Tuesday, May 18, 2010

Polytopes: Part IV

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

All the regular polytopes up through the fourth dimension have been discussed, and we couldn't directly visualize these elaborate structures, but we could understand their construction, and their polyhedral components. However, when one goes to the fifth dimension, all direct understanding is out of reach. And what of the sixth dimension? And the seventh? And the hundredth? How can we possibly deal with the polytopes in these dimensions? However, rather than increasing in complexity, regular polytopes become much simpler in higher dimensions, which allows us to generalize to n-dimensions in many aspects.

Starting with five dimensional space, we must consider a further extension of the Schlafli system. Since the structures in four dimensions were denoted by {p,q,r}, we now have five-dimensional polytopes {p,q,r,s}. Based on the sixteen regular polychora, ({3,3,3}, {4,3,3}, {3,4,3}, {3,3,4}, {5,3,3}, {3,3,5}, {5/2,3,3}, {3,3,5/2}, {5/2,5,3}, {5/2,3,5}, {5,5/2,5}, {3,5,5/2}, {5,3,5/2}, {5/2,5,5/2}, {5,5/2,3}, and {3,5/2,5}) one arrives at a staggering 34 possible forms:

{3,3,3,3}, {3,3,3,4}, {3,3,4,3}, {3,4,3,3}, {4,3,3,3}, {3,3,3,5}, {5,3,3,3}, {4,3,3,4}, {4,3,3,5}, {5,3,3,4}, {5,3,3,5}, {3,3,3,5/2}, {5/2,3,3,3}, {4,3,3,5/2}, {5/2,3,3,4}, {5,3,3,5/2}, {5,5/2,3,3}, {5/2,3,3,5}, {5,5/2,5,3}, {3,5/2,5,3}, {3,5,5/2,5}, {5,3,5/2,5}, {3,3,5,5/2}, {5/2,5,3,3}, {5,5/2,3,5}, {3,5,5/2,3}, {5/2,5,5.2,5}, {5/2,5,5/2,3}, {3,5/2,5,5/2}, {5,5/2,5,5/2}, {5/2,3,3,5/2}, {5/2,3,5,5/2}, {5/2,5,3,5/2}, {3,3,5/2,5}

11 of these involve only convex polychora, while the remaining 23 involve star polychora. Despite the vast range of possible forms, very few actually create polytopes in the fifth dimension. These are called 5-polytopes, or polytera (singular: polyteron). The curvature of these polytera is defined by determining whether the four dimensional solid angle around each vertex adds up to more than, less than, or exactly equal to, the four dimensional sphere. To find the curvature equation in higher dimensions, it becomes useful to use the general form of the equation in n dimensions. The function to find the formula is known as the (delta) equation. The curvature formula for a polytope {p,q,r...,y,z} (with any number of letters in between r and y) is expressed {p,q,r...,y,z}, and depends whether the resulting formula is greater than, less than, or equal to zero. The equation is defined recursively, or that each formula depends on the previous one counting up in dimensions. Assuming the trivial cases in one and two dimensions as follows (with {} implying the straight line as the universal polytope in one dimension):

∆{}=1 (this formula never changes in value, because all polytopes in one dimension are lines and are all basically identical)
and
∆{p}=(sin(π/p))^2 (this formula is always positive, corresponding to the fact that polygons always have positive curvature and are finite)

one can find the formula for any number of dimensions greater than two using:

{p,q,r...,y,z}={q,r,...,y,z}-{r,...,y,z}*(cos(π/p))^2

For example, to find ∆{p,q}:

∆{p,q}=∆{q}-∆{}*((cos(π/p))^2)=
(sin(π/q))^2-1*((cos(π/p))^2)=
(sin(π/q))^2-(cos(π/p))^2

By setting this greater than zero (solutions would then be finite polyhedra)

(sin(π/q))^2-(cos(π/p))^2>0
(sin(π/q))^2>(cos(π/p))^2
sin(π/q)>cos(π/p)

which, by a property of trigonometry, (for p,q>2, which, conveniently is what is required for true polyhedra) equals

sin(π/q)>sin(π/2-π/p)
π/q>π/2-π/p
π/p+π/q>π/2
1/p+1/q>1/2

The final formula seems very familiar, as it is the curvature formula from the second part of this post for polyhedra, that we have successfully derived using the formula! Using the same equation for {p,q,r,s} (I won't show all the work this time), we obtain the curvature formula for polytera. For a finite polyteron,

