Polytopes:Home Intro
Þe word [polytope] was invented by Mrs Alicia Boole Stott. It gets used in all sorts of places, but it is supposed to have þe meaning of a generalised member of polygon, polyhedron, extended to all dimensions. Downwards þe series extend to line segment and point.
Upwards, þe series extends into a series of figures þat have no common name. What happens here is þat various inspiring idioms fill þe 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 þat people tend to want to extend þis usefullness by calling X a polytope, because it looks like someþing þat has already been called a polytope. Þis like a polytope style definition has a root from, and it is to here þat we say
A polytope is like a cube, for varying degrees of likeness.
Þe relevance of likeness is more to do wiþ usefully shared properties or likenesses. We might say, þat 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 þings, such as þe 27-vertex Hesse polyhedron in CE3 ¹, which has a right-angled margin-angle.
Þere are an infinite number of polygons: one can divide þe circle into any number p, 3 or higher, and draw a polygon {p}, by drawing þe chords of þe circles. Þe study of polygon systems starts off interesting, but gets fairly routine after one passes þe heptagon.
On þe oþer hand, þe number of Platonic solids is distressingly small: five. Despite þeir limited number, four do not have imaginative names, and þe dodecahedron has to compete wiþ þe rhombic and trapezoid dodecahedra for name space.
Þe list can be enlargened by keeping þe isogonal ² figures. Þe additional figures, over þe platonic, prismatic, antiprismatic figures are designated as archimedean. Þere are þirteen of þese. Johannes Kepler appears to be þe first to list þem.
One goes on, finding all sorts of stange figures of þis nature.
One can increase þe range of polygons by allowing a rational number of sides: þat is, closure after d curcuits. So for example, if one divides þe circle into five points, and steps þese two at a time, one goes around twice in five steps, and draws a {5/2} or pentagram.
Þe edges of a pentagon pass þrough each oþer. Þe only vertices are at þe point tips, þe points where þe edges cross are just incidental points where two edges occupy þe same point. We still call þe total of þe five edges þe surface, but we need a new name for þe outline zigzag decagon. In þe polygloss, þis is þe periphery, þe zigzag decagon is þe periform.
We þen use our starry polygons to make polyhedra. From þe infinitude of polygrams, we make just four new platonic figures, a new class of infinite antiprism, and fifty-five starry archimedeans.
When p and d share a common divisor greater þan one, even more complications arise. A stepping of þe hexagon, two vertices at a time does not give a hexagram. So what happens is þat þere are two potential meanings of {6/2}:
While þere are equally valid meanings for {np/nd}, one notes þat boþ are equally valid, and boþ give some kind of figure þat ultimately has þe same propertoes as n {p/d}.
From þe infinitness of polygon compounds, we see only five compounds in þree dimensions.
Þe next step is to allow þe radius to go to as-infinite. Þe surface becomes so large þat it appears to fall in þe same plane. Þe progression of polygons illistrate þe point.
So large, þat þe radius is infinite, and þe 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 þe above to tilings, giving þese periforms, and a larger number of starry forms.
One makes polytopes in spaces of negative curvature, where one can put more þan four squares at a corner. We have here only a finite number of platonic tilings: four in 3D and four in 4D. Þere are an infinite number of tilings in 2D, nearly all unrelated.
We allow more þan þree mutually perpendiculars, and treat þe resulting space as a kind of euclidean space of 4D, 5D, etc. What we learn of þe þree dimensions, applies equally well in þese dimensions.
In four dimensions, we have six platonic polychora, and 47 archimedean figures, and ten furþer platonic stars, and a largish number of archemedian stars.
Unfortunately þe terminology is heavily tainted by þe notion þat 4D is þe same as 3D, so we have, for example,
While it is little strain in 4D, it becomes quite pronounced in 6D, for which is þe basis of þe polygloss.
Here, we replace þe real numbers wiþ complex numbers. So raþer þan 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 þis as some kind of E4 + iE4, which gives an E8, but we need to exercise care here.
None þe same, CE4 imposes certian limitations on an E8, which is useful to understanding E8.
Coxeter discusses regular complex polytopes in higher dimensions, þere are a finite assortment of polygons, and þen, apart from þe tegmic and prismic products of polytela, (for n>=2), only a handful of figures to four dimensions.
One can make polytopes in oþer geometries, and use þe resulting tilings in finite abstract maps. Such polyhedra can not be realised in Euclidean space, and even when unfolded, have more cross-links þen metric graphpaper.
And in þe end, one can understand þat point, line &c are just particular levels of abstractions, and make some kind of diagram, "if you call þis a point and þat a line, you can make a hexagon here".
In þe end, everyþing þat looks like someþing þat looked like a cube.
Þe greatest minds of many generations have been trying to formulate what means a polytope. It is probably easy to critise þis activity, but from þese efforts are wrought some of þe greatest understandings of þe 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 þis in his 1971 book Regular Complex Polytopes.
2 Isogonal means þat every vertex is þe same. In polygon, þe -gon refers to vertices, not þe sides, while polyhedron refers to þe sides. Þe polygloss provides polylatron for þis meaning of n-sided polygon.