| # | Face | Group |
| 0 |
|
D4 |
| 1 |
|
D4 |
| 2 |
|
D2d |
| 3 |
|
D2d |
| 4 |
|
D4 |
| 5 |
|
D4 |
| 6 |
|
D2x |
| 7 |
|
D2x |
| 8 |
|
D4 |
| 9 |
|
D4 |
The die faces' symmetry groups are the symmetry
group of the square and two of its subgroups.
They are all rotation-reflection (dihedral) groups.
The die itself, ignoring face markers, has octahedral symmetry, Oh.
| Group | Elements |
| D2x | Rotations: 0°, 180° |
| Reflections: axial: horizontal and vertical | |
| D2d | Rotations: 0°, 180° |
| Reflections: diagonal: both diagonals | |
| D4 | Rotations: 0°, 90°, 180°, 270° |
| Reflections: both axes and both diagonals |
The die itself, ignoring face markers, has octahedral symmetry, Oh.
Faces on a Die: Their Placement
Opposite faces usually add up to 7,
but within this constraint,
there are 24 = 16 possible arrangements:
- 2: Reflection of the die
- 2: Rotation of face 2 by 90°
- 2: Rotation of face 3 by 90°
- 2: Rotation of face 6 by 90°
Non-Cubical Dice
The most common kind is the coin. Flipping a coin is eseentially using it as a d2 die.
Players of tabletop role-playing games have used numerous kinds of dice, distinguished by them as d<number>. They include:
Degeneracies:
The most common kind is the coin. Flipping a coin is eseentially using it as a d2 die.
Players of tabletop role-playing games have used numerous kinds of dice, distinguished by them as d<number>. They include:
| Family | Number of Sides |
| Platonic solids | 4, 6, 8, 12, 20 |
| Catalan solids | 12, 24, 30, 48, 60, 120 |
| Bipyramids | Even numbers ≥ 6 |
| Trapezohedra | Even numbers ≥ 6 |
| Prisms | Integers ≥ 3 |
| Antiprisms | Even numbers ≥ 2 |
Degeneracies:
- 2-antiprism: tetrahedron
- 3-antiprism: octahedron
- 3-trapezohedron, 4-prism: cube
To my symmetry index page