# -: Like a cube :-

Polytopes:Home Intro

## Polytopes

The word [polytope] was invented by Mrs Alicia Boole Stott. It gets used in all sorts of places, but it is supposed to have the meaning of a generalised member of polygon, polyhedron, extended to all dimensions. Downwards the series extend to line segment and point.

Upwards, the series extends into a series of figures that have no common name. What happens here is that various inspiring idioms fill the gap, such as polycell, polyhedroid, and even to a stranger array of names.

Polytopes are facinatingly useful in all sorts of fields, apart from being also very pretty. What happens, is that people tend to want to extend this usefullness by calling X a polytope, because it looks like something that has already been called a polytope. This like a polytope style definition has a root from, and it is to here that we say

A polytope is like a cube, for varying degrees of likeness.

The relevance of likeness is more to do with usefully shared properties or likenesses. We might say, that a hexagonal tiling is like a cube, because it is made of polygons, fitted to completely cover a surface.

One gets into more and more abstract things, such as the 27-vertex Hesse polyhedron in CE3 ¹, which has a right-angled margin-angle.

## Polygons and Polyhedra

There are an infinite number of polygons: one can divide the circle into any number p, 3 or higher, and draw a polygon {p}, by drawing the chords of the circles. The study of polygon systems starts off interesting, but gets fairly routine after one passes the heptagon.

On the other hand, the number of Platonic solids is distressingly small: five. Despite their limited number, four do not have imaginative names, and the dodecahedron has to compete with the rhombic and trapezoid dodecahedra for name space.

The list can be enlargened by keeping the isogonal ² figures. The additional figures, over the platonic, prismatic, antiprismatic figures are designated as archimedean. There are thirteen of these. Johannes Kepler appears to be the first to list them.

One goes on, finding all sorts of stange figures of this nature.

## Starry figures

One can increase the range of polygons by allowing a rational number of sides: that is, closure after d curcuits. So for example, if one divides the circle into five points, and steps these two at a time, one goes around twice in five steps, and draws a {5/2} or pentagram.

The edges of a pentagon pass through each other. The only vertices are at the point tips, the points where the edges cross are just incidental points where two edges occupy the same point. We still call the total of the five edges the surface, but we need a new name for the outline zigzag decagon. In the polygloss, this is the periphery, the zigzag decagon is the periform.

We then use our starry polygons to make polyhedra. From the infinitude of polygrams, we make just four new platonic figures, a new class of infinite antiprism, and fifty-five starry archimedeans.

## Compounds

When p and d share a common divisor greater than one, even more complications arise. A stepping of the hexagon, two vertices at a time does not give a hexagram. So what happens is that there are two potential meanings of {6/2}:

• Starry, made by linking 0-2, 1-3, 2-4, 3-5, 4-0, 5-1, ie x -> x+2.

While there are equally valid meanings for {np/nd}, one notes that both are equally valid, and both give some kind of figure that ultimately has the same propertoes as n {p/d}.

From the infinitness of polygon compounds, we see only five compounds in three dimensions.

## Tilings

The next step is to allow the radius to go to as-infinite. The surface becomes so large that it appears to fall in the same plane. The progression of polygons illistrate the point.

So large, that the radius is infinite, and the surface appears as a tiling in a lower dimension. We convert an infinite polygon to an apeirogon, a tiling of line-segments in a line, and an infinite polyhedron into an apeirohedron, or tiling of polygons.

We apply all of the above to tilings, giving these periforms, and a larger number of starry forms.

• 3 platonic apeirohedra
• 11 archemedian apeirohedra

## Hyperbolic Space

One makes polytopes in spaces of negative curvature, where one can put more than four squares at a corner. We have here only a finite number of platonic tilings: four in 3D and four in 4D. There are an infinite number of tilings in 2D, nearly all unrelated.

## Higher Dimensions

We allow more than three mutually perpendiculars, and treat the resulting space as a kind of euclidean space of 4D, 5D, etc. What we learn of the three dimensions, applies equally well in these dimensions.

In four dimensions, we have six platonic polychora, and 47 archimedean figures, and ten further platonic stars, and a largish number of archemedian stars.

Unfortunately the terminology is heavily tainted by the notion that 4D is the same as 3D, so we have, for example,

• dihedral angles which have nothing to do with hedra. We use margin angles
• cells being both 3D and N D
• faces being both 2D and N-1 D

While it is little strain in 4D, it becomes quite pronounced in 6D, for which is the basis of the polygloss.

## Complex Space

Here, we replace the real numbers with complex numbers. So rather than considering a E4 (w,x,y,z) in reals, we make a CE4 (w,x,y,z) in complex numbers. Of course, we can write this as some kind of E4 + iE4, which gives an E8, but we need to exercise care here.

None the same, CE4 imposes certian limitations on an E8, which is useful to understanding E8.

Coxeter discusses regular complex polytopes in higher dimensions, there are a finite assortment of polygons, and then, apart from the tegmic and prismic products of polytela, (for n>=2), only a handful of figures to four dimensions.

## Other geometries

One can make polytopes in other geometries, and use the resulting tilings in finite abstract maps. Such polyhedra can not be realised in Euclidean space, and even when unfolded, have more cross-links then metric graphpaper.

## Incidence maps

And in the end, one can understand that point, line &c are just particular levels of abstractions, and make some kind of diagram, "if you call this a point and that a line, you can make a hexagon here".

## And in the end

In the end, everything that looks like something that looked like a cube.

The greatest minds of many generations have been trying to formulate what means a polytope. It is probably easy to critise this activity, but from these efforts are wrought some of the greatest understandings of the world around us.

1 CE3 is complex-euclidean 3-dimensional space, which might be represented by (x,y,z), where x, y, and z are complex numbers. Coxeter described this in his 1971 book Regular Complex Polytopes.

2 Isogonal means that every vertex is the same. In polygon, the -gon refers to vertices, not the sides, while polyhedron refers to the sides. The polygloss provides polylatron for this meaning of n-sided polygon.