Polychoron categories . . . Home Page . . . My other web pages . . . George Olshevsky's polychoron site . . . Mason Green's scaliform site . . . Alkaline's site and forum on the 4th dimension . . . Robert Webb's Stella software website . . . Wikichoron website
Above shows a few sections and vertex figure of sixhidy (sik SHI dee) which is one of the podiumverts, and is one of over 1000 picture files that I have rendered using Pov-Ray . Sixhidy is short for "small 600-120dis120choron".
The reason for this polychoron reduction is due to two reasons. First, there was some disagreement as to which objects should be called true polychora and which objects should be considered more degenerate. My original definition allowed for exotic-celled figures, coinciding-faced figures, and fissary cases. Norman Johnson defined a polychoron in a more traditional way which actually excluded all three of the above cases (which will now be considered as polychoroids). Second there was the "Pandora's box" effect going on in higher dimensions - if we were to allow exotic-celled cases in 4-D, they could outnumber traditional cases a trillion to one by the time you get to 12-D - thus completely drowning out traditional cases. Conciding cases done the same, and many of them would be copy-cats. Because of this, there was an agreement to exclude these types from being considered as true polytopes.
Then there's the fissary cases, these are polytopes which have compound vertex figures, edge figures, or face figures, etc... These cases would be excluded by Norman's definition, even though they do not cause a pandora's box effect. Fissary cases could be allowed in a somewhat looser definition of polychoron, since the only problem is a compound-like effect. In some sense, fissary polychora are half way between true polychora and compound polychora. Exotic-celled polychoroids are half way between true polychora and exotic (degenerate) polychoroids.
A polychoron is uniform if its vertices are transitive (all alike in an isogonal sort of way) and all of it's cells are uniform polyhedra.
A polychoron is a four 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.
Also since my old web site days, another feat has been accomplished, the first sections of an idcossid as well as a dircospid has been done, not by myself, but by Michael Roedel - who has also wrote code to determine the number of pieces a polychoron has. So far the most extreme case is the dircospid Gadros Daskydox which has 29,310,000 pieces!!!. The cells of Gadros Daskydox are 1200 gikes (acting like 600 compounds of 2 gikes), 4800 ohoes (acting like 2400 2-oho compounds), and 6000 tuts (acting like 2400 2-tut compounds and 120 10-tut compounds). Notice that I said "so far", there may still be a polychoron with a larger piece count, not all of them has had their pieces counted - it is very likely that a dircospid will win.
My polychoron search began back in 1990, when I searched for them using vertex figures (verfs), faceting techniques, and a "digging-in-the-verf" technique. I used a blue note book and filled it with verf drawings, long names (many have changed since then), and cell lists. I carried that blue note book with me nearly every day to college, either to show people or to write more info into it. By 1993 there were over 1000 polychora in that book, although many of them were fissary or exotic-celled. That year also brought the discovery of the first (and presently only) non-prismattic uniform polychoron known to contain a snub polyhedron - Rapsady - rapsady contains 120 sirsids (also known as yog-sothoths), 120 sesides, and 1440 paps (pentagonal antiprisms). Later I started to hand draw sections of some of the more simpler polychora, mainly pentachorics and tessics, and wound up with approximately 100 polychora in hand drawn sections. Also during that time, I invented my short names (which have lately been called Bowers Style Acronyms by Richard Klitzing). These short names were the result of writing the polyhedra in an abbreviated form, where I later mentally pronounced the abbreviation, this lead me to change the long name abbreviation into a pronouncible short name. An example is the quasitruncated small stellated dodecahedron - abbreviated to QTSSD, mentally pronounced as quit SIS sid, later spelled as quit sissid. I now have short names for all of the 1849 known uniform as well as the additional scaliform polychora with the exception of many idcossids and dircospids. Also all uniform polyhedra and many of the uniform polytera (5 dimensional) have short names.
It wasn't until 1997, that I contacted other polyhedronist, starting with the legendary Magnus Wenninger, after that contact, I got a letter from another polyhedronist Vincent Matsko who got wind of my discoveries, this letter came immediately after I discovered the massive idcossid and dircospid regiments (as well as their lesser counter parts, the sidtaps and gidtaps) which is one of the biggest discoveries so far. Not long after that, I searched the web for 4-D polytopes and found George Olshevsky's web site and later contacted him. He also got wind of my discoveries before hand, so we compared info and teamed up to search for more polychora. In 1998 I discovered the blends (also known as the sabbadipady regiment), George later found Sto and Gotto, two members of the rit (rectified tesseract) regiment that have demitessic symmetry. George and I later allowed for coinciding cases which brought the polychoron count into the 8000s. In 1999 George found the most unusual polychora to date, the swirlprisms and later I started to investigate another unