For an arbitrary die with opposite faces (the tetrahedral four sided dice fails here), it is standard for a dice to have the sum of the numbers on the two faces to be one greater than the sides of the dice. This is true for the cube (d6), octahedron (d8), decahedron (d10 - not a platonic solid but still obeys the rules) dodecahedron (d12), and the icosahedron (d20).

As the cube is the most classic of dice, there are a few more numerical bits in it than just the sum to seven. Looking at a corner of the dice, the numbers go around it counter clockwise:

  ____    ____
 / 1 /|  / 4 /|
+---+3| +---+6|
| 2 | / | 5 | /
+---+/  +---+/
From this, there is only one way to arrange a dice:
    | 2 |
| 3 | 1 | 4 |
    | 5 |
    | 6 |

On dice that use points (dots) as opposed to the number (most frequently inked on rather than engraved - the engraving process removes some mass and thus causes the dice to be slightly weighted). The 1, 4, and 5 are symmetric, however the 2, 3 and 6 are not. There are a total of 8 possible orientations for these points. The most common one is:

  /*      /|
 /   *   / |
/      */ *|
--------  *|
|*     |* *|
|      |* /
|     *|*/

Abstracting the dice out there are certain properties that all dice have:

  • Equal chance of landing on each face
  • All faces must be identical or a mirror image
  • All faces must be placed identically in relation to each other
  • A dice must be convex
From this, the platonic solids clearly fall into these rules, however as any RPGer knows there are other dice.

The most well known of these other dice is the Rhombic Triacontahedron or d30. There are some other interesting versions of dice that are not used such as the Rhombic Dodecahedron which is a d12 made out of rhombic faces rather than pentagonal.

For much more information on the topology of dice that I can't quite get my brain around (and thus would either mangle the information or copy it wholesale - neither of which is good), I highly recommend looking at where Euler gets invoked and the quest for the d100 is explored.