((cos(π/q))^2)/((sin(π/p))^2)+((cos(π/r))^2)/((sin(π/s))^2)<1

12 of the 34 total forms satisfy this: {3,3,3,3}, {3,3,3,4}, {4,3,3,3}, {3,3,3,5/2}, {5/2,3,3,3}, {4,3,3,5/2}, {5/2,3,3,4}, {5,5/2,3,3}, {3,3,5,5/2}, {5/2,5,5/2,3}, {3,5/2,5,5/2}, and {5/2,3,3,5/2}. However, all nine of these that are star polytera can be calculated to have infinite density, meaning that there are infinite planes in the polyteron. However, this is impossible in regular finite polytera, and therefore all but the first three can be eliminated. We will return to more general forms of {3,3,3,3}, {4,3,3,3} and {3,3,3,4} later.

There can also be tilings of Euclidean four dimensional space and these are the only ones that can be understood in four dimensions. The simplest example, {4,3,3,4}, also known as the tesseractic honeycomb, has four tesseracts (8-cells) at each face, and a three dimensional projection is shown below.



With these figures, it is difficult to see any recognizable structure, but the 8-cells in this picture can vaguely be seen.

Similarly, {3,4,3,3} has three 24-cells at each face, and {3,3,4,3} has three 16-cells at each. No star polytera exist that are tilings of the Euclidean four dimensional plane, although {5,3,3,5/2}, {5/2,3,3,5}, {3,5/2,5,3} {3,5,5/2,3}, {5/2,5,5/2,5}, {5,5/2,5,2/2}, {5/2,3,5,5/2} and {5/2,5,3,5/2} all satisfy

((cos(π/q))^2)/((sin(π/p))^2)+((cos(π/r))^2)/((sin(π/s))^2)=1

However, all nine possible four dimensional hyperbolic tilings exist, namely: {3,3,3,5}, {5,3,3,3}, {4,3,3,5}, {5,3,3,4}, {5,3,3,5}, {3,5,5/2,5}, {5,5/2,5,3}, {3,3,5,5/2}, and {5/2,5,3,3} and all these satisfy

((cos(π/q))^2)/((sin(π/p))^2)+((cos(π/r))^2)/((sin(π/s))^2)>1

Finally, returning to the three finite regular polytera {3,3,3,3}, {3,3,3,4} and {4,3,3,3}, one can see that only four forms are possible in six dimensions: {3,3,3,3,3}, {3,3,3,3,4}, {4,3,3,3,3} and {4,3,3,3,4}. The last of these is a tiling (as we will soon see) and the only seven dimensional regular figures are {3,3,3,3,3,3}, {3,3,3,3,3,4}, {4,3,3,3,3,3} and {4,3,3,3,3,4}. The last of this is also a tiling, and the pattern continues. Therefore, for finite regular polytopes existing in n dimensional Euclidean space (n-1 dimensional elliptic space) there are only three forms:

{3^(n-1)}
The general polytope in n dimensions with n-1 3's in its symbol is known as the n-simplex. It has n+1 vertices and the rest of its elements, known as i-faces, come in numbers discussed below. The n-simplex is always a regular finite polytope in any number of dimensions.

Construction in n dimensions: Start with a point. This is the 0-simplex. Choose another distinct point and connect them. The result is a line segment, which is the 1-simplex. Choose a point outside of this line that is equidistant from the two existing points and connect them. The result is the regular triangle {3}, which is the 2-simplex. Choose another point outside this plane, that is equidistant from all three points and connect each pair of points with an edge. The result is the tetrahedron {3,3} which is the 3-simplex. Continue this procedure for any number of dimensions.

The number of i-faces (0-face=vertex, 1-face=edge, 2-face=face, 3-face=cell, etc.) in an n-simplex is based on the binomial theorem and Pascal's triangle. For an n-simplex, the number of i-faces is

(n+1)!/((i+1)!(n-i)!)

where ! is the factorial function (n!=1*2*3*4...*n).

For example, the number of faces on a 4-simplex is 5!/((3!)(4-2)!)=10, and the number of 5-faces in a 9-simplex is 10!/((6!)(9-5)!)=210.

