4D crap game

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4D crap game

Postby PatrickPowers » Thu Dec 31, 2015 4:48 am

Construction workers working as guest laborers in the fourth dimension introduced the game of craps. They found to their amazement and consternation that 4D dice had eight faces. There was no way around it: the traditional rules had to be changed. They cleverly found a way to keep the chance of winning very similar while changing as few of the rules as possible.

The rules of craps are like this. Roll two dice. If the dots total seven or eleven, you win. If the dots total 2, 3, or 12, you lose. Otherwise, the number you rolled is your "point." Continue to roll. Should you roll your point, you win. Should you roll a seven, lose. If neither of these occur, roll again. The sequence of rolls may theoretically be infinite, but terminates with probability one.

The first design goal is that the overall chance for the shooter to win be as close as possible to 50%.
The second design goal is to minimize changes to the rules and introduce minimal complexity.
The third goal is for the number of throws until one wins or loses to average about the same as in 3D.

Here's a proposed solution:

Number the faces of one die from 0 to 7 and the other from 1 to 8.
A total of 7, 11, or 14 on the first roll wins.
A total of 1,2,3,12,13, or 15 on the first roll loses.


This is a superset of the old rules and it gets rid of the long-odds points like 1 and 15 that would tend to drag on for too many rolls. It is also in conformance with superstition for 13 to lose, and for a lucky double seven totaling 14 to win.

Odds of total of 7, 11,14 on first roll: 14/64 = .2344
Odds of total of 1, 2, 3, or 12,13,15 on first roll: 14/64 = .2344

The chance of winning vs. losing on the opening roll are 0.50. The chance of winning the point must then also be very close to 50%. Let's follow the traditional rules where one fails with 7 only.

Set of possible points: 4,5,6,8,9,10
Number of winning combinations per point: 4,5,6,8,7,6

Sum the chance of winning for each point.
4/(4+7)*4/36 + 5/(5+7)*5/36 + 6/(6+7)*6/36 + ....

Each term in this series is n^2/((n+7)36)

(42/11 + 52/12 + 62/13 + 82/15 + 72/14 + 62/13)/36 = 0.468

The overall chance of winning overall is then .2344 + 0.468*(1-2(.2344)) = 0.483. For traditional craps it is 0.493. Quite close.

For 3D craps the chance of winning or losing on any one toss for point is about 30%. In 4D the chance is about 13/64=20%. One would expect that on average 50% more tosses would be needed to win or lose the point. This can be ameliorated by adding the rule that one wins with a roll of 1 or 15 and loses with a 14. Then the point is won or lost in roughly 28% of tosses, similar to the situation in the 3D game. With this addition to the rules the chance of the shooter winning increases slightly to .2344 + 0.476*(1-2(.2344)) = .487, which is closer to the 3D odds of .493. Close enough, I say.

Rules of craps with 4 dimensional dice.

The faces of one die are numbered from 0 to 7 and the other from 1 to 8.
A total of 7, 11, or 14 on the first roll wins.
A total of 1,2,3,12,13, or 15 on the first roll loses.
Otherwise the shooter must try to equal the "point," the total just rolled, by rolling the dice again.
Win if a total of 1, the point, or 15 is rolled.
Lose if a total of 7 or 14 is rolled.
If neither occurs, roll for point again.


-----

Craps comes from the French crapaud[frog], since those playing craps on the ground tend to squat in the pose of a frog. This enables them to retrieve the dice readily. Casino craps shooters avoid this inelegant pose via use of a purpose-built craps table.
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Re: 4D crap game

Postby Klitzing » Thu Dec 31, 2015 11:03 am

You even could play that "4D games of crap" within 3D already:
just apply those numbers onto 2 octahedra. :nod:
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Re: 4D crap game

Postby PatrickPowers » Thu Dec 31, 2015 1:23 pm

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Re: 4D crap game

Postby PatrickPowers » Thu Dec 31, 2015 1:24 pm

Klitzing wrote:You even could play that "4D games of crap" within 3D already:
just apply those numbers onto 2 octahedra. :nod:
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IMO t would be more fun to watch tumbling hypercubes on an app. Though you'd have to trust the owner that the dice were honest. ;-)
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Re: 4D crap game

Postby wendy » Thu Dec 31, 2015 3:00 pm

Were it a complex game, you could have a twelve sided tesseractic dice in 4D, or something bigger,

But dare i say it, there is a 240-sided (decimal: twe = 200), that is about the same size, if not smaller.
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Re: 4D crap game

Postby PatrickPowers » Sat Jan 02, 2016 5:07 pm

The faces of 4D dice are cubes, so how would the spots be arranged?

4 vertices of tetrahedron
5 vertices of tetrahedron + center
6 vertices of octahedron
7 vertices of octahedron + center
8 vertices of cube
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Re: 4D crap game

Postby wendy » Mon Jan 04, 2016 11:46 am

The opposites amount to nine on each case, and this set can be made from a cube + centre.

1, 8 Centre vs cube
2,7 Opposite points vs petrie polygon + centre.
3,6 Make a diagonal vs petrie polygon
4, 5 tetrahedron vs centred tetrahedron
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Re: 4D crap game

Postby Klitzing » Mon Jan 04, 2016 12:13 pm

... and Wendy here intends that those tetrahedra will be considered as alternated cubes, for sure - rather than something completely different (by symmetry). :nod:
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Re: 4D crap game

Postby foursquare » Wed Jan 13, 2016 8:16 pm

There are 8 sided octahedral dice, they are used in dungeons and dragons. All that needs doing for the craps game suggested is to alter the numbers on one of them. All the platonic solids are used as fair dice, and all the Catalan or Archimedian duals could be used in 3 dimensions, not to mention the infinite series of dipyramids and trapezohedra (also called antidipyramids)
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Re: 4D crap game

Postby Klitzing » Thu Jan 14, 2016 10:35 am

Well, all you need, to have a fair dice, would be that it is convex, that all the facets (i.e. (d-1)-dim. boundaries) ought have the same (d-1)-dim. volume (q.e. area for d=3), and that the bodycenter of mass has the same height above all of them.

The same volume requirement surely is achieved whenever the facets all are congruent or even belong to a single orbit of overall symmetry.
On the other hand, when the vertices would belong to a single orbit of overall symmetry, then you'll have a circum(hyper)sphere and its center would coincide with that bodycenter of mass.

Therefore esp. all convex "noble polytopes" are of special interest here. As the set of noble polytopes just requires these 2 properties, having a single type of facets and a single type of vertices wrt. the action of the overall symmetry of that polytope.

The convex regular polytopes clearly are the most trivial subset of the noble ones. In fact, those just require additionally that for any other dimension inbetween there ought be a single orbit of according elements under that very symmetry as well. - Thus regular and noble polyhedra would just differ in the count of allowed edge classes. In 2D, i.e. for polygons both notions would coincide. But the higher the dimensional number d, the more freedom comes in...

Thus e.g. any N-gonal duoprism (i.e. xNo xNo) would belong here. In fact, it is convex, uniform, has a single class of facets (N-gonal prisms: x xNo), but for any N>2 and not equal to 4 you'll have at least 2 different involved 2-dimensional objects... Similarily any multiprism of equal regular components would belong here, kind like a doedecahedral quadprism x5o3o x5o3o x5o3o x5o3o - which lives then in 12D. - Could you guess the shape of their (equivalent) facets?

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Re: 4D crap game

Postby quickfur » Thu Jan 14, 2016 6:22 pm

Among the "dice" polytopes are those that can be produced from self-dual regular polytopes. Given a (regular) polytope that's self-dual, the polytope obtained midway between the starting polytope and its dual, produced by truncation, must be facet-transitive. So in 4D you have the bitruncated 5-cell, with 10 equivalent truncated tetrahedral cells, serving as a 10-sided fair die, and the bitruncated 24-cell, with 48 equivalent truncated cube cells, that serves as a 48-sided fair die.

The 24-cell family only exists in 4D, but the simplex family extends to n dimensions for all n. The meso-truncate of the simplex family would therefore qualify as fair dice across the dimensions. In Coxeter-Dynkin symbols, these would be, in order of increasing dimensions:

xx (hexagon, as truncated triangle) - 6 equivalent edges
oxo (octahedron, as rectified tetrahedron) - 8 equivalent triangles
oxxo (bitruncated 5-cell) - 10 equivalent truncated tetrahedra
ooxoo (mesotruncated 5-simplex) - 12 equivalent rectified 5-cells
ooxxoo (mesotruncated 6-simplex) - 14 equivalent facets
oooxooo (mesotruncated 7-simplex) - 16 equivalent facets
... etc.

In even dimensions, the CD diagram has two ringed nodes in the middle; in odd dimensions, a single ringed node in the center.

In general, the simplex family produces dice of 2(n+1) facets in n dimensions.

Of course, there are many other fair dice. In 3D, for example, the family of n-sided bipyramids also serve as fair dice, even though they have two different classes of vertices; this is because the faces are transitive under the symmetry group, thus ensuring equal probability of landing on any one face. (They are the duals of the uniform n-gonal prisms -- the vertex transitivity of the uniform prisms ensure the equivalence of the faces of the duals.) In 4D, besides the mesotruncates mentioned above, there are also a host of unusual shapes that nevertheless constitute fair dice, such as bidex (bi-24-diminished 600-cell, consisting of 48 equivalent tridiminished icosahedra), which is chiral, and the various non-uniform (but still cell-transitive) polychora with swirlprism symmetry, the duoprisms as Klitzing mentioned, etc.. Bidex is quite unusual, in that it's both cell-transitive and vertex-transitive, so both it and its dual are fair dice (and noble polychora), but neither are among the uniform or regular polychora!

Also included among the "dice" polytopes are the n-dimensional Catalans (duals of the uniform polytopes). In 5D, one of the interesting Catalans is the dual of the 120-cell prism, which has 1200 equivalent facets(!), each of which is in the shape of an elongated (non-regular) 5-cell (basically a 600-cell bipyramid).

So as you see, as you go up the dimensions, the number of possible fair dice shapes increases a lot.
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Re: 4D crap game

Postby Polyhedron Dude » Tue Feb 02, 2016 11:03 am

In 4-D there's quite a smorgasbord of fair dice, many have a prime number of sides - one of my favorites is the tridecachoron where its 13 congruent facets is an 8-faced polyhedron with 4 isosceles pentagons and 4 kites as faces. When rolled on a 4-D table a corner will point upwards. Another cool dice has 250 facets that form two orthogonal spiral configurations where the facets resemble the tablets of Moses. I also like to include some curved objects as dice as long as the contact regions (part in contact with surface) are congruent such as the regular convex polytwisters.
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Re: 4D crap game

Postby quickfur » Tue Feb 02, 2016 6:28 pm

Polyhedron Dude wrote:In 4-D there's quite a smorgasbord of fair dice, many have a prime number of sides - one of my favorites is the tridecachoron where its 13 congruent facets is an 8-faced polyhedron with 4 isosceles pentagons and 4 kites as faces. When rolled on a 4-D table a corner will point upwards. Another cool dice has 250 facets that form two orthogonal spiral configurations where the facets resemble the tablets of Moses. I also like to include some curved objects as dice as long as the contact regions (part in contact with surface) are congruent such as the regular convex polytwisters.

The tridecachoron interests me. How do you construct it?
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Re: 4D crap game

Postby Polyhedron Dude » Wed Feb 03, 2016 2:40 am

quickfur wrote:The tridecachoron interests me. How do you construct it?

Take a 13 by 13 grid and put a point in the first square on the first row. Then go down one and over 5 - like an extended knights move in chess. Keep doing this till you reach the bottom row allowing horizontal wrapping (like in the game Asteroids). Take the square grid and curve it horizontally into a cylinder and then the vertical into a duoring (hedrix of the tiger). The 13 points are the vertices of the dual of the tridecachoron. This shape can easily be made using Stella4D - there is now a feature on Stella4D to make these "gyrochora", which was a suggestion I made to Rob Webb. The tridecachoron is gyrochoron 13,5.
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Re: 4D crap game

Postby quickfur » Wed Feb 03, 2016 6:46 am

Polyhedron Dude wrote:
quickfur wrote:The tridecachoron interests me. How do you construct it?

Take a 13 by 13 grid and put a point in the first square on the first row. Then go down one and over 5 - like an extended knights move in chess. Keep doing this till you reach the bottom row allowing horizontal wrapping (like in the game Asteroids). Take the square grid and curve it horizontally into a cylinder and then the vertical into a duoring (hedrix of the tiger). The 13 points are the vertices of the dual of the tridecachoron. This shape can easily be made using Stella4D - there is now a feature on Stella4D to make these "gyrochora", which was a suggestion I made to Rob Webb. The tridecachoron is gyrochoron 13,5.

Cool! This construction is straightforward enough that I can probably easily construct a model of it in my polytope viewer. Should prove interesting :D
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Re: 4D crap game

Postby wendy » Wed Feb 03, 2016 7:51 am

Going at steps of 1,2 on a 13*13 grid won't do. You have to use 2,3 to get just 13 points. 2²+3² = 13.

Doing 1,2 on a 5*5 grid gives a pentachoron.
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Re: 4D crap game

Postby quickfur » Wed Feb 03, 2016 5:24 pm

I used the following grid:
Code: Select all
X............
..X..........
....X........
......X......
........X....
..........X..
............X
.X...........
...X.........
.....X.......
.......X.....
.........X...
...........X.

and generated a 13-choron using those points as hyperplane normals. I got an interesting little polychoron with a most interesting spiralling structure, almost like a conch shell. It's rather hard to get a good view of it, but here are some preliminary projections:
Image
This projection shows a single cell. It has a rather sharp, wedge-like shape. The green strip is another cell that lies on the far side, but touches cells on the near side. The next image shows the projection on the far side of this yellow cell:
Image
The red dot in the middle is the vertex that lies antipodal to the yellow cell. The next projection shows the face between the yellow and green cells:
Image
Here you can see the kite-shaped face shared between the two cells.

Very interesting little polychoron indeed!
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Re: 4D crap game

Postby Polyhedron Dude » Thu Feb 04, 2016 9:01 am

I call that one Mobius 13. Rocky 13 does a 1,3 step while the Tridecachoron (most symmetric of the three) has a 1,5 step.
Great pics by the way. Another interesting feature about gyrochora is that all of their 2-faces are vertex isosceles.
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