Polytopes: Part III

This is the third part of a four part post. For the first part, see here. For the second part, see here.

Our entire world, and our entire Universe, all that we know, is limited to the third dimension. How, then, can anyone understand the fourth? The fourth dimension, which eludes our visual understanding forever, can only be thought of mathematically. Yet so many wonders arise from it. In the fourth dimension, you can see the inside of a three-dimensional object. This is easily visualized by a lower dimensional analogy. Imagine two shapes on a two-dimensional plane. From their perspective, they can see only the outlines of shapes, and their inside remains hidden. However, someone in the third dimension looking from above could easily see the inside of such a shape, and identify it as a triangle, for instance. The same thing happens from the fourth to the third dimension. Other wonders include two infinite planes intersecting at a single point, and two cubes at an edge. To attempt to gain an understanding of this Universe's inhabitants, we must understand the structures in Euclidean 4-space. Although the actual structures are tilings of three dimensional planes, all but the tiling of the Euclidean plane can only be understood from the vantage point of the fourth dimension. The result is a higher class of polytope: the polychora (singular: polychoron).

Again extending the system used before, a regular polychora can be represented by {p,q,r}, which represents r polyhedra {p,q} at every edge. However, before one randomly assigns numbers for p, q, and r, one must take into account the restrictions provided by polyhedra. For {p,q,r} must be made up of finite polyhedra, and {p,q} must therefore be one of the nine regular polyhedra that existed previously. In addition, {q,r} represents the configuration around each vertex of the polychoron, called the vertex figure, and this must also be finite for the polychoron to exist. Taking all this into account, and using the nine regular finite polyhedra {3,3}, {3,4}, {4,3}, {3,5}, {5,3}, {5/2,5}, {5,5/2}, {5/2,3}, and {3,5/2}, one ends up with 25 possible regular polychora:

{3,3,3}, {4,3,3}, {3,4,3}, {3,3,4}, {4,3,4}, {5,3,3}, {3,5,3}, {3,3,5}, {5,3,5}, {4,3,5}, {5,3,4}, {5/2,3,3}, {3,5/2,3}, {3,3,5/2}, {5/2,3,5/2}, {5/2,5,3}, {5/2,3,5}, {5,5/2,5}, {3,5,5/2}, {4,3,5/2}, {5,3,5/2}, {5/2,5,5/2}, {5,5/2,3}, {3,5/2,5}, and {5/2,3,4}.

The curvature of these figures is determined by the measure of the solid angle around each vertex. The solid angle is an extension of the regular angle that exists in three dimensions. A "full" angle is a sphere surrounding a point, and a partial angle is one that only fills a portion of the surrounding sphere. An example of a solid angle is shown below.



This angle, represented by the omega symbol, fills only a portion A of the entire sphere of radius r. It is apparent that r and A are proportional in this equation, because the size of A increases as r increases, but the value of omega remains the same. Note that the value "r" below is the value in {p,q,r}, not the radius of the sphere above.

The equation for the curvature of {p,q,r} is

((sin(π/p))^2)((sin(π/r))^2)>(cos(π/q))^2

which can be square rooted and adjusted to obtain

(sin(π/p))(sin(π/r))>cos(π/q)
(sin(π/p))(sin(π/r)-cos(π/q)>0

If this equation results in a value greater than 0, the polychoron is a tiling of elliptic three-dimensional space. As a result, they are finite, and can only be accurately expressed in Euclidean four-dimensional space. Six convex forms satisfy this formula, and five of these are extensions of the Platonic Solids. The polychora are constructed with polyhedra, and each polyhedron in a polychoron is known as a cell. The other elements are faces, edges, and vertices, just as before.

{3,3,3}
(sin(π/3))(sin(π/3))-cos(π/3)=((√3)/2)((√3/2))-1/2=3/4-1/2=1/4>0
The triplet {3,3,3} defines a convex regular polychoron with tetrahedra as cells with three at each edge. The resulting figure has five tetrahedral cells, ten triangular faces, ten edges, and five vertices. It is known as the pentachoron, or 5-cell. It is also the four-dimensional analog of the triangle and the tetrahedron. It is shown below.



The above projection shows the wireframe Schlegel diagram of the pentachoron projected in the third dimension. This projection allows one to see each individual cell of a polychoron, although three of the five tetrahedra in the image are distorted, without being in the fourth dimension.



Another projection of the pentachoron showing it as the tiling of a four dimensional sphere in elliptic geometry, The projection curves the faces and cells of the figure. This projection "unfolds" the three dimensional sphere onto a two dimensional plane, just like a flat map of the globe, and it is extended here to unfold the four dimensional sphere into a distorted three dimensional figure.

{4,3,3}
(sin(π/4))(sin(π/3)-cos(π/3)=((√2)/2)((√3)/2)-1/2=(√6)/4-1/2~.1123...
The triplet {4,3,3} defines a convex regular polychoron with cubes as cells and three at each edge. The resulting figure has eight cubic cells, 24 square faces, 32 edges, and 16 vertices. It is known as the octachoron, the 8-cell, or the tesseract. This polychoron is the four dimensional analog of the square and the cube. It is pictured below.



The wireframe projection of the 8-cell. In the fourth dimension, all eight cubes are of equal size and shape, but they are distorted in this image.



The elliptic projection of the 8-cell.

{3,3,4}
(sin(π/3))(sin(π/4)-cos(π/3)=((√3)/2)((√2)/2)-1/2=(√6)/4-1/2~.1123...
The triplet {3,3,4} defines a regular convex polychoron with tetrahedra as cells and four at each edge. The resulting figure has 16 tetrahedral cells, 32 triangular faces, 24 edges, and eight vertices. It is known as the hexadecachoron, or 16-cell. It is the four dimensional analog of a square and an octahedron. Note that the octahedron and cube are both based on the square, because the square is {4}, and the corresponding forms of the three and four dimensional analogs are {3,4}, {4,3}, {3,3,4}, and {4,3,3}, with a four at either the beginning or end of the symbol. WIth one dimension, these two cases are equivalent. because {4} with a 4 at the beginning is the same as {4} with a four at the end. The 16-cell is shown below.



The wireframe projection of the 16-cell. As before, all 16 cells are the same size in the fourth dimension, but appear distorted when projected.



Elliptic projection of the 16-cell.

{3,4,3}
(sin(π/3))(sin(π/3))-cos(π/3)=((√3)/2)((√3)/2)-(√2)/2=3/4-(√2)/2=.04289...
The triplet {3,4,3} defines a convex regular polychoron with octahedra as cells and three at each edge. The resulting figure has 24 octahedral cells, 96 triangular faces, 96 edges, and 24 vertices. It is known as the icositetrachoron, or the 24-cell. It is the only regular convex polychoron of the six that is not based on a lower polytope, and is my personal favorite (if I'm allowed to have one) of the six. It is shown below.



The wireframe projection of the 24-cell. This is the only convex polychoron with octahedral cells.



The elliptic 24-cell. This projection is the same as previous ones, except for the fact that each face of the figure is "bubbled".

{5,3,3}
(sin(π/5))(sin(π/3))-cos(π/3)=.00903696...
The triplet {5,3,3} defines a regular convex polychoron with dodecahedra as cells and three at each edge. The resulting figure has 120 dodecahedral cells, 720 pentagonal faces, 1200 edges and 600 vertices. It is known as the hecatonicosichoron, or 120-cell. It is the four dimensional analog of the pentagon and dodecahedron. The 120-cell is shown below.



The wireframe projection of the 120-cell. Due to the projection, the polychoron is distorted so that the nearest cells actually appear to be the "smallest" and "farthest", but in reality the ones closest to the center, i.e. the smallest, are the closest from a four-dimensional viewpoint.



Elliptic "bubbled" 120-cell. The same apparent paradox with the distorted viewpoint applies, as discussed above.

{3,3,5}
(sin(π/3))(sin(π/5))-cos(π/3)=.00903696...
The triplet {3,3,5} defines a regular convex polychoron with tetrahedra as cells and five at each edge. The resulting figure has 600 tetrahedral cells, 1200 triangular faces, 720 edges and 120 vertices. It is known as the hexacosichoron or 600-cell. It is the four dimensional analog of the icosahedron.



The wireframe projection of the 600-cell.



Elliptic projection of 600-cell. For both of the above projections the same viewpoint "paradox" that applies to the 120-cell applies here.