The n-simplex is the simplest polytope that needs n dimensions to define, and is the general form of the sequence: point, line segment, triangle, tetrahedron, pentachoron... etc. In addition, n-simplices are most often represented by symmetric graphs that map out the vertices and show the connections between them. However, the drawback of this representation is that only vertices and edges can be mapped, and there is no easy way to see higher i-faces. As a result, symmetric graphs give few implications to the actual structure of the polytope. The symmetric graph of the 5-simplex {3,3,3,3} is shown below.



The symmetric graph of the 5-simplex (or hexateron). The six vertices and 15 edges are visible, as is the fact that every pair of points is connected, but this view lacks any higher features. Also, for an n-simplex, the symmetric graph is always based on the regular polygon {n+1}.

{4,3^(n-2)}
The general polytope in n dimensions with a 4 followed by n-2 3's in its Schlafli symbol is known as the n-cube. It is always a finite regular polytope in any number of dimensions. It has 2^n vertices and the number of i-faces again depends on the binomial theorem but with an extra term in front of it. In general, to find the number of i-faces of a n-cube, the formula is

(2^(n-i))((n!)/((i!)(n-i)!))

For example, the number of edges (1-faces) on a 3-cube is (2^(3-1))((3!)/((1!)(3-1)!))=12, and the number of 7-faces on an 18-cube is (2^(18-7))((18!)/((7!)(18-7)!))=65175552.

Construction in n dimensions: To construct a regular n-cube, begin with a point. This is the 0-cube. Choose another distinct point and connect them. The result is a line segment, which is the 1-cube. Define another line segment as the shifting of the original one which is parallel to the original and so the distance between the lines is the same as the length of each line segment. Connect the corresponding vertices on the two line segments, and the result is the square {4}, which is the 2-cube. Take this square, and shift it out of the existing plane up or down the distance between any two points, to obtain two parallel squares. Connect each original point to its corresponding shifted point. The result is the 3-cube {4,3}. Continue for any number of dimensions to obtain any n-cube. To demonstrate this procedure, the construction for the 4-cube {4,3,3}, also called the 8-cell, is shown below.



n-cubes, just like n-simplices, can be expressed with symmetric graphs. For example, the 6-cube's symmetric graph is shown below.



The n-cube is also called the measure polytope in n dimensions, and is the general form of the sequence point, line segment, square, hexahedron, octachoron... etc.

{3^(n-2),4}
The general polytope in n dimensions with n-2 3's followed by a final 4 in its Schlafli symbol is known as the n-orthoplex, or the cross polytope. It is the general dual of the n-cube, and has 2n vertices. A general form for the number of i-faces is once again based on the binomial theorem. In general, the number of i-faces in an n-orthoplex is

(2^(i+1))((n!)/((i+1)!(n-(i+1))!))

For example, the number of cells in a 5-orthoplex is (2^(4))((5!)/((4)!(5-(5))!))=80, and the number of 6-faces in a 10-orthoplex is (2^(7))((10!)/((7)!(10-(7))!))=15360.

Construction in n dimensions: Start with a point. This is the 0-orthoplex. Choose another distinct point and connect the two. The result is the 1-orthoplex, or a line segment. Then choose two points equidistant from the existing two and connect them to the existing points in such a way that a square is formed (eliminate the line between the original two points). The square {4} is the 2-orthoplex. Then, choose a point equidistant from all four of the existing points that is the same distance from each point as each point is from another adjacent to it. Also choose this point's reflection through the square. Connect all points, except those directly opposite from each other. The result is the octahedron {3,4} which is the 3-orthoplex. This is a more complex construction process, but it is easy to see with a little thought that it produces the orthoplexes. This process can be continued for any number of dimensions.

As with the first two general polytopes, the n-orthoplex can be represented with a symmetric diagram. For example, the layout of the 7-orthoplex's 14 vertices into the regular polygon {14} looks like this:



This symmetric graph looks similar to that of a simplex, but with one exception. The vertices in a simplex are all connected in every possible way, but in an orthoplex, points are connected to every other point except the one directly opposite it. In conclusion, the n-orthoplex is a convex regular finite polytope in any number of dimensions, and is the general term of the sequence point, line segment, square, octahedron, 16-cell, etc.

{4,3^(n-2),4}
The fourth and final regular polytope in n-dimensions is represented by a 4, followed by n-2 3's, and then another 4. It is an infinite cubic honeycomb in Euclidean geometry, which is why {4,3^(n-2),4} needs only n dimensions to be visualized. If it were an elliptic or hyperbolic polytope, it would need n+1 dimensions, and the symbol would be {4,3^(n-3),4}. The elements of the n-cubic honeycomb are simply n-cubes.

