Uniform Polytera and Other Five Dimensional Shapes


Welcome to the all new polyteron web site. This web site will focus on the uniform polytopes of five dimensions. Uniform polyteron count at 1251 plus many fissaries, 1774 prisms, and infinities of duoprisms. Last updated June 10, 2012.

2012 June 10 - Added basic shapes, polyteron count freed of fissaries, various minor changes.

2008 August 27 - Page created.


Back to the Fourth Dimension . . . Home Page . . . Glossary . . . On to the Sixth Dimension


Oifpeyax

Above shows a planar grid of "poke-sections" of the polyteron oifpeyax(OIF pee yaks) which is one of the fastegiumverts - its terons are 6 pinnips (blue), 6 pinnipdips (yellow), 15 ohopes (green), and 20 triddips (red), it was rendered using Pov-Ray .


Definition of Uniform Polyteron

A polyteron is uniform if its vertices are transitive (all alike in an isogonal sort of way) and all of it's facets (called terons) are uniform polychora.

A polyteron is a five dimensional polytope, where a polytope must be monal, dyadic, and properly connected. Monal means that every element is represented only once (two vertices cant be in the same place), dyadic means that if you take an n dimensional element and an n-2 dimensional element, then there are either 0 or 2 n-1 dimensional elements adjacent to both (in other words, edges have two vertices, only two facets meet at a ridge, and this is true for all of it's elements and element-figures (vertex figure and the like)). Properly connected means that it nor any of it's elements or element-figures are compounds (even though the compound of 5 cubes can act as a cell, it is actually five cells which are cubes). Properly connected also means that if any two n dimensional elements are in the same n space, then all of their common elements together must be limited to an n-1 dimensional space (in other words, if two cells are in the same realm, they are not rigidly locked together by common elements, like an ike-gad combo would be (ike and gad would share all 30 edges which is a 3-D arrangement instead of a 2-D or less arrangement, this ike-gad combo causes the object to be a coinciding case). Now consider 5 cubes in the same realm (like in the 5-cube compound), if you take any two of the cubes, they have no element in common (well technically they share the -1 dimensional nulloid in common which is sometimes called the empty polytope) - so therefore they are not rigidly locked like the ike-gad case.

There are also a few non-uniform scaliform polytera known, but the search for these has yet to be underway.

Basic Five Dimensional Shapes

These seven shapes can be considered as the basic 5-D shapes, they are either flat in each dimension or they join them in a uniform curve. The seven shapes are:

Penteract - which can be generalized as a variety of pentablocks when the dimensions are of different length. This shape would be the basic building block in five dimensional space. It could be represented by |||||.

Tessinder - This is the cross product of a disk and a cube (or 3-D block for variants). It has six cubinders as sides and one curved side. It is flat in three dimensions, the other two form a curve and it has rollability of 1. It could be represented as |||().

Ducyclinder - This is the prism of the duocylinder. It has two flat sides and two curved sides with rollability of 1. It could be represented as |()().

Cusphinder - It has two flat dimension and three curved ones. It has rollability of 2 on curved sides. This is the cross product of the sphere and a square. It could roll like a ball on four sides. It could be represented as ||(|).

Glominder - It has one flat dimension and four curved ones. It has rollability of 3 on the curved side. This is the prism of the glome and may be the can shape of 5 dimensions. It could roll like a glome. It could be represented as |(||).

Cyclosphere - It has two dimension that curve in a cycle and three that curve in a ball. It has rollability of 1 and 2 depending on which of the two sides it sits on. This is the cross product of the sphere and a circle. It could roll like a ball and like a circle. It could be represented as ()(|).

Pentorb - This is the five dimensional sphere, can also be called pentasphere. It has rollability of 5, for all five dimensions are curved. It could be represented as (|||).


Uniform Polyteron Categories

I started my main search for the star uniform polytera in 2000, finding well over 1000 which included fissaries and coincidic ones - many of them has yet to be named. The convex ones were likely known by H.S.M. Coxeter, since they can easily be derived using Dynkin diagrams. The current count of the uniform polytera is 1251 (which excludes infinite categories and the gigantic prism category, and of course undiscovered ones) - Before eliminating the fissaries, I had a count of 1410. I would suggest visitors to get familiar with the uniform polychora with pennic and tessic symmetry, as well as the prismattic ones with tepe, ope, triddip, and tisdip symmetry from categories 19 and A from the polychoron site.

Polyteron pics can be viewed by simply clicking on the polyteron's name within each category file.

Here is a list of the 19 categories plus the four prismattic categories.

Category A: Prisms - There are 1774 prisms - a prism for every uniform polychoron (excluding the tesseract and those of category 19, the prisms), their verfs are pyramids of the polychoron's verf.

Category B: Duoprisms - There are 75 infinite sets of duoprisms. For every uniform polyhedron A there is a series of duoprisms AxB, where B can be any regular polygon. The cube-square duoprism is the penteract and belongs to category 1. Their verfs are dyad disphenoids of the verf of the polyhedron component.

Category C: Antiprism-Polygon Duoprisms - This is the doubly infinite set of the cross products AxB, where A is a polygonal antiprism and B is a polygon. Their verfs are trapezoid (normal and crossed)-dyad disphenoids.

Category D: Duoprism Prisms - for every polygon-polygon duoprism (polychoron category A), there is a prism of it here. This category is doubly infinite, there verfs are disphenoid pyramids.

Category 1: Primaries - (Polytera 1 - 12) These are the 3 regular polytera (Hix (oooox), Pent (oooo'x), and Tac (xooo'o)) and the demipenteract (Hin - (ooo9)) along with their regiments. Verfs are regular or semiregular polychora and facetings.

Category 2: Truncates - (Polytera 13 - 24) These are the truncates (oooxx) and the bitruncates (ooxxo). Verfs are pyramids of tet, oct, and trip (and facetings) or regular polygon-dyad disphenoids.

Category 3: Rectates - (Polytera 25 - 47) These are the rectified (oooxo) and birectified (ooxoo) hix and the rectified penteract (ooox'o) and their regiments. These three regiments (rix, dot, and rin) have 7, 5, and 11 members respectively. There verfs are facetings of tet prisms (for rix and rin) and facetings of triddip (for dot).

Category 4: Rat Regiment - (Polytera 48 - 91) Rat (oxoo'o) is the rectified pentacross (triacontiditeron), it has 44 members. It's verf is an ope (octahedral prism).

Category 5: Nit Regiment - (Polytera 92 - 240) Nit (ooxo'o) is the birectified penteract which has a tisdip (trigonal square duoprism) verf. There are 149 members in this regiment with many fissaries, if we counted fissaries there would be 288 known members. This regiment looks like it is the largest.

Category 6: Sphenoverts - (Polytera 241 - 378) There are four regiments, the first three (sarx, sirn, and wavinant) have "trip-wedge" verfs - which is a wedge with a trigon prism base - they have 15 members each, the other regiment is sart which has a cubic wedge verf, which has 93 members. These are the ooxox cases.

Category 7: Birhombates - (Polytera 379 - 448) These have verfs shaped like "duowedges" and their facetings. There are two regiments - sibrid (with 23 members) and sibrant (with 47 members). These are the oxoxo cases.

Category 8: Fastegiumverts - (Polytera 449 - 668) There are five regiments, four with 37 members and one with 72, their verfs are facetings of trigonal antifastegiums and trigonal fastegiums - a fastegium contains a podium-prism-podium trigonic structure, while the antifastegium has an antipodium-prism-antipodium trigonic structure. These polytera are sure to be attractive looking. These are the oxoox cases.

Category 9: Podiumverts - (Polytera 669 - 709) There are three regiments, two have 15 members and one has 11 - their verfs look like tetrahedron podiums and tetrahedral antipodiums and their facetings. These are the xooox cases.

Category 10: Greater Truncates - (Polytera 710 - 735) These 26 polytera include the great rhombates (ooxxx), great prismates (oxxxx), great birhombates (oxxxo), and great cellates (xxxxx).

Category 11: Sphenopyriverts - (Polytera 736 - 774) Also called prismatotruncates. There are four regiments of 7 and one of 11, their verfs are wedge pyramids. These are the oxoxx cases.

Category 12: Podipyriverts - (Polytera 775 - 816) There are six regiments of seven, their verfs are podium and antipodium pyramids. These are the xooxx cases.

Category 13: Prismatorhombates and Kin - (Polytera 817 - 858) There are 14 regiments of three. These have trapezoid disphenoid, and trapezoid pyramid pyramid verfs. The prismatorhombates are the oxxox cases, also here are the xxxox and the xxoxx cases.

Category 14: Antisphenoverts - (Polytera 859 - 930) There are three regiments, one with 18 members (card) and two with 27 (carnit and wacbinant), each regiment also has 9 fissaries. Their verfs are best described as two wedges stuck together at the bases, but turned 90 degrees to each other and folded into 4th dimension - and their facetings of course. These are the xoxox cases.

Category 15: Siphin Regiment - (Polytera 931 - 986) This regiment has 56 members plus 20 fissaries. Siphin is the small prismated hin - its verf is a tet || oct - which is like a tet cupola (looks like a stretched rap). Its symbol is xoo9.

Category 16: Skivbadant and Gikvacadint Regiments - (Polytera 987 - 1112) Both of these regiments have 63 members each and form what Norman Johnson calls a "battalion". Their verfs are skewed duowedges.

Category 17: Sibacadint and Gidacadint Regiments - (Polytera 1113 - 1142) These regiments have 15 members each, their verfs are skewed wedge pyramids.

Category 18: Skatbacadint Regiment - (Polytera 1143 - 1249) This huge regiment with 107 troops is the ultimate skewed regiment in 5-D, its verf is a skewed antifastegium, its symbol is xo(ox"x').

Category 19: Miscellaneous - (Polytera 1250 - 1251) The only two miscellaneous ones known are the Johnson antiprisms of five dimensions, hopefully more polytera will be found in this category.


Following is a list of some of my other polytope related pages.

Home Page - This is my home page, it has links to some of my non-polytope pages such as Array Notation, Infinity Scrapers, Elements, Existence of God, etc.

Polytwisters - This page lists the 194 known uniform polytwisters with pics of many of them (and many more pics to come). Polytwisters are like rollable twisted "polyhedra" in 4 dimensions. They dont have vertices, they have equators (rings). Their sides aren't flat like polyhedra, they are cylindrical. If placed on a 4-D table, it could roll no matter which way you set it.

Uniform Polychora - This web site deals with the four dimensional stuff.

Polytopes of Various Dimensions - This page lists the names of what polytopes are called in various dimensions, all the way up to a tridecillion dimensions! Example: in 10-D they are called polyxenna.

Dice of the Dimensions - This page describes the fair dice of variant dimensions, including those with curved sides.

Regiments - Here's something we've all been waiting for, a list of the various regiments with their members listed as well. I'm working on up the dimensions (at dimension 6 at the moment) and hope to get to dimension 8.

Special thanks to Andrew Weimholt, who has let me use his polytope.net domain to store my polyteron pics. Without the domain there wouldn't of been enough room on my site to store all the pics.


Polyhedron Dude
Page created by Jonathan Bowers, © 2008-2012
e-mail = hedrondude at suddenlink dot net