Dice of the dimensions

[Polyhedron Dude]

Here we can discuss the various dice shapes that show up in the variant dimensions - I'll define a die (plural dice) to be a figure with identical sides and each side is carved by an isocurve (an isocurve includes points, circles, spheres, duocircles, or any other structure with all points identical).

A die that is carved by points means that it is carved by hyper-planes located at those point (the hyper-plane is tangent to some sphere centered at the center of the dice). This type of dice include various polytopes. A die carved by circles means that it is carved by a hyper-plane that carves in a circle pattern to create cylinder-like sides. The cylinder, by the way, is carved by a circle and two points (I call the areas of carvature - the base) - all points on the base of a die are indistinguishable (or exactly alike) - a cylinder is not a die since the two points (which leads to the circle sides) are distinguishable from the points on the circle (which leads to the tube side). Think of a sphere with the two poles and the equator marked - these markings taken together form the base, imagine that a plane is carving the space away from the sphere - it only carves at the marked locations - the poles and equator - at the equator it would carve out the curved side, at the poles it would cut off flat faces. Since the points of the equator are distinguishable from the two pole points, the shape carved by this base would not be a die. The duocircle is a die however since it's base are the solar and polar equators - every point of this base structure is indistinguishable. The duocircle would make a good die to randomly select between two objects and to randomly select a color from the color wheel for the object. A duosphere would be good to randomly select a point on one of two globes. Regular polytwisters also make good dice. All regular convex polytopes are dice. I would challenge the reader to try to find some of the more stranger looking dice.

Polyhedron Dude

[Keiji]

Whatever. I don't understand any of that, but on the subject of dice...

a bionian dice has 1-4 (a square)
The interesting thing here is that the dice will keep moving in a straight line until either friction or another object stops it. It won't be able to backtrack, as to do so it would have to stop first, and then rotate.

a trionian dice has 1-6 (a cube)
this can backtrack either by turning left or right.

a tetronian dice has 1-8 (a tesseract)
this can, by analogy, backtrack by turning left, right, lambda or rho. (by the way, what happened to lambda and rho in the glossary...?) Since there's so many directions the dice can not only shade out a circle but also any circle on a sphere.

[Aale de Winkel]

In order to be a real die each face must have equal probability of turning up topside, these are quit a limited number of objects of regular polyhedra covered by:
http://mathworld.wolfram.com/RegularPolyhedron.html

However it seems the polygedron dude can have duocircular dice(?)

of course one could have irregular shaped dice where some faces have different possibility of turning topside, but these would surve the same purpose as loaded dice, (not to be used for serious gamblers )

[alkaline]

zant and wint are the same thing as lambda and rho, and i saw no need in keeping lambda in rho.

[Aale de Winkel]

zant and wint are the same thing as lambda and rho, and i saw no need in keeping lambda in rho.

Ain't zant and wint the same thing as delta and upsilon, according to the table with the glossary item "direction" this is the case.

Also a tetra-global substitution of tridth --> trength seems still be needed :twisted:
Whether the duocylinder is still a tetra-object is up to Alkaline and Polyhedron Dude, but for me it is a penta-object.

[alkaline]

well delta and upsilon are directions from realmspace into tetraspace, and zant/wint and directions relative to a tetronian object.

where is there still tridth somewhere? i thought i did a replacement.

[BClaw]

All the regular polytopes would make good dice, so the 4D&D player has a 5-sider, an 8, a 16, a 24, a 120, and the occasional 600-sided die for those really weighty decision-making rolls. :twisted: I didn't understand a lot of the terminology used in the initial post, so this might have been accounted for. For the dice to be fair, as was mentioned, there needs to be an equal chance for any side to come up. So, the sides should all be the same shape and size, do the dihedral angles all need to be the same? What about the lengths of all the edges? Also, I think some of the duals need to be considered. A rhombicosihedron isn't a suitable design for a die, but the 30-sider (rhombic triacontahedron, and one of my favorites) is it's dual. One last thing. The dihedral angles should not be so large that the die rolls forever. In 2D, any die you need is a simple polygon, but a 100-sided die would take forever to stop rolling. I wonder what would be a good rule for the maximum size of the dihedral angle?

[Polyhedron Dude]

All of the duals of the archimedean solids would make fair dice, I like the dual of sirco the best, it is the one with 24 kite shapes for sides. Also the duals of the prisms and antiprisms work. The sphere could also be considered as a dice - it has one spherical side, it can be rolled to fairly choose a location on the Earth's surface.