After five dimensions, the above four are the only regular polytopes for each dimension, but there are many other polytopes with regular elements, and these are known as uniform polytopes. One subset of uniform polytopes is the regular ones discussed above, and another is the quasi-regular polytopes, which are based on two types of i-faces. There are limited possibilities for these, and these come in a few different families, each represented by a special Schlafli symbol {3^{a,b,c}}. Of the four families, only one provides an infinite number of quasi-regular polytopes. Note that the {a,b,c} doesn't tell the number of 3's, as before, but has a different meaning, which is not discussed here.

{3^{1,b,1}}
The first family of quasi-regular polytopes is represented {3^{1,b,1}}, where the variable b is used just to match up with the {a,b,c} above. These polytopes are known as n-demicubes. In n dimensions, the n-demicube is {3^{1,n-3,1}}. They are constructed by connecting alternating vertices of an n-cube with edges. To demonstrate this process, the 3-cube and 3-demicube are shown below.



This image shows a transparent cube (3-cube) with its two possible demicubes. One connects alternate vertices, and the second simply connects all those not covered by the first. The two demicubes of a cube are always identical, and in this case, are two tetrahedra. Therefore, the tetrahedron, as well as being {3,3}, is also {3^{1,0,1}}. The n-demicube is quasi-regular beginning with the fifth dimension, as the 4-demicube is the 16-cell. Each n-demicube is made up of (n-1)-simplices and (n-1)-demicubes, and therefore is also defined recursively, with each demicube depending on the one before it. Each n-demicube has exactly 2n (n-1)-demicubes and 2^(n-1) (n-1)-simplices as (n-1)-faces. These are the highest faces before the polytope itself, and are sometimes called facets.

Since the 3-demicube is a tetrahedron, and the 3-simplex is also a tetrahedron, the 4-demicube is made of 8 tetrahedra and 8 tetrahedra, and the result is 16 tetrahedral cells, which is a 16-cell. However, the 5-demicube is made out of 10 4-demicubes, which are 16-cells, and 16 4-simplices, which are 5-cells, and the 5-demicube is therefore the first quasi-regular demicube. Continuing this pattern, one finds that there are infinite demicubes, all of which are finite and quasi-regular. As before, n-demicubes are also represented with symmetric graphs, and the 7-demicube is shown below in symmetric graph form.



The 7-demicube has only half the vertices of the 7-cube, but over 200 more edges. Also, red dots represent single vertices, while orange represent two overlapping and yellow four.

{3^{1,b,2}}
The second family of quasi-regular polytopes is the {3^{1,b,2}} family. The {3^{1,b,2}} polytope only is distinct from aforementioned polytopes when b takes the values 2, 3, 4 and 5, corresponding to polytopes in 6, 7, 8, and 8 dimensions, as we will see shortly. When b takes the value 0, the {3^{1,0,2}} polytope exists in four dimensions, and is simply {3,3,3}, or the 4-simplex. From there, the system is defined recursively, with each {3^{1,b,2}} in n dimensions (with one exception, see below) having {3^{1,b-1,2}'s and (n-1)-demicubes as facets. The polytope {3^{1,1,2}} has 4-simplices and 4-demicubes as facets, and this has already been discussed above as being the 5-demicube. Therefore,

5-demicube={3^{1,2,1}={3^{1,1,2}}

However, the {3^{1,2,2}} polytope has {3^{1,1,2}}'s and 5-demicubes as facets, which are both equivalent, and the {3^{1,2,2}} polytope is therefore a new polytope, made of 54 5-demicube facets. The next polytope {3^{1,3,2}} has {3^{1,2,2}}'s and 6-demicubes as facets, and the pattern continues. However, when one gets to {3^{1,5,2}} polytope, composed of {3^{1,4,2}}'s (in eight Euclidean dimensions, seven elliptic) and 8-demicubes, one finds not an elliptic polytope, but a new Euclidean tiling! Therefore, this infinite polytope also exists in eight dimensions. However, having a Euclidean tiling in the polytope family ends it, for if the {3^{1,6,2}} polytope existed, it would have to have {3^{1,5,2}}'s as facets, and these are infinite, which is not allowed in polytopes. Therefore, only four new polytopes arise from this family. The symmetric graph of {3^{1,3,2}} is pictured below.



In this image, blue vertices are single and red have a multiplicity of 2 (having 2,4,6 or some even number of vertices coinciding).

{3^{2,b,1}}
The third of four families of quasi-regular polytopes is the {3^{2,b,1}} family. Again, only four new polytopes are generated by this family and these are when b=2, 3, 4, or 5. The facets of such a polytope in n dimensions (again, with one exception) are {3^{2,b-1,1}}'s and (n-1)-simplices. The {3^{2,0,1}} polytope is the 5-cell again and the {3^{2,1,1}} has 32 4-simplices, making it the 5-orthoplex. The first new polytope is {3^{2,2,1}} which has 5-orthoplexes and 5-simplices as facets. The pattern continues again until {3^{2,5,1}} which is another Euclidean tiling, made of an infinite number of 8-simplices and {3^{2,4,1}}'s. As before, the pattern must end there, since there cannot be a polytope with infinite facets, although a polytope can have an infinite number of facets. The {3^{2,4,1}} polytope is shown below.



This polytope exists in eight dimensional Euclidean space, and only its 2162 vertices are presented in this graph. Although it may seem that presenting only vertices doesn't present much of the polytope, the addition of edges would over clutter the image, as the {3^{2,4,1}} has over 69000!

{3^{a,2,1}}
The final quasi-regular polytope family is the {3^{a,2,1}} family and it is the only one of the four in which the first number a of the symbol varies and not the second, b. This family provides only 3 new polytopes in 5 dimensions and up, but yields a few interesting cases in lower dimensions as well. In n dimensions, the facets are simply (n-1)-simplices and (n-1)-orthoplexes. The sequence begins with the {3^{-1,2,1}} polytope, which exists in three dimensions and has triangles and squares as faces. The resulting figure is a triangular prism, which is simply two parallel triangles connected by three squares. The next polytope has 3-simplexes and 3-cubes as cells and is represented {3^{0,2,1}}. This polytope is equivalent to the rectified 5-cell. The next two, {3^{1,2,1}} and {3^{2,2,1}}, have been covered already. The first of these is the 5-demicube, and the second is discussed above and is a member of the {3^{2,b,1}} family.

The first distinct polytope is the {3^{3,2,1}} polytope which is made up of {3^{2,2,1}}'s and 6-orthoplexes. The pattern continues until, as before, the {3^{5,2,1}} polytope is a Euclidean eight dimensional tessellation with infinite {3^{4,2,1}}'s and 8-orthoplexes as facets. As before, {3^{6,2,1}} cannot exist as a result.



The symmetric graph of the eight-dimensional (seven-dimensional elliptic) {3^{4,2,1}} polytope.

No other major families of quasi-regular polytopes exist, because all of the possibilities of recursive dependance, i.e. being based on the previous polytope and the simplex, or the orthoplex and simplex, etc. None involve square faces or any n-cubes at all, except of course, the n-cubes themselves, and are all based on triangles. However, there are many other uniform polytopes that may be obtained by operations on the three regular polytopes in n-dimensions. It has been discussed numerous times in the previous posts that truncation, or the slicing of vertices, and cantellation, or the slicing of edges. Also, in the previous part of this post, the idea of the slicing of cells in four dimensions and up, called runcination, was discussed. These operations can be extended, however, to the slicing of any i-face, and there are more possibilities in every dimension. For example, in five dimensions, the sterication operator, or slicing of 4-faces with respect to the fifth dimension, can be added to the four other operators, and any combination of the five can be applied. By the sixth dimension, there are hundreds of possibilities with the addition of the pentellation operator, and soon there are way too many to keep track of. Names of the polytopes also become unwieldy, as the combination of truncation, cantellation, sterication, and pentellation on a 6-simplex is named the pentistericantitruncated 6-simplex. Due to their diverse natrue, the uniform polytopes in seven dimensions and up have not been properly classified.

With any number of dimensions, and an infinity of possibilities in each, the world of polytopes is a limitless, beautiful branch of mathematics that defines much that we see in nature and what we find in abstraction.

Sources: Regular Polytopes by H.S.M. Coxeter, and various wikipedia titles: List of Regular Polytopes, k21 familiy, Uniform polyteron, Uniform polypeton, Truncation (geometry), etc.

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