3fold rotational symmetry: 3+9+9+9+3.
Symmetry type D3h (3fold prismatic symmetry)
All faces are parallel.
Numbered such that all numbers are by groups of 3 that sum to 51: the 3 numbers at the North and South pole the 9 numbers at the equator and at the 2 tropic taken 3 by 3 (any 3 numbers at 120°) All numbers sum to 34 by group of 2 (except obviously 17, since only 17 itself can sum to 17 to give 34): all the numbers of the poles and of the tropics with their symmetric number relatively to the equator  all the numbers of the equator (except 17) with their ymmetric number relatively at number 17 
D33 / C3Optimum 3fold 
No design available yet 

Polyhedron consists of 12 pentagons and 21 hexagons
Optimum in the sense of “packing”.
Phi= 36.254553°.
C3 (3fold rotational symmetry) 
D36 / unk 
McTrivia / Shapeways 

Unknown design
Numbered like 2D6 
D40 / unk 
McTrivia / Shapeways 

Unknown design

D50 / D6h6fold 
Alan Davies 

Symmetrical construction with 6fold rotational symmetry: 1+6+12+12+12+6+1.
Symmetry type D6h (6fold prismatic symmetry)
Numbered 0 (empty),1..49 
D50 / D4hWMF 
WMF

North pole
South pole
Equator 
Symmetrical construction with 4fold rotational symmetry: 1+4+4+4+8+8+8+4+4+4+1
Symmetry type D4h (4fold prismatic symmetry)
Numbered 1..49 plus symbol WMF (0 or 50). Used for German Lotto with numbers 1..49.

D50 / Oh 
friz / Shapeways Missing 
North pole
South pole 
Symmetrical design, based
Symmetry type Oh (full octahedral symmetry).
Roulette die, labelled 1, 2, 3, ..., 36, 0, 00, ODD, EVEN, HIGH, LOW, BLACK, RED, 112, 1324, 2536, Col 1, Col 2, Col 3. 
D50 / unk 
Magic / Shapeways Not yet available


This is a Truncated Sphere D50 based on a special polyhedron that is the intersection of a cube (6 squares), and octahedron (8 hexagons with a 3fold rotational symmetry), a rhombic dodecahdron (12 hexagons with a 2fold rotational symmetry) and a nonusual icositetrahedron (24 hexagons with a simple symmetry). You can check: 6 + 8 + 12 + 24 = 50.

D60 / unk 
McTrivia / Shapeways 

Unknown design
Numbered 0..59 
D100 / imp“Zocchihedron” 
GameScience 

Symmetrical construction: 1+6+11+15+17+17+15+11+6+1
Symmetry type improper rotation / rotoreflection: Lower hemisphere is mirrored at equatorial plane and rotated by 360/34°.
Numbered 1..100
Zocchihedron is the trademark of the most common 100sided die, which was invented by Lou Zocchi, and debuted in 1985. 
D144 / unk 
McTrivia / Shapeways 

Unknown design

D1 


Truncated sphere with one circle at the south pole such that the die stops with a high probability with the “1” up. 
D4 


Colours instead of numbers. Easier to roll than a tetrahedron 