Many of the projections may be confusing, and this is because the cells closest in the four-dimensional viewpoint are bunched up in the center of the projection and are small. The reason for this can be easily seen with the projection of the dodecahedron into two dimensions.



This is the projection of the dodecahedron onto a two dimensional plane. The faces "on top" are actually the nearest, while the bottom faces, i.e. the ones that would be the farthest away in three dimensions are on the edges. These are the largest, but must distorted.



This image shows a similar wireframe projection of the 24-cell to the one shown above, but with the "closest" cell highlighted in red. This allows one to see that the closest cell is oriented in the center of the projection and is the least distorted.

Another way to show the polychora is by presenting their net. The net of a polytope is the "unfolded" polytope into its faces, if it is a polyhedron, or cells, if it is a polychoron.



The net of the dodecahedron. If this net is folded into the third dimension, the pentagons line up as the faces of the dodecahedron.

The net of a polychoron is made of polyhedra. For example, the net of the 8-cell is eight cubes, as shown below. When this figure is folded into the fourth dimension, it becomes the 8-cell. For the nets of all of the six polychora above, see here.



Of the six, four come in two dual pairs, the 8-cell and 16-cell, and 120-cell and 600-cell, and two are their own duals: the 5-cell, and the 24-cell. Also, for the six shown above, all have the property that the number of cells plus the number of edges is equal to the number of faces plus the number of vertices.

There are also non-convex regular finite polychora, but we will skip all that involve star polygons and therefore any star polyhedra as cells for now. Of the 25 cases above, 11 do not include the star polygon symbol 5/2. These are {3,3,3}, {4,3,3}, {3,4,3}, {3,3,4}, {4,3,4}, {5,3,3}, {3,5,3}, {3,3,5}, {5,3,5}, {4,3,5}, and {5,3,4}. Six of these have already been dealt with, (the regular convex finite polychora) and this leaves only five: {4,3,4}, {3,5,3}, {5,3,5}, {4,3,5}, and {5,3,4}. Using the formula above but with the case

(sin(π/p))(sin(π/r)-cos(π/q)=0,

where the curvature of {p,q,r} is zero, one can see that any solution to this equation would tile Euclidean three dimensional space. Only one of the above pairs fits that definition, and that is the only tiling of Euclidean three dimensional space, {4,3,4}. It is the only polychoron that most can truly visualize, because it can be expressed in three-dimensional Euclidean space, as opposed to the above polychora, which are in elliptic three dimensional space, and can only be truly expressed in Euclidean geometry with the use of four dimensions. {4,3,4} is shown below.

{4,3,4}
(sin(π/4))(sin(π/4))-sin(π/3)=((√2)/2)((√2)/2)-1/2=2/4-1/2=0


This infinite polychoron has four cubes at each edge, and is also known as the cubic honeycomb.

The one remaining case of regular convex polychora is

(sin(π/p))(sin(π/r)-cos(π/q)<0

Four polychora satisfy this definition, {5,3,5}, {5,3,4}, {3,5,3}, and {4,3,5}. The formula states that these are negatively curved, and therefore are tilings in hyperbolic geometry. The first two have dodecahedra as cells, the third icosahedra, and the fourth, cubes. As an example, the case {3,5,3} is shown below.

{3,5,3}
(sin(π/3))(sin(π/3))-cos(π/5)=-.86803...


This infinite polychora is a tiling of negatively curved space, or hyperbolic space. Each edge is surrounded by three icosahedra. The projection here is an extension of the bounding circle used in three dimensional models to a bounding sphere, which contains the entire hyperbolic universe in one finite sphere. As a result all of the icosahedral cells appear distorted but are all regular and of the same size from a hyperbolic viewpoint. The center icosahedron is highlighted to emphasize a cell. Other than this, however, the cells are hard to identify. The other three dimensional hyperbolic tilings are similar in structure to the above one.

14 polychora have not been discussed out of the 25 above. The remaining ones have at least one 5/2 in their symbols, making them non-convex star polychora. The 14 possible forms for regular star polychora are:

{5/2,3,3}, {3,5/2,3}, {3,3,5/2}, {5/2,3,5/2}, {5/2,5,3}, {5/2,3,5}, {5,5/2,5}, {3,5,5/2}, {4,3,5/2}, {5,3,5/2}, {5/2,5,5/2}, {5,5/2,3}, {3,5/2,5}, and {5/2,3,4}.

{3,5,5/2}
(sin(π/3))(sin(π/(5/2)))-cos(π/5)=.014622...
The triplet {3,5,5/2} defines a non-convex regular polychoron with icosahedra as cells, and five at each vertex, but with each cell penatrating the interior of the polychoron. The resulting figure has 120 icosahedral cells, 1200 triangular faces, 720 edges, and 120 vertices. It is known as the icosahedral 120-cell, and is shown below.



The drawback of this projection is that the faces are not transparent, and it is therefore impossible to view the interior of the polychoron, as opposed to the wireframe view, where the interior is revealed. This polychoron appears convex, but is not in four dimensional space.

{5/2,5,3}
(sin(π/(5/2)))(sin(π/3))-cos(π/5)=.014622...
The triplet {5/2,5,3} defines a non-convex regular polychoron with small stellated dodecahedra as cells, with three at each edge. The resulting figure has 120 small stellated dodecahedral cells, 720 pentagrammic faces, 1200 edges, and 120 vertices. It is known as the small stellated 120-cell, and is pictured below.



The solid cell projection of the small stellated 120-cell. A few pentagrammic {5/2} faces are clearly visible, but it is difficult to isolate those that penetrate the polychoron's interior.

{5,5/2,5}
(sin(π/5))(sin(π/5))-cos(π/(5/2))=.0364745...
The triplet {5,5/2,5} defines a non-convex regular polychoron with great dodecahedra as faces and five at each edge. The resulting figure has 120 great dodecahedral cells, 720 pentagonal faces, 720 edges and 120 vertices. It is known as the great 120-cell, and is shown below.



The solid cell projection of the great 120-cell. The great dodcahedra are evident, along with the pentagons as faces. The easiest one to spot is in the front, and is colored light purple.

{5,3,5/2}
(sin(π/5)(sin(π/(5/2)))-cos(π/3)=.059010699...
The triplet {5,3,5/2} defines a non-convex regular polychoron with dodecahedra as cells and five at each edge, with some penetrating the interior of the polychoron. The resulting figure has 120 dodecahedral cells, 720 pentagonal faces, 720 edges, and 120 vertices. It is known as the grand 120-cell, and is pictured below.



The projection of the grand 120-cell is exactly the same as that of the great 120-cell for reasons discussed below. However, the dodecahedra in this image are visible as well as the great dodecahedra, and one can spot two types of cells in this same image!

{5/2,3,5}
(sin(π/(5/2)))(sin(π/5))-cos(π/3)=.059010699...
The triplet {5/2,3,5} defines a non-convex regular polychoron with great stellated dodecahedra as cells, and five at each edge. The resulting figure has 120 great stellated dodecahedral cells, 720 pentagrammic faces, 720 edges, and 120 vertices. It is known as the great stellated 120-cell and is shown below.



The solid face projection of the great stellated 120-cell. The pentagrammic faces and great stellated dodecahedral cells are easily identified.

{5/2,5,5/2}
(sin(π/(5/2)))(sin(π/(5/2)))-cos(π/5)=.0954915...
The triplet {5/2,5,5/2} defines a non-convex regular polychoron with small stellated dodecahedra as cells, and five at each edge, with each penetrating the interior of the polychoron. The resulting figure has 120 small stellated dodecahedral cells, 720 pentagrammic faces, 720 edges, and 120 vertices. It is known as the grand stellated 120-cell, and is shown below.



The solid face projection of the grand stellated 120-cell, which is exactly the same as that of the great stellated 120-cell for the same reason as above, and it is discussed below.

{5,5/2,3}
(sin(π/5))(sin(π/3))-cos(π/(5/2))=.200019966...
The triplet {5,5/2,3} defines a non-convex regular polychoron with great dodecahedra as cells and three at each edge. The resulting figure has 120 great dodecahedral cells, 720 pentagonal faces, 1200 edges, and 120 vertices. It is known as the great grand 120-cell, and is pictured below.



The solid face projection of the great grand 120-cell. The pentagonal faces are hard to spot, because they pass through the interior of the polychoron, but they are there, as are the great dodecahedral cells.

{3,5/2,5}
(sin(π/3))(sin(π/5))-cos(π/(5/2))=.200019966...
The triplet {3,5/2,5} defines a non-convex regular polychoron with great icosahedra as cells and five at each edge. The resulting figure has 120 great icosahedral cells, 1200 triangular faces, 720 edges, and 120 vertices. It is known as the great icosahedral 120-cell, and is pictured below.



The solid face projection of the great icosahedral 120-cell. The triangular faces are exceptionally hard to spot here, but they can be seen by connecting three points on opposite sides of the polychoron. The points of the great icosahedra can be seen, but their structure is hidden in the interior.

{3,3,5/2}
(sin(π/3))(sin(π/(5/2)))-cos(π/3)=.323639....
The triplet {3,3,5/2} defines a non-convex regular polychoron with tetrahedra as cells and five at each edge, but with each penetrating the polychoron's interior. The resulting figure has 600 tetrahedral cells, 1200 triangular faces, 720 edges, and 120 vertices. It is known as the grand 600-cell and is shown below.



The solid face projection of the grand 600-cell. For the final time, this projection is identical to that of the great icosahedral 120-cell, despite having different numbers of cells.

{5/2,3,3}
(sin(π/(5/2)))(sin(π/3))-cos(π/3)=.323639....
The triplet {5/2,3,3} defines a non-convex regular polychoron with great stellated dodecahedra as cells, and three at each edge. The resulting figure has 120 great stellated dodecahedral cells, 720 pentagrammic faces, 1200 edges, and 600 vertices. It is known as the great grand stellated 120-cell, and is pictured below.



The solid face projection of the great grand stellated 120-cell. The pentagrammic faces are fairly evident, as are the great stellated dodecahedra.

The ten above polychora come in four dual pairs:
Icosahedral 120-cell & Small Stellated 120-cell
Grand 120-cell & Great Stellated 120-cell
Great Grand 120-cell & Great Icosahedral 120-cell
Grand 600-cell & Great Grand Stellated 120-cell
and the remaining two, the Great 120-cell, and Grand Stellated 120-cell, are their own duals.

In addition, there are many similarities between the above polychora. In terms of vertex arrangement, the first nine polychora shown share the vertices of the 600-cell, and there are 120 of them, while only the tenth has 600 vertices and corresponds to the vertex arrangement of the 120-cell. In terms of edge arrangements, the 600-cell, the icosahedral 120-cell, the grand 120-cell, and the great 120-cell all share an edge and vertex arrangement, as do the small stellated 120-cell and the great grand 120-cell, the great stellated 120-cell, the great 600-cell, the grand stellated 120-cell and the great icosahedral 120-cell, leaving the great grand stellated 120-cell all by itself. Finally, there are three pairs above whose projections are identical because they have the same vertex, edges and face arrangement. Only the cells are connected differently, and this can only be seen in four dimensions. This applies to the great 120-cell and grand 120-cell, the great stellated 120-cell and the grand stellated 120-cell, and the great icosahedral 120-cell and the grand 600-cell.

The naming of these polychora comes from the processes used to create them. When one extends the edges of an existing polytope until it intersects itself (if it ever does) and forms another polytope, such as the pentagon to the pentagram, this is called stellation. Anything with "stellated" in its name has gone through this process. When the faces are extended from a polytope (polyhedron or higher) until they intersect themselves (if they ever do), and another polytope is formed, this is called greatening. Any polyhedron or polychoron with "great" in its name has undergone this process. Finally, if the cells (of a polychora or higher) are extended until they intersect themselves and form another polytope, this is known as aggrandizement. Any polychoron with "grand" in its name has undergone this process. These three processes are used to construct the four star polyhedra and 10 star polychora from the icosahedron and dodecahedron, and 600-cell and 120-cell, respectively.

After all these forms have been dealt with, there are only four possible pairs remaining: {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}, and {3,5/2,3}. These all have positive curvature, but none of them form regular star polychora. One can always begin to construct them, but they never close up and an infinite number of cells result, which is an impossibility in finite polychora.

All of the above deals with regular polychora, but one can also find polychora with more than one type of regular cell. These are known as uniform polychora. As with polyhedra truncation and cantellation are used to obtain these figures. However, since truncation is the slicing of vertices and can be used on polygons and higher, and cantellation is the slicing of edges and can be used on polyhedra and higher, there is a third operation, known as runcination. This process slices cells with respect to the fourth dimension, a process which is hard to understand from our third dimensional viewpoint.

When a polychoron is truncated, it turns from the ordinary {p,q,r} into what is written as t0,1{p,q,r}. The truncated 120-cell is shown below.



Wireframe projection of the truncated 120-cell, where new cells are shown in red.

When a polychoron {p,q,r} is rectified, or truncated to the point that the original edges disappear it is written t1{p,q,r}. If the resulting polychoron is truncated again, it is bitruncated, and if it is rectified again, it is known as birectified. A polychoron {p,q,r} must be rectified three times before resulting in its dual, {r,q,p}. As a result, there are five intermediate steps between a regular polychoron and its dual when truncating that result in a uniform polychoron:

Parent: {p,q,r}
Truncated: t0,1{p,q,r}
Rectified: t1{p,q,r}
Bitruncated: t1,2{p,q,r}
Birectified: t2{p,q,r}, same as t1{r,q,p}
Tritruncated: t2,3{p,q,r}, same as t0,1{r,q,p}
Trirectified or Dual: t3{p,q,r}, same as {r,q,p}

With cantellation, there are two intermediate steps between a polyhedron and its dual:

Cantellated t1,2{p,q,r},
Bicantellated t2,3{p,q,r}, same as t1,2{r,q,p}

For example, the cantellated 8-cell looks like this:



As before, new cells formed by the cantellation are shown in red and yellow.

With rucination, there is only one intermediate step, and it is the runcinated polychoron t0,3{p,q,r}. The rucinated 5-cell is shown below.



This operation separates cells that were previously connected, and puts new cells in their place. This image shows the new cells from the rucination of the four distorted original cells in red, and highlights the new cell from the fifth, which makes up the tetrahedral envelope, or the outside of the figure, in green.

Finally, any combination of the above operators can be used. The most special case is when all three operators, truncation, cantellation, and rucination are all done to a polychoron. Then, the polychoron is omnitruncated. This is expressed t0,1,2,3{p,q,r}. As an example, the omnitruncated 8-cell, which is written t0,1,2,3{4,3,3}, is shown below.



One can also apply these operations to infinite polychora, such as the cubic honeycomb {4,3,4}. For example, the truncated cubic honeycomb t0,1{4,3,4}, replaces cubes with truncated cubes, and fills the resulting spaces with regular octahedra. As a result, this uniform honeycomb still covers the entire Euclidean three dimensional space. A section of this honeycomb is shown below.



The same processes can be applied to hyperbolic tilings as well.

In addition, there is an "inverse" of an omnitruncated polychoron, called a snub polychoron. It is creating by applying birectification to alternate vertices of a polytope. It is denoted s{p,q,r} for polychora. The snub 24-cell is shown below.



This polychoron appears to have edges on flat faces, but in reality the flattened dark blue areas that are not triangular are distorted icosahedra. The structure of the snub 24-cell is more easily examined by use of its net, which is shown below.



In the net of the snub 24-cell, the tetrahedra and icosahedra that make it are clearly visible.

The above operations on convex regular polychora plus their snubs make up 46 out of the 47 distinct convex uniform polychora. The 47th is a four dimensional prism that cannot be expressed with a Schalfli symbol {p,q,r}, and is unique in that aspect. It is made of tetrahedral and pentagonal antiprismic cells. The pentagonal antiprism is the polyhedron with three triangles and one pentagon around each vertex.



The 47th uniform polychoron, or the grand antiprism, is a composition of 300 tetrahedral cells and 20 of the type above. It is the only irregular pattern in uniform polychora.

It has been proved that there are exactly 47 uniform convex polychora, but the total number of non-convex uniform polychora is not known, and there may be thousands! There are many polychora not based on regular ones, but these are not directly related to regular polytopes, and are not mentioned. The only exception to this is the grand antiprism, which is based on prisms which are not regular, but the polychoron itself is still uniform.

Sources: http://mathforum.org, http://weimholt.com/andrew/polytope.shtml, Regular Polytopes by H.S.M. Coxeter, a variety of wikipedia titles: Uniform Polychora, Schlafli-Hess polychoron, List of Regular polytopes, etc.

This post is finished in Polytopes: Part IV.