# Dice of the Dimensions

This page will deal with shapes that would act as fair dice in their dimension. A die (plural - dice) is a convex solid with congruent "contact zones". A contact zone is the part of the solid's surface that is flush against a surface (like a table top) under a particular stable orientation - example: the squares of a cube are it's 6 contact zones. Another example: the cylinder has two types of contact zones, a circular type (when sitting up) and a line segment (when sitting in a rollable position). Yet another example is a sphere - it's contact zones are points. Consider the previous three examples, the cube has 6 identical square contact zones - it is convex and it is solid in 3 dimensions - so it is a 3-D die, the cylinder has two types of contact zones, so it is not a die, the sphere is convex and solid in 3-space, while all of its contact zones (points) are congruent, so it is a die. In 4-space, the cylinder would appear flat, like a polygon does in 3-space, it would only have two contact zones in 4-space (both shaped like the entire cylinder itself) - sort of like a coin, but since it is not solid in 4-space, it can't be considered to be a die there either.

## Dice of 0, 1, and 2 Dimensions

The only dice in 0 and 1 dimensions are the monad (single point) and dyad (line segment) respectively. The monad has no contact zones, so it is a die by default. The dyad has two endpoints as it's contact zones, however it can't be rolled in 1-space.

In 2-space, all of the regular polygons will work as dice, as well as variations of the even numbered polygons - for instance a variation of the square die is a rhombus die. Their contact zones are edges.

We can't forget the circle (disk), it acts as a one sided roley die - it's contact zones are points.

## Dice of 3 Dimensions

### Platonics

The 5 Platonic solids: tet, cube, oct, doe, and ike are all dice, as well as their variations. These five are normally encountered as dice.

Tet can be stretched or squashed into a disphenoid and then twisted into what I like to call a "scalenoid" - these all have 4 triangular sides (equalateral for tet, isosceles for disphenoids, and scalene for scalenoids).

The cube can be stretched or squashed corner to corner to get crystal shaped dice and then it can also be twisted - these all have 6 sides (square shaped for cube, rhombus shaped for crystals, and tetragons for twisted crystals).

Oct can be stretched or squashed on all three axises to get square tegums (fusils, or dipyramids) and dyad tritegums - these all have 8 triangular sides (equalateral for oct, isosceles for square tegums, and scalene for dyad tritegums).

Doe can be altered to take on pyrital symmetry or chiral tet symmetry - these have 12 pentagonal sides (regular for doe, isosceles pentagon for pyrital, and "scalene pentagon" for chiral).

Ike on the otherhand can't be altered without taking away it's die status, it has 20 equalateral triagles for it's sides.

### Tegums

Tegums are the duals of prisms, they are also called dipyramids or fusils. The term tegum was coined by Wendy Krieger, fusil was coined by Norman Johnson. Tegums can be constructed by taking a polygon and putting a dyad through it's center, then cover it with triangles (edge of polygon to an endpoint of the dyad). Tegums can be stretched or squashed, tegums of 2-D dice are also dice. Oct and its variations take care of the square and rhombus tegums. Other tegum examples are below:

Trit - the trigonal tegum has 6 isosceles triangles ("isots" for short) for sides. The version with equalateral triangles ("equits" for short) is the Johnson solid "trigonal dipyrmid" (short name = tridpy).

Pit - the pentagonal tegum has 10 isots for sides. The version with equits is the Johnson solid "pentagonal dipyramid (or pedpy).

Hat - the hexagonal tegum has 12 isots for sides. Not only can it be stretched or squashed, it can also be altered so that it is a triambus tegum (a triambus is a hexagon with trigonal symmetry and congruent edges).

n-gonal tegums has 2n isots for sides. "Evengonal" tegums can be altered as well as stretched, while "oddgonal" tegums can only be stretched (or squashed) to get dice variations.

### Antitegums

Also called antifusils - are the duals of the antiprisms, they can be stretched or squashed as well as twisted to get variations. There is one for every polygon, where the trigonal case has been dealt with under the cube and it's variations. They have 2n sides which have a kite shape (or a tetragon for the twisted ones). The pentagonal antitegum is used to make the 10 sided dice found in game stores.

### Catalans

Also called duals, are the duals of the Archimedian solids. For now I'll give them short names as follows: dual of sirco = du sirco. The Catalans are as follows:

Du Co is the rhombic dodecahedron, it has 12 rhombuses and its tetrahedral symmetric variations have kite shaped faces. Makes an attractive die.

Du Id is the rhombic triacontahedron, it has 30 rhombuses and no variations. It is used as the 30 sided dice found in game stores.

Du Tut is the triakis tetrahedron, it has 12 isots. It has one degree of variations.

Du Tic is the triakis octahedron, it has 24 isots and one degree of variations.

Du Toe is the tetrakis hexahedron, it has 24 isots and two degrees of variations, some variations have scalenes instead of isots.

Du Tid is the triakis icosahedron, it has 60 isots and one degree of variations.

Du Ti is the pentakis dodecahedron, it has 60 isots and one degree of variations.

Du Sirco is also called the kited-24, it has 24 kites and two degrees of variations, some variations have pyrital symmetry with tetragon faces. Du sirco makes a very attractive 24-sided die.

Du Srid is also called the kited-60, it has 60 kites and one degree of variations, it would make a very attractive 60-sided die.

Du Girco has 48 scalenes and two degrees of variations.

Du Grid has 120 scalenes and two degrees of variations.

Du Snic has 24 isosceles pentagons and two degrees of variations which will turn the pentagons scalene or irregular, this would make a strange looking die, for it looks a bit cock-eyed when rolled, no face is pointing up, instead it is pointing up and a bit to the side.

Du Snid has 60 isosceles pentagons and two degrees of variations which will turn the pentagons scalene or irregular, it is cock-eyed also.

### Roley Dice

There are only two types of roley dice in 3-space:

The spindle is the dual of the cylinder, it looks like two cones fused together into a roller shape, it has one degree of variation (can be stretched or squashed). The contact regions are dyads, it has two sides which look like the curved side of a cone. When rolling it will trace out a circle. A good use for this die would be to randomly choose one of two objects and what color it takes on (using the color wheel). It can also be considered to be the tegum of a circle.

The sphere (or ball) has one side, its entire surface. It's contact regions are points. It can roll all over the surface of a table. A good use for a sphere die would be to randomly choose a location on the surface of the Earth.

## Dice of 4 Dimensions

### Regular Polychora

There are 6 convex regular polychora: They are pen, tes, hex, ico, hi, and ex.

Pen (or pentachoron) has 5 tetrahedral sides. It can also be squashed or stretched planarly to get a chiral figure with a subsymmetry of the pentagon duoprism - to picture this, imagine it being squashed towards it's 2-D projection which looks like a complete pentagon - the stages along the way are the variations.

Tes (or tesseract) has 8 cubes for its sides. It can be stretched or squashed from corner to corner to get polychora with stretched cubes (crystals) for sides, you could also then squash or stretch it sideways in such a way as to generate disphenoid verfs at the top and bottom vertices. There are also rhombus duoprism variations. It can also be distorted like the duoprisms, see triddip below.

Hex (or 16-cell) has 16 tets for its sides. It can be stretched or squashed via any of it's four axises, resulting in oct tegums, square duotegums, square tegum tegums, and dyad tetrategums - where their sides are trigon pyramids, disphenoids, scalenoids, and irregular tets respectively.

Ico (or 24-cell) has 24 octs for its sides. I suspect there are several ways to variate this in a swirlprismattic sort of way.

Hi (or 120-cell) has 120 does for its sides, this would make an awesome dice. There are also swirlprismattic variations which have a pyritoswirlprism symmetry, as well as chirotetra-swirlprism symmetry.

Ex (or 600-cell) has 600 tets for its sides. I'm not aware of any variations.

### Uniform Polychora

Two more uniform polychora will be dice:

Deca (or decachoron) has 10 tuts and no apparent dice variations (but I could be wrong).

Cont (also called octagonny or 48-cell) has 48 tics. This one looks like it can be deformed swirlprismattically in two different ways, both with two degrees of variation - but I'll need to check it closer to make sure. If it works, the sides will be variant distorted tics.

### Duoprisms

Any n-gonal duoprism will work and have 2n sides shaped like n-gonal prisms. (note that n,m-duoprisms don't work as dice).

Triddip is the trigonal duoprism with 6 trips for its cells, it also has two degrees of variation - To picture the variations, mark the centers of each trip of triddip, the result would be the vertices of two orthogonal triangles, now take these two triangles and twirl them closer to each other (one of the degrees), then you can rotate one of the triangles within it's plane (second degree). The actual facets probably look like angle-skewed trips.

Pedip is the pentagonal duoprism with 10 pips (pentagon prism) for its sides. It can be deformed like triddip.

Hiddip is the hexagon duoprism with 12 hips. It can be deformed like triddip, there's also triambus duoprism dice.

Even duoprisms act like hiddip, while odd ones act like pedip. These duoprisms are actually the square of their base polygon and are therefore powertopes.

### Tegums

Let B be a die of dimension 3, then the B tegum is also a die, so there are dice shaped like ike tegums, du sirco tegums, du co tegums, and antitegum tegums - they have twice the number of sides as the original polyhedron did. Tegums of tegums will be discussed more in the duotegum section, while the tegum of roley dice will be mentioned in the roley dice section below.

### Duotegums

These are the duals of the duoprisms. Let A and B be 2-D dice, then the A,B duotegum is a die also. The sides are disphenoids and the number of sides is the product of the number of sides of A and B - you can also change the sizes of the two base polygons to get more variations. If A and B are the same shape as well as the same size, then the A duotegum is the "diamond" of A and it is a powertope. The vertices of the duotegums are on two orthogonal planes - with the vertices of A on one and B on the other. Examples are the 25 sided pentagon duotegum, the 12 sided triangle-square duotegum, and the 42 sided hexagon-heptagon duotegum. The A tegum tegums are none other than A-rhombus tegums.

### Du Oc Power

The du oc is the dual of the octagon (or in other words an octagon standing on its corner). This section will deal with powertopes that take a base polygon to the du oc power. The only 4-D powertopes that will create dice will either have a square, diamond, or du oc as the power - squares and diamonds have been mentioned in the duoprism and duotegum sections respectively, so the du ocs are the only powertopes left for 4-D dice. If A is a 2-D die, then the du oc of A which is also called A^du oc is also a die. The number of sides are twice the square of A's sides. The sides are wedge shaped. Examples are the du oc of trigon (sounds like a role playing game), the du oc of square (has rhombus variations), the du oc of heptagon (has 98 sides), and the du oc of octagon (which has tetrambus variations, du oc itself also has tetrambus variations!). To picture what one of these looks like, we'll use the du oc of hexagon - start with a compound of a hexagon duoprism and a hexagon duotegum, where they have the same circumradii and where the hexagons of the first are in the same orientation of the hexagon equators of the second - then - the du oc of hexagon is the convex hull of this compound.

### 4-D Catalans

These are the duals of the "Archimedean" polychora, many of them are morphable:

Du Rap has 10 trigon tegums for sides.

Du Rit has 32 somewhat flattened trigon tegum sides.

Du Rico has 96 somewhat stretched trigon tegum sides.

Du Rahi has 1200 flattened trigon tegum sides.

Du Rox has 720 pentagon tegum sides.

Du Tip has 20 trigon pyramid sides.

Du Tat has 64 trigon pyramid sides.

Du Thex has 48 square pyramid sides.

Du Tico has 192 trigon pyramid sides.

Du Thi has 2400 flat trigon sides.

Du Tex has 1440 pentagon pyramid sides.

Du Srip has 30 isot tegum sides (the isot tegum is the dual of a wedge).

Du Srit has 96 isot tegum sides.

Du Srico has 288 isot tegum sides.

Du Srahi has 3600 isot tegum sides.

Du Srix has 3600 isot tegum sides.

Du Deca has 30 disphenoid sides.

Du Tah has 96 disphenoid sides.

Du Cont has 288 disphenoid sides.

Du Xhi has 3600 disphenoid sides.

Du Grip has 60 scalenoid sides.

Du Grit has 192 scalenoid sides.

Du Grico has 576 scalenoid sides.

Du Grahi has 7200 scalenoid sides.

Du Grix has 7200 scalenoid sides.

Du Gippid has 120 irregular tet sides.

Du Gidpith has 384 irregular tet sides.

Du Gippic has 1152 irregular tet sides.

Du Gidpixhi has 14400 irregular tet sides.

Du Prip has 60 kite pyramid sides.

Du Prit has 192 kite pyramid sides.

Du Proh has 192 kite pyramid sides.

Du Prico has 576 kite pyramid sides.

Du Prahi has 7200 kite pyramid sides.

Du Prix has 7200 kite pyramid sides.

Du Spid has 30 flat trigon antitegum (flat crystal shaped) sides.

Du Sidpith has 64 trigon gem shaped sides (looks like trigon antitegum, but with one half stetched, it is the dual of the trigon antipodium).

Du Spic has 144 square antitegum sides. This one is sure to be a fascinating die.

Du Sidpixhi has 2400 trigon gem shaped sides.

Du Sadi has 96 sides. The shape of the side has trigon symmetry - start with a trigon antipodium, blend a trigon pyramid onto its base, then blend a small trigon pyramid onto its top face - this causing the adjacent isots to turn into kites. Du sadi has pyrital ico symmetry.

Du Gap has 100 sides. It's sides look like a sort of dodecahedron-wedge fusion, it has 4 pentagons on the back end, 4 kites (2 on the top, 2 on bottom), and 2 trapezoids which form the wedge part.

### Catalan Wannabes

There are some polychora that "try" to be uniform, but they have too many parameters to satisfy to make uniformity (these are mainly things like the snub tesseract), their duals however do qualify as dice:

Du Snip (dual of snub pentachoron) has 60 sides. The sides have chiral symmetry, with 2 scalene pentagons, 2 kites, and 4 triangles for it's sides.

Du Snit (dual of snub tesseract) has 192 sides. Sides similar to du snip's.

Du Sni (dual of snub ico) has 576 sides. Sides similar to du snip's.

Du Snahi (dual of snub 120-cell) has 7200 sides. Sides similar to du snip's.

### Scaliform and Duals

One scaliform is a die, as well as the duals of four.

Bidex has 48 tridiminished icosahedra for its sides, it has a directed tetraswirlprism symmetry.

Du Bidex has 72 chiral sides which have 2 trapezoids and 4 isots.

Du Tutcup has 24 rhombus skew pyramids. Tutcup is the result of taking two parallel tuts in opposite orientations and filling in the space between them with trigon cupolas, du tutcup is it's dual.

Du Spidrox has 600 sides, it is the dual of spidrox which is resulted by cutting off 120 vertices off of rox in a swirlprismattic way. It's sides look like a twisted kite antitegum which has 6 tetragons and 2 isots.

Du Prissi has 288 sides, it is the dual of prissi where prissi has 24 ikes, 96 trips, 24 tuts, and 96 tricus (trigon cupolas) for cells. Du prissi's sides look like a pentagon pyramid with the pentagon sliced off in a folded sort of way, resulting in a kite and four triangles that meet at the apex and two tetragons on the bottom.

### Duoantitegums

In 4-D there are duoantiprisms, but nearly all of them are not uniform, but they are isogonal - so therefore, their duals - the duoantitegums - are dice if they are convex: The m,n-duoantitegum has 2*m*n sides which looks like an expanded disphenoid which resembles a cube but with opposite squares looking like half disphenoids, m and n can be 2 or greater. If m or n equals 2, then it's "disphenoid part" turns back into a square (more like a rhombus), the 2,2-duoantitegum is none other than the tesseract.

### Du Gap's Relatives

Du gap which is the dual of the grand antiprism, has relatives - gap can be thought of as the pentagon super duoantiprism, so du gap would be the pentagon super duoantitegum - so why not change "pentagon" to some other n-gon, even though the gap relatives are no longer uniform, they are still isogonal - so the duals will be dice - n can be 2 or greater. They will have 4*n^2 sides which are shaped like du gap's side, unless n=2, then the trapezoids of the du gap side will transform into trigons.

### Du Sidpith's Relatives

Sidpith can be thought of as the square duoexpandoprism, which has two orthogonal girdles of 8 cubes (square prisms) and square prisms and wedges (the trips) connecting the two and disphenoids, so its dual can be considered as the square duoexpandotegum. The other duoexpandotegums are dice, they have 4*n^2 sides which resemble isot shifted tegums - n can be 3 or larger, 4 has been counted as du sidpith.

### Phased Duoexpandotegums

These are like the last batch but with a phase change. Consider their duals, which are a phase change of the duoexpandoprisms - instead of having two girdles of 2n n-prisms, they have two girdles of n stretched out 2n-prisms.

### Anti-Du Ocs

Consider the duals of the du oc powertopes, they will have two girdles of n stretched n-prisms connected by rectangle trapezoprisms, now do a phase change on the girdles such that they are anti-aligned to each other and now with rectangle pyramids and two types of disphenoids to connect them - then dual back - these "anti-du-ocs" will also be dice.

### Gyrotegums

Also called step tegums by Wendy Krieger. To generate one, start with an m by n array of squares (or rectangles) but make sure m and n are not relatively prime, they can also be equal. Select a vertex on the first row, then move over a number of spaces to select one on the next row - continue till you reach the last row (be aware that the rectangle array wraps like in a 2-D videogame screen) - continue to the first row to see if it closes (reaches the first vertex), if not - keep going until it is closed - now fold up the array on both axises into a duoprism - remove all vertices except for the selected ones, all vertices should be congruent if done correctly - now take the dual of the convex hull of those vertices - it is a gyrotegum die. The number of sides will be the number of vertices selected, there are 7 and 13 sided die amongst the gyrotegums. Note that some of the results will be polygons, these are not 4-D dice. As a matter of fact there are many other ways to generate congruent vertices on the array, including compounding several copies of a working vertex set (also including compounding those that only generate "slanted" polygons).

### Double Symmetry Dice

Since pentachoric and icoic symmetries can be doubled to get decachoric and contic symmetry, more dice can be found based on this doubling. The intersection of 2 polychora mentioned below are always oriented so that the symmetry doubles:

The decastellated spid or Tudu Rap - take spid and put pyramids on all ten of its tets, so that it's trips (triangle prisms) expand to have pyramid caps on both triangles - the result is a 20 sided die. This is the intersection of 2 du raps.

The 48 stellated spic or Tudu Rico - take spic (which has 48 octs and 192 trips) and put pyramids on all of its octs, so that its trips have pyramids on both ends - this leads to a 192 sided die. This is the intersection of 2 du ricoes.

Tudu Tip - the intersection of 2 du tips has 60 sides.

Tudu Tico - the intersection of 2 du ticoes has 576 sides.

Tudu Srip - the intersection of 2 du srips has 60 sides.

Altudu Srip - the intersection of 2 altered du srips has 60 sides - altered so that the srip's coes are further out than the octs.

Tudu Srico - the intersection of 2 du sricoes has 576 sides.

Altudu Srico - altered so that the srico's sircoes are further out than the coes.

Tudu Deca - intersection of two stretched decas, has 60 sides.

Tudu Cont - intersection of two stretched conts, has 576 sides.

Tudu Grip - intersection of two grips, has 120 sides.

Tudu Grico - intersection of two gricoes, has 1152 sides.

Altudu Grip - altered so that the grip's toes are further out than the tuts.

Altudu Grico - altered so that the grico's gircoes are further out than the tics.

Tudu Gippid - intersection of two stretched gippids, has 240 sides which are irregular tets.

Tudu Gippic - intersection of two stretched gippics, has 2304 sides which are irregular tets.

Tudu Prip - intersection of two prips, has 120 sides.

Tudu Prico - intersection of two pricoes, has 1152 sides.

Altudu Prip Version 1 - altered so that prip's coes are further out than the tuts.

Altudu Prip Version 2 - altered so that prip's hips are further out than the trips.

Altudu Prip Version 3 - altered so that prip's coes and hips are further out than the tuts and trips respectively.

Altudu Prico Version 1 - altered so that prico's toes are further out than the sircoes.

Altudu Prico Version 2 - altered so that prico's hips are further out than the trips.

Altudu Prico Version 3 - altered so that prico's toes and hips are further out than the sircoes and trips respectively.

Tudu Spid intersection of two stretched spids, has 60 sides.

Tudu Spic intersection of two stretched spics, has 288 sides.

### Roley Dice

These dice can roll, and they're not the only ones!

Glome is the 4-D sphere, it has one side - its surface. Its contact regions are single points, it can roll all over a 3-D surface.

Spherindle is the sphere tegum, it has two sides which look like the curved side of a sphone (sphere pyramid), it can roll like a ball, but will only trace out small spheres instead of planes. It is the dual of the spherinder. Contact regions are dyads.

Duospindle is the circle-circle tegum (can be a powertope if both circles are the same) - it has one side, its surface - which curves like circles on two axises. Contact regions are dyads, it can roll in two orthogonal directions, but would try to trace out circles (unless it changed it's rolling direction).

Duocylinder is the square of a circle (or a circle duoprism) - it also has skewed variations. It has two sides which curve like a cylinder, contact regions are circles. It can roll like a cylinder. It and the duospindle are duals.

Polygon Spindles are the polygon-circle tegums, where "polygon" can be any polygon that qualifies for a die. These have the same number of sides as the polygon itself, the sides are shaped like the curved surface of a dyadic cone (cone with a dyad as its apex). Contact regions are triangles. They roll like cones and therefore trace out circles (the triangle contact region will trace out a spindle).

Roley Du Oc is the du oc of circle, yes you can take powertopes of roley objects. It has two sides which roll like the duospindle, contact regions are dyads. The sides are like the following, start with a square based wedge (wedge part is sq2 times longer than squares edge length, height of wedge is sq2*pi times smaller), now curve both axises of the square around into a circle, while making sure that the wedge part is curved into a circle with cone like surroundings.

### Regular Polytwisters

Any convex regular polytwister is a roley die, they roll like cylinders - but when tipped to a different side, they will roll off at a different angle:

Tetratwister has 4 cylindrically curved sides that swirl around each other, its contact regions are bowed out triangles.

Cube Twister has 6 cylindrically curved sides which resemble a 4 sided twisted rod that was curved into a ring, contact regions are bowed out squares.

Octatwister has 8 cylindrically curved trigon twister sides, contact region is a bowed triangle.

Dodecatwister has 12 cylindrically curved pentagonal twister sides, contact region is a bowed pentagon.

Icosatwister has 20 cylindrically curved trigon twister sides, contact region is a bowed triangle.

n-gonal Dysters has n cylindrically curved dyadic twister sides, contact region is a bowed dyad (2-D football shape). n is 3 or more.

### Catalan Polytwisters

These roll like the regular polytwisters and have cylindrically curved twister sides that swirl around each other.

Du Co Twister has 12 roley sides, contact region is a bowed rhombus.

Du Id Twister has 30 roley sides, contact region is a bowed rhombus.

Du Tut Twister has 12 roley sides, contact region is a bowed isot.

Du Tic Twister has 24 roley sides, contact region is a bowed isot.

Du Toe Twister has 24 roley sides, contact region is a bowed isot.

Du Tid Twister has 60 roley sides, contact region is a bowed isot.

Du Ti Twister has 60 roley sides, contact region is a bowed isot.

Du Sirco Twister has 24 roley sides, contact region is a bowed kite.

Du Srid Twister has 60 roley sides, contact region is a bowed kite.

Du Girco Twister has 48 roley sides, contact region is a bowed scalene.

Du Grid Twister has 120 roley sides, contact region is a bowed scalene.

Du Snic Twister has 24 roley sides, contact region is a bowed isosceles pentagon.

Du Snid Twister has 60 roley sides, contact region is a bowed isosceles pentagon.

### Tegum and Antitegum Twisters

For each tegum and antitegum in 3-space, there is a polytwister version with the same number of sides.

### Screwballs

These things are a bit crazy, they roll like a cylinder but trace out a screw like spiral as it rolls on the 3-D surface. To generate one, take a screw like spiral (like the spring in a notebook) and curve it in a ring using the 4th dimension, you can also wind it several times before connecting for more compact spirals. To top things off, we can also compound as many of these spirals as we want, as long as they are identical and are positioned in such a way to make them congruent - Now, every point on this spiral (or set of spirals) are congruent - consider the glome that contains all of these points, then carve out a section of 4-space with tangent realms (3-D hyperplanes) at every point - this will carve out a screwball. Screwballs can have any number of sides, this number depends on how many spirals we started with - if we started with a compound of 7 spirals, we will have a 7 sided screwball, if we started with only one spiral, we will have a one sided screwball - these are the curved versions of gyrotegums. What determined which screwball it is are the following numbers: the number of times it is wound on the x-y axis, the number of times it is wound on the z-w axis, and the number of spirals - they can be varied by the angle of latitude of the spirals, 45 degree being mid-way.

Back to polychoron page