For a "dice" to be a "dice" it should have the same probability of landing on each of it's faces. Therefore, let us assume that each face has edges of uniform length throughout the dice and all the faces are flat. That is, it is made up of however many regular shapes, and all these shaps must be the same.

for instance, 6 3x3 squares would form a cube. each face has area 9, each edge has length 3, and all the faces are flat.

A sphere of radius Pi would not as it has a curved surface...

Anyway, how many types of dice are we able to make. Obviously, we have a cube, but are there others? are there infinitely many others? or finitely? and please provide a proof ðŸ˜›

Originally posted by genius For a "dice" to be a "dice" it should have the same probability of landing on each of it's faces. Therefore, let us assume that each face has edges of uniform length throughout the dice and all the faces are flat. That is, it is made up of however many regular shapes, and all these shaps must be the same.

for instance, 6 3x3 squares would form a cube. each ...[text shortened]... there others? are there infinitely many others? or finitely? and please provide a proof ðŸ˜›

There are many other kinds of dice, as many of the nerds on this site can tell you. But aside from the 6 types of dice standard to D&D and other role-playing games, yes, there should be infinitely many dice. I'll provide a proof when I think of one, but there is the answer now, in case this problem was important or something.

Originally posted by genius For a "dice" to be a "dice" it should have the same probability of landing on each of it's faces. Therefore, let us assume that each face has edges of uniform length throughout the dice and all the faces are flat. That is, it is made up of however many regular shapes, and all these shaps must be the same.

for instance, 6 3x3 squares would form a cube. each ...[text shortened]... there others? are there infinitely many others? or finitely? and please provide a proof ðŸ˜›

There can be an infinite number of types of die but as you add faces you also get less surface area for the dice to stop on, leading up to a spherical shape. So there is a practical limit in that each die needs to stop in a reasonable length of time and for the digits to be readable since its the opposite side that counts. That last requirement for the top to be readable adds its own restriction too, you have to have an even # of faces, if odd # there will never be top facing dots. I assume you will use dots. But adding dots to represent larger #'s like 10 would make them difficult to easily be read off by the uneducated slobs most likely to use them. So it would seem unpractical to use more than 12 faces, needs to be even #'s so there is only one top face.
I would think 12 faces would be max for practical reasons then.

Originally posted by genius For a "dice" to be a "dice" it should have the same probability of landing on each of it's faces. Therefore, let us assume that each face has edges of uniform length throughout the dice and all the faces are flat. That is, it is made up of however many regular shapes, and all these shaps must be the same.

for instance, 6 3x3 squares would form a cube. each ...[text shortened]... there others? are there infinitely many others? or finitely? and please provide a proof ðŸ˜›

Assuming I have understood your question correctly, you are looking for the Platonic Solids (they are just good friends).

The Platonic Solids have the same regular shape on every face.
To prove how many there are, imagine the net of each one. If you start with equilateral triangles for the faces, you can't make a solid with only two triangles meeting at each vertex. You can with 3 meeting at each vertex - that is a tetrahedron (4 faces). 4 equilateral triangles meeting at a point gives an octahedron (8 faces) and 5 triangles round each point gives an icosahedron (20 faces). 6 equilateral triangles together round a point makes 360 degrees, so it won't fold up to make a solid. Now start again with squares round a point. 3 squares is the only number that works - this gives a cube (6 faces). With pentagons, 3 is the only number that works - this gives a dodecahedron (12 faces). 3 regular hexagons meeting at a point have angles of 360 degrees, so they won't fold up. Any polygon with more than 6 sides won't work because 3 of their angles sum to more than 360 degrees. Thus there are 5 Platonic Solids.

Originally posted by sonhouse There can be an infinite number of types of die but as you add faces you also get less surface area for the dice to stop on, leading up to a spherical shape. So there is a practical limit in that each die needs to stop in a reasonable length of time and for the digits to be readable since its the opposite side that counts. That last requirement for the top ...[text shortened]... o there is only one top face.
I would think 12 faces would be max for practical reasons then.

There is no need to have the requirement of reading the dice from the top! A common 4 sided die has the number read along the bottom edge (a variation has it read on the top verticies).

Additionally, there is no requirement for ALL sides to be equal, just for the possible landing sides to have an equal probability. An easy way to make an 'infinite' die is to make a barrel with smooth spherical ends on which the die could not stop from a roll. The slats of the barrel are the various numbers. To minimize the diameter required for a given face, the slats can be made as triangles (for even numbers). I believe this type of die is available and solves the many faces problem.

A d10 can essentially be divided into 2 halves, but it's faces aren't regular polygons. The design for it can be extended for as many faces on each half as you desire, although it does suffer from being divided into 2 halves, and this would probably be more pronounced the more sides you add.

You also have a d30 which has an identical but odd shape on each side (I forget how the faces look, but I have seen one or two.)

There are also "crystal" versions of dice now that were designed for better bounce. These versions have triangular faces interlaced along the center like teeth, but they do have a cone on either end in addition to this. The dice cannot stand stably on these cone ends though, and as such the cone ends cane be the final stop for the dice.

Originally posted by geepamoogle A d10 can essentially be divided into 2 halves, but it's faces aren't regular polygons. The design for it can be extended for as many faces on each half as you desire, although it does suffer from being divided into 2 halves, and this would probably be more pronounced the more sides you add.

You also have a d30 which has an identical but odd shape on ...[text shortened]... ly on these cone ends though, and as such the cone ends cane be the final stop for the dice.

Originally posted by XanthosNZ Dice is plural, the singular is die.

Thus, i entitled the thread "dice"...

Diapason is quite correct, there are only 5. Quite counterintuitive, i thought ðŸ™‚

I've got another nice counterintuitive maths problem to post (and no, it has nothing to do with goats and 3 doors...). Although i think it may deal with AC, which is another curious concept in maths but one i would rather not get too involved with...

Originally posted by FabianFnas Isn't all dice with six sides? Otherwise it's not a die, it's a polyeder. A die is polyeder with six sides. Also known as a cube.

Sorry I brought it up. Perhaps it has only to do with definitions. Sorry.

You might have a point though.

When one asks another to descibe a die, he/she will most likely describe a cube.

I'm not certain if a die is defined that way though.

In my opinion a 4-sided die is just as much die as a 6-sided die.