Commonly used $10$-sided dice are pentagonal trapezohedrons, as opposed to pentagonal bipyramids. Given that bipyramids are a more "obvious" shape for a fair die with an even number of faces, it's curious to me that the trapezohedrons are the more commonly used shape.

So, what are the advantages, if any, of trapezohedrons over bipyramids for making fair dice? Specifically, are there any meaningful differences in the symmetry properties of these shapes?

Note: the name "trapezohedron" is misleading, at least in the USA. The faces are actually kites, as if we had removed two opposing faces of a pentagonal (regular) dodecahedron and extended the remaining faces to close the gap.

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    $\begingroup$ Intuitively I would think that the trapezohedron has a slightly larger space to write a number on each side, relative to the overall size of polyhedron. $\endgroup$ Jun 12 '19 at 11:50
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    $\begingroup$ Maybe they roll better? (And I don't mean "you get better numbers".) $\endgroup$
    – Arthur
    Jun 12 '19 at 11:51
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    $\begingroup$ They used to be bi-pyramids, and then the manufacturers decided that the current shape is more aesthetic. Neither are particularly fair dice. All of the "D & D" dice are subject to more manufacturing irregularities than 6 sided dice (with casino dice facing the most rigorous standard). And the 10 is the most easily manipulated by the person throwing the dice. $\endgroup$
    – Doug M
    Jun 12 '19 at 12:06
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    $\begingroup$ @Doug M do you have a picture of a bipyramid die with 10 faces? Did they print the numbers on the edge? I'd love to see it! $\endgroup$ Jun 12 '19 at 12:47
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    $\begingroup$ Edited for clarity, see last paragraph. $\endgroup$ Jun 12 '19 at 15:40

A pentagonal bipyramid would work fine. The problem is that reading the result would be difficult.

Dice roll on a surface and land on one of the faces, then you read the result (usually) on the face on top of the die (the standard d4 would be an exception).

If you make a pentagonal bipyramid and it lands on one of the faces, however, the top of the die is an edge!

This holds for any shape that has an odd number of sides along is mid-section: if you make a bipyramid, it's going to end up with an edge up when you roll it.

This is what such a die would look like pentagonal bipyramid die In the left picture, the result is a 4, in the right it's a 5. Not very convenient; but it works!

  • $\begingroup$ Oh duh that's so obvious when I think about it! $\endgroup$
    – Syncrossus
    Jun 12 '19 at 12:50

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