Polyhedron Dude

[BClaw]

The sphere could also be considered as a dice - it has one spherical side, it can be rolled to fairly choose a location on the Earth's surface.

Polyhedron Dude

And don't forget the disc! A nice 2-sided die, but is there a 3-sided die in any dimension besides the 2nd?

[elpenmaster]

on the three sided die:
in the third dimension, a 3 sided die is possible:
take a regular triangular prism, and warp (elongate) the triangular ends until they are points. then you have 3 sides. they are curved, but are equal. another way to imagine it is to take the earth (sphere) and cut it by longitude every 120 degrees. then angle these peices so that there is 3 ridges on the sphere.
:wink: [/quote]

[pat]

another way to imagine it is to take the earth (sphere) and cut it by longitude every 120 degrees. then angle these peices so that there is 3 ridges on the sphere.
:wink:

Yet another way to word this.... inscribe a regular n-gon in the equator. Then, inscribe a regular n-gon in every plane parallel to the equator such that the corners of the n-gons are all on the same longitudes as the one cut out of the equator.

It isn't technically necessary to keep the n-gons vertexes all on the same longitudes. But, the path a vertex will have to be continuous to keep the die n-sided. If the path of a vertex (as a function of lattitude) is continuous and either symmetric or antisymmetric w.r.t. the equator, then I'm sure it will form a die. I think it will form a die any time the path is continuous. (Here, I'm assuming that each lattitude slice is a regular n-gon.)

Of course, if the path of a vertex is continous but not smooth, there may be some way that there will be places the die could stop that are distinguishable from other places. Of course, this begs the question. Could we not take a 6-sided die and label two faces '1' and two faces '2' and two faces '3' and call it a 3-sided die?

[elpenmaster]

for gambling, that would work, but when disputing the existence of various types of die, it would be cheating!


by the way, you ever heard that joke about the con artistt and the one sided dice? this con artist goes up to somebody, takes out a small sphere, and says, "lets roll this dice, if it lands heads up, you give me a million dollars. if it doesnt, i'll give you a million dollars" the other person agrees and rolls the dice. since a sphere has only one side, the con artist won
i know that joke isnt very funny
:twisted:

[Keiji]

Of course, this begs the question. Could we not take a 6-sided die and label two faces '1' and two faces '2' and two faces '3' and call it a 3-sided die?

It still has 6 sides, but they are in 3 pairs of sides with the same value.

Therefore, 6 sides, but 3 values.

[wendy]

In four dimensions, one can have a dice with as many sides as one chooses. The appropriate shape for this is a {p}{p/d} mod-tegum. You take a p-p duotegum, and you select d such that hcf(p,d) = 1.

You then tick off the faces of the duotegum, so that one goes one step one way, and d steps the other way. The marked faces are then stellated, and you get a iso-facial polytope that is ready to rumble as a p-sided dice.

W

[Paul]

Hello Wendy,

<cough><cough>... clearing my throat...

That was quite a voracious posting fury...

Are you sure we're all talking about the same thing here?

You seem most... creative... though, Wendy.

Take care,

[wendy]

A tegum is the dual of a prism, so if you extend the face-planes in the perpendicular to the centre-vertex-direction, you get the same thing.

Jonathan Bowers showed me the seven-sided and thirteen0sided ones.

W

P.S. It's pretty impressive to fill out the whole page on the first day.

[Rkyeun]

And if you want to play DnD in 4 dimensions with your old D20...
Well, you can. Our 3D D20 works like a coin flip if you only give it the tiniest bit of 4D extrusion.

[wendy]

In four dimensions, you can have dice of 10 and 48 sides as well, eg the bitruncated bt{3,3,3} and bt{3,4,3}.

In practice, you can make a dice from a modtegum, with any number of sides, eg 7. To do this, start with a {kp}{kp} prism, and number the vertices in k cycles (0...p-1). Select then a pair of coordinates of the form 0,0 1,n 2,2n, 3,3n modulo p, as long as n != +/- 1. This will then give you k^2 p points. One then extends the polar planes, and you get a dice with k^2 p sides.

When p=5, k=1, n=2 you get the pentachoron. (5 sides)

when p=10, k=1, n=3 you get the bitruncated pentahedron (10 sides),

The sides are remarkably uniform when one can get a sum of squares going. This is because we can have two different dice merged into one. So p=26, k=1, n=5 gives a most remarkable polytope with 26 sides. A dice with p=13, k=1, n=5 also works, since 13 divides 1^2+n^2.