Welcome to the all new polypeton web site. This web site will focus on the uniform polytopes of six dimensions. This count is still underway and I hope to have all of the typical ones enumerated soon. The main symmetries are hoppic (simplex), axic (hexeract), haxic (demihexeract), and jakic (E6 polytope Jak). Hoppic and jakic can be doubled to get feic and moic symmetry. All of the typical hoppics (includes feics) are now counted, there are 916 of them. So far only the largest regiments have yet to be counted of the other symmetries and the total count is already breaking 10,328 - which only includes the fully counted and defissed non-prismattic regiments! Chosidap was discovered Dec. 16, 2016 along with higher dimensional versions of hosiap. In October 2019, I found 135 new members in the hax regiment along with new compounds and fissaries there, increasing its size by an order of magnitude. Below is a breakdown of the count along with uncounted regiments listed after them:

Hopic - 916

Axic - 5832 - brox, brag, spog, siborg, sobpoxog, gipkix, sipkix

Haxic - 1268 - sophax

Jakic - 2291 - rojak, ram, sirjak, shopjak, spojak, spam, sabrim

Misc - 21

Last updated **February 29, 2020**.

**2020 February 29** - Category 1 added with new discoveries.

2018 June 9 - Chosidap update.

2016 February 1 - Updated some counts.

2014 April 22 - Page released online.

2014 January 25 - Page created.

Back to the Fifth Dimension . . . Home Page . . . Glossary . . . On to the Seventh Dimension

A polypeton is **uniform** if its vertices are transitive (all alike in an isogonal sort of way) and all of it's facets (called petons) are uniform polytera.

A **polypeton** is a six 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. 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 6-D shapes, they are either flat in each dimension or they join them in a uniform curve. The seven shapes are:

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

**Pentinder** - This is the cross product of a disk and a tesseract. It has 8 tessinders as sides and one curved side. It is flat in four dimensions, the other two form a curve and it has rollability of 1. It could be represented as **||||()**.

**Ducyclosquare** - This is the cross product of the duocylinder and a square, also the cross product of two cylinders. It has four flat sides and two curved sides with rollability of 1. It could be represented as **||()()**.

**Triocylinder** - This is the cross product of three disks. It has three curved sides with rollability of 1. It could be represented as **()()()**.

**Sphericube** - It has three flat dimensions and three curved ones. It has rollability of 2 on curved side. This is the cross product of the sphere and a cube. It could roll like a ball on its curved side and it has six flat sides. It could be represented as **|||(|)**.

**Cylindisphere** - It has one flat dimension and two that curve in a cycle and three that curve in a ball. It has rollability of 2 on one of the curved sides, and rollability of one on the other. This is the cross product of the sphere and a cylinder. It has two flat sides. It could be represented as **|()(|)**.

**Duospherinder** - It has six curved dimensions. It has rollability of 2 on both curved sides. This is the cross product of two spheres. It could roll like a ball no matter how it is set. It could be represented as **(|)(|)**.

**Glomosquare** - It has two flat dimensions and four curved ones. It has rollability of 3 on the curved side. This is the cross product of the glome and a square. It could roll like a glome on the curved side and set flat on four sides. It could be represented as **||(||)**.

**Cycloglome** - It has two dimension that curve in a cycle and four that curve in a glome. It has rollability of 1 and 3 depending on which of the two sides it sits on. This is the cross product of the glome and a circle. It could roll like a glome and like a circle. It could be represented as **()(||)**.

**Pentorbinder** - It has one flat dimension and five curved ones. It has rollability of 4 on the curved side. This is the prism of the pentorb and may be the can shape of 6 dimensions. It could roll like a pentorb. It could be represented as **|(|||)**.

**Hexorb** - This is the six dimensional sphere, can also be called hexasphere. It has rollability of 6, for all six dimensions are curved. It could be represented as **(||||)**.

Currently I'm searching out the uniform polypeta and hope to completely enumerate the typical ones and find as many atypical ones as possible. Over 8000 uniform polypeta are currently known and I suspect the final count of the typical non-prismatic uniforms (typicals are those that are not snubs, swirlprisms, or other weird sort) will be a five digit number. Known atypical cases include those like iquipadah and those from category 38 and there are likely many others that are waiting to be found, could there be something like the idcossids but with jak or mo symmetry? - who knows! The Wythoffian regiments (categories 1-37) 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: oooooo (hoppics), ooo8o (haxics), ooooo'o (axics), oooox"x (star axics), ooo(o'x"x) (blocky goccoics), oo(o'x"x)o (medial blocky goccoics), o(o'x"x)oo (medial spiky goccoics), (o'x"x)ooo (spiky goccoics), and oo8oo (jakics).

Here is a list of the 38 categories plus the 8 prismattic categories.

**Category A: Prisms** - There are 1292 prisms - a prism for every non-prismatic uniform polyteron (excluding the penteract which leads to the hexeract), their verfs are pyramids of the polyteron's verf.

**Category B: Chorigonal Duoprisms** - For all of the non-prismatic uniform polychora (1774 so far), there is an infinite set of "chorigonal" duoprisms. Their verfs are dyad disphenoids of the verf of the polychoron component.

**Category C: Hedral Duoprisms** - This set contains 2849 polyhedron-polyhedron duoprisms. They are the cross products AxB, where A and B are uniform polyhedra. Their verfs are X-Y disphenoids, where X and Y are the verfs of A and B.

**Category D: Hedrigonal Dippips** - This infinite set are simply the prisms of polyteron category B. The verfs disphenoid pyramids of the polyhedron verfs.

**Category E: Hedral Antiprism Duoprisms** - This infinite set are the cross product of the 3-D antiprisms and the uniform polyhedra. Their verfs are a-b dispheonids where a is the verf of an antiprism and b is the verf of a uniform polyhedron.

**Category F: Antiprism Duoprisms** - This infinite set are the cross product of two 3-D antiprisms. Their verfs are trapezoid-trapezoid disphenoids and their crossed versions.

**Category G: Antiprismigonal Dippips** - This infinite set are the prisms of polyteron castegory C. Their verfs are disphenoid pyramids of trapezoids and their crossed versions.

**Category H: Trioprisms** - These are the cross product of three polygons, there are w^{3} (w denotes omega) of these. Their verfs are "trisphenoids".

**Category 1: Primaries** - *New for 2020*(Polypeta 1 - 312) These are the 3 regular polypeta (hop (ooooox), ax (ooooo'x), and gee (xoooo'o)), the demihexeract (hax (ooo6o)), the E6 polytope (jak - oo8ox), and 1_{22} (mo - oo6oo) along with their regiments. Verfs are regular or semiregular polypeta and facetings.

**Category 2: Truncates** - (Polypeta 313 - 345) These 33 polypeta are the truncates (ooooxx), the bitruncates (oooxxo), and the tritruncates (ooxxoo). Verfs are pyramids of pen, hex, tepe, rap, and triddip (and facetings), (tet, oct, or trip)-dyad disphenoids, and trigon-trigon and trigon-square disphenoids.

**Category 3: Rectates** - (Polypeta 346 - 421) These are the rectified (ooooxo) and birectified (oooxoo) hop and the rectified ax and gee (oooox'o and oxooo'o) and their regiments. These four regiments (ril, bril, rax, and rag) have 7, 18, 19, and 132 members respectively. Their verfs are facetings of pen prisms (for ril and rax), facetings of trigon-tetrahedral duoprism (for bril) and facetings of hex prism (for rag).

**Category 4: Brox Regiment** - (Polypeta 422 - ??) Brox (oooxo'o) is the birectified hexeract. It likely has hundreds if not thousands of members. Its verf is an squatet (square-tetrahedral duoprism).

**Category 5: Brag Regiment** - (Polypeta ?? - ??) Brag (ooxoo'o) is the birectified hexacross. It will likely have between 500-800 members when defissed. Its verf is a troct (trigon-octahedral duoprism).

**Category 6: Jak Rectates** - (Polypeta ?? - ??) These are the rojak (oo8xo) and the ram (oo9oo) regiments. These two colossal regiments likely have thousands of members each! Rojak's verf is a rappip (rap prism) and ram's verf is a tratrip (triddip prism).

**Category 7: Tet Sphenoverts** - (Polypeta ??+1 - ??+93) These are the cantellated simplex (sril - oooxox), cantellated hexeract (srox - oooxo'x), and the quasicantellated hexeract (wavaxixog - (o'x"x)ooo = Gooo for short, G for gocco) regiments, each of these have 31 members and their verfs looks like tepe wedges (tet prism wedge).

**Category 8: Grand Sphenoverts** - (Polypeta ?? - ??) There are the cantellated hexacross (srog - xoxoo'o) and the cantellated jak (sirjak - oo9ox) regiments. The srog regiment has 290 members plus two pure compounds and 63 fissaries and its verf is an ope wedge (oct prism wedge). Sirjak likely has thousands or tens of thousands of members, its verf is a tisdip wedge (trip prism wedge).

**Category 9: Birhombates** - (Polypeta ?? - ??) These are the bicantellations of the three regulars along with their regiments. They are sabril (ooxoxo), saborx (ooxox'o), and siborg (oxoxo'o). Their verfs are shaped like trigon (first two) or square (siborg) duowedges and their facetings. Sabril has 63 members, saborx has 95 members, and siborg likely has around a thousand give or take.

**Category 10: Podiumverts** - (Polypeta ??+1 - ??+97) These three regiments are the expanded simplex (staf - xoooox), expanded hexeract (stoxog - xoooo'x), and the quasiexpanded hexeract (goxaxog - oooG) regiments. Their verfs look like pentachoron antipodiums (first two) and pentachoron podiums (goxaxog) and their facetings. Staf has 19 members, the other two have 39 each, where 31 each are typical members and the remaining are like iquipadah from 4-D.

**Category 11: Fastegiumverts** - (Polypeta ??+1 - ??+953) There are five regiments, four with 125 members (plus 11 fissaries) and one (scag) has 453 members. Their verfs are facetings of tetrahedral antifastegiums (first three) and tetrahedral fastegiums (last two). The five regiments are scal (oxooox), scox (oxooo'x), scag (xooox'o), gagdex (ooGo), and harjak (oo6xo).

**Category 12: Dippifastegiumverts** - (Polypeta ??+954 - ??) These are the runcinates and those with similar verfs. There are five regiments: spil (ooxoox), spox (ooxoo'x), spog (xooxo'o), gadoxog (oGoo), and sirhax (oox8x). Spog likely has thousands if not tens of thousands of members, its verf looks like a tisdip antifastegium (ox o'x || xo o'o). The others have 138 members plus a few fissaries each. Spil's verf is a triddip antifastegium (ox ox || xo oo), the other two have a triddip fastegium verf (ox ox || ox oo).

**Category 13: Duofastegiumverts** - (Polypeta ?? - ??) These three regiments have verfs that are facetings of a strange 5-D shape we can call the duoantifastegium (duofastegium for the last regiment), in Klitzing's atop notation it would be a ox x o || xo o x for the duoantifastegiums and ox x o || ox o x for the duofastegium. The three regiments are sibpof (oxooxo), sobpoxog (oxoox'o), and barm (ox8xo). Sibpof and barm have 143 members each, sobpoxog likely has several hundred members.

**Category 14: Cupoliverts** - (Polypeta ??+1 - ??+1933) These three regiments are the sochax (xoo8x), hejak (oo6ox), and trim (xo8ox). Their verfs are pen-rap anticupola (sochax), pen-rap cupola (hejak), and hex antiprism (trim). In atop notation, the verfs are ooox || oxoo, ooox || ooxo, and o6o || o9o respectively. Sochax and hejak has 688 members and trim has 557 members, all three have many fissaries.

**Category 15: Cupogiumverts** - (Polypeta ??+1934 - ??) These are the members of two colossal regiments (likely has thousands of members) - the sophax regiment (oxo8x) and the shopjak regiment (ox8ox). The verf of sophax is a tet-oct cupogium (cupola fastegium - aka oox x || oxo o). The verf of shopjak is an oct-tet pucogium ("pucola" fastegium - aka oxo x || oox o).

**Category 16: Greater Truncates** - (Polypeta ??+1 - ??+57) These 57 polypeta include the great rhombates (oooxxx), great prismates (ooxxxx), great birhombates (ooxxxo), great cellates (oxxxxx), great biprismates (oxxxxo), and great terates (xxxxxx) and anything with similar verfs which includes tet scalenes, oct scalenes, trig tettenes, square tettenes, pennenes (bilateral hixes), and hixenes (irregular hix).

**Category 17: Prismatotruncates** - (Polypeta ??+58 - ??+241) There are six regiments, their verfs are polygonal wedge pyramids or duowedge pyramid (pithax). These are the ooxoxx cases. The regiments are patal (ooxoxx - 15 members), potax (ooxox'x - 15 members), potag (xxoxo'o - 93 members), quoptax (ooxox"x - 15 members), pithax (oxo9x - 31 members), and shorjak (oo$ox - 15 members).

**Category 18: Prismatorhombates and Kin** - (Polypeta ??+242 - ??+343) There are many three (or six) member regiments here, their verfs are various trapezoid pyramid products. They include the ooxxox, oxxoxx, oxxxox, xoxxxx, and xxoxxx cases and similar.

**Category 19: Celliprismates and Kin** - (Polypeta ??+344 - ??+458) There are many seven (or five) member regiments here, their verfs are podium / antipodium disphenoids, or scalenes. They include the oxxoox, xooxxx, and xxooxx cases and similar.

**Category 20: Sphenic Pyriverts** - (Polypeta ??+459 - ??+550) There are twelve regiments with seven (or eleven) members. Their verfs are wedge disphenoids or wedge scalenes. These are the oxxoxo and oxoxxx cases.

**Category 21: Tericellates** - (Polypeta ??+551 - ??+640) These six regiments have 15 members each. These are the xxooox cases. Their verfs are tet podium / antipodium pyramids. The regiments are tocal (xxooox), tacog (xooox'x), tacox (xxooo'x), quotacog (xooox"x), gogradex (ooGx), hotjak (oo6xx).

**Category 22: Cellitruncates** - (Polypeta ??+641 - ??+910) There are six regiments, five with 37 members, one with 85 (catog). Their verfs are trigon fastegium / antifastegium pyramids. The regiments are catal (oxooxx), catax (oxoox'x), catog (xxoox'o), quactix (oxoox"x), cathix (xoo9x), and hopitjak (ox8xx).

**Category 23: Cellirhombates** - (Polypeta ??+911 - ??+1258) There are four regiments here, their verfs are "antiduowedge wedges". The regiments are cral (oxoxox), crax (oxoxo'x), crag (xoxox'o), and gapgodex (Goxo). Except for crag (which has 123 members), they have 75 members each.

**Category 24: Teraprismates** - (Polypeta ??+1259 - ??+2093) These six regiments have 139 members plus one fissary each. Their verfs are trigonal fastegium / antifastegium wedges. The regiments are topal (xoxoox), topag (xooxo'x), tapox (xoxoo'x), gocagdex (Goox), gopoxagox (oGox), and crohax (xox8x).

**Category 25: Antisphenopyriverts** - (Polypeta ??+2094 - ??+2246) These six regiments have 27 members each (except for patom which has 18). Their verfs are antiduowedge pyramids. The regiments are tocral (xoxoxx), tocrag (xoxox'x), tocrax (xxoxo'x), quitcrag (xoxox"x), gicpogadex (Goxx), and patom (xo$ox).

**Category 26: Trapezoidal Duodispheniverts** - (Polypeta ??+2247 - ??+2270) These three regiments have trapezoid-trapezoid disphenoid verfs. The regiments are taporf (xoxxox), tiprixog (xoxxo'x), and gocragadex (Gxox). Taporf has six members, the other two have nine.

**Category 27: Cophix and Catjak Regiments** - (Polypeta ??+2271 - ??+2382) These two regiments have a tet-oct cupola pyramid verf and have 56 members plus 20 fissaries each. Cophax is xxo8x, catjak is xo8xx. The siphin regiment is amongst its facets.

**Category 28: Skewed Sphenic Pyriverts** - (Polypeta ??+2383 - ??+2468) These six regiments have 15 members (except for prom which has 11). Their verfs are skewed wedge disphenoids or scalenes. The regiments are sacprokix (xxxG), capartakix (xxGx), gacprokix (xGxx), sparkix (oxxG), gaparkix (xGxo), and prom (xx6xx).

**Category 29: Skewed Prismatotruncates** - (Polypeta ??+2469 - ??+2657) These three regiments have 63 members each and have skewed duowedge pyramid verfs. The regiments are spatakix (oxGx), goptakix (xxGo), and potjak (ox6xx).

**Category 30: Skewed Fastegipyriverts** - (Polypeta ??+2658 - ??+3056) These three regiments have 133 members each and have skewed trigon fastegium / antifastegium pyramid verfs. The regiments are scopkix (xxoG), gacpakix (xoGx), and hictijik (xo6xx).

**Category 31: Scrokix and Gacrokix Regiments** - (Polypeta ??+3057 - ??+3398) These two regiments, xoxG and xGox respectively, have an "antiduowedge skewed wedge" verf and have 171 members each.

**Category 32: Drokix, Srokix, and Grokix Regiments** - (Polypeta ??+3399 - ??+4419) These three regiments are oxGo, ooxG, and xGoo. Drokix has 511 members, the others have 255 members each. Their verfs are various skewed trigonal duowedges and their facetings.

**Category 33: Gipkix, Sipkix, and Spojak Regiments** - (Polypeta ??+4420 - ??) These three regiments are xoGo, oxoG, and ox6ox. Their verfs have a trigonal fastegic/antifastegic fategium/antifastegium shape - a sort of trip-trigon-trigon scalene. They likely have thousands of members each.

**Category 34: Cakix Regiment** - (Polypeta ??+1 - ??+781) This regiment has 781 members. Its symbol is xooG. Its verf is a skewed tet antifastegium.

**Category 35: Sram Regiment** - (Polypeta ??+782 - ??+1068) This regiment has 287 members. Its symbol is ox6xo. Its verf is a duowedge wedge.

**Category 36: Spam Regiment** - (Polypeta ??+1069 - ??) This regiment likely has thousands of members, in other words, lots of spam. Its symbol is xo6ox. Its verf is a tet antipodium - oct fastegium.

**Category 37: Sabrim Regiment** - (Polypeta ?? - ??) This regiment likely has thousands of members. Its symbol is xo9ox. Its verf is a square antiduowedge.

**Category 38: Miscellaneous** - (Polypeta ?? - ??). So far only three regiments are known - ontip and gontip, the 6-D versions of ondip and gondip which have 8 members each and of course many scaliforms. There's also the newly found chosidap regiment which is formed by blending six haxes together and has 4 hosiaps amongst its facets, it has squico (square x ico) symmetry.

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.

**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 house my website. Without the domain there wouldn't of been enough room on my site to store all the pics. *

Page created by Jonathan Bowers, © 2014-2020 e-mail = hedrondude at suddenlink dot net