Welcome to the all new polyteron web site. This web site will focus on the uniform polytopes of five dimensions. There are currently 1292 known uniform polytera (used to be 1251) plus many fissaries, 1774 prisms, and infinities of duoprisms. In 2013, I discovered a new regiment - Hosiap. Hosiap is the blend of 6 hins (demipenteracts). Last updated May 10, 2021.

**2021 May 10, 2021** - Updated count from 1293 to 1292, due to fissary member in sart regiment.

**2020 February 29** - Added scaliform categories and revealed the verfpages.

**2019 September 23** - Verfpages are now available. These are large picture files showing verfs of many of the polytera. I currently have 120 pages made with several polyteron verfs on each page.

2014 April 22 - *Largest update for the polyteron site!* Added updates, a new regiment is found! (hosiap from category 19), added verfs and fields of sections to the category pages. Added categories 4, 9, 10, 13, and 19. Also added a quicref page to display potential facets (uniform polychora with lower symmetry groups).

2013 September 12 - Created Polyteron of the Day website.

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 . . . Verfpages . . . On to the Sixth Dimension

**Above shows a field of "poke-sections" of the noble uniform polyteron Nat which is from the nit regiment - its terons (facets) are 32 garpops, it was rendered using Pov-Ray .**

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). Norman Johnson prefers that if 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). I decided to relax this restriction to allow the compound of 3 tesseracts (gico) to show up as a combo-facet, for it would of been rigidly locked, two tesses share 8 vertices in common which are arranged like the vertices of hex. My new version of this restriction would be that any n-facet regiment can only be activated once by either a true polytope or a pure compound in the regiment (as a combofacet), or a fissary if they are included in the count. This means that if the facet regiment is ico, then the compound gico is allowed - but the ico-gico combo is not since it will activate the ico regiment twice instead of once.

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 **|||()**.

**Duocyclinder** - 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 **|()()**.

**Spherisquare** - 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 its curved side and has four flat sides. 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 **()(|)**.

**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 **|(||)**.

**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 **(|||)**.

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 1292 (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 recently hit a big milestone where I have rendered cross sections in all known regiments (except for prismatics and the miscellaneous category). The Wythoffian regiments (categories 1-18) have symbols of the following forms where any of the o's can be either an o or an x and the x's stay as is: ooooo (hixics), oo8o (hinnics), oooo'o (pentics), ooox"x (star pentics), oo(o'x"x) (blocky goccoics), o(o'x"x)o (medial goccoics), and (o'x"x)oo (spiky goccoics). 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. They can be viewed in this quicref page as a series of cross sections.

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** - *New pics, April 2014* (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** - *New pics, April 2014* (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** - *New pics, April 2014* (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** - *New, April 2014* (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 - 377) 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 92 members. These are the ooxox cases.

**Category 7: Birhombates** - (Polytera 378 - 447) 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 448 - 680) There are five regiments, four with 37 members and one with 85, 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** - *New, April 2014* (Polytera 681 - 721) 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** - *New, April 2014* (Polytera 722 - 747) These 26 polytera include the great rhombates (ooxxx), great prismates (oxxxx), great birhombates (oxxxo), and great cellates (xxxxx).

**Category 11: Sphenopyriverts** - (Polytera 748 - 786) 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 787 - 828) There are six regiments of seven, their verfs are podium and antipodium pyramids. These are the xooxx cases.

**Category 13: Prismatorhombates and Kin** - *New, April 2014* (Polytera 829 - 870) 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 871 - 942) There are three regiments, one with 18 members (card) and two with 27 (carnit and wacbinant). 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 943 - 998) 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 999 - 1124) 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 1125 - 1154) These regiments have 15 members each, their verfs are skewed wedge pyramids.

**Category 18: Skatbacadint Regiment** - (Polytera 1155 - 1287) This huge regiment with 133 troops is the ultimate skewed regiment in 5-D, its verf is a skewed antifastegium, its symbol is xo(ox"x').

**Category 19: Miscellaneous** - *New, April 2014* (Polytera 1288 - 1292) This category includes the two 5-D Johnson antiprisms and the newly found hosiap regiment which has 3 uniform members as well as some scaliform and compound members. Hopefully more will be found here.

Scaliforms are known in five dimensions as well, here is a list of known categories, not many categories known yet.

**Category SA: Scaliform Prisms** - These are the prisms of the 4-D scaliforms, this category has several sub-categories.

**Category SB: Disphenoids** - These are pyramid products of two equal polygons, polygons must have an angle less than 90 degrees, so all of these except for the trigonal one (hix itself) are starry. This category is infinite.

**Category SC: Duoantifastegiums** - These strange wedge-like objects have antifastegia as its sides. This category is infinite.

**Category S1: Alterprisms** - similar to tutcup, these have top and bottom faces flipped in alternate orientations, they work on polychora with pennic, icoic, and demitessic symmetry.

**Category S2: Uniform Party Crashers** - these are found in uniform regiments like the five in the hosiap regiment or are formed by cutting vertices off uniforms.

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.

**Polyteron of the Day** - Here are cross sections of many uniform polytera, a sort of warm up for the future of this webpage.

**Verfpages** - This page leads to high resolution pics of the polyteron vertex figures.

**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.

**Uniform Polypeta** - This web site deals with the six 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 house this website. Without the domain there wouldn't of been enough room on my site to store all the pics. *

*To God, the Source of Truth, be the glory. *

Page created by Jonathan Bowers, © 2008-2020 e-mail = hedrondude at suddenlink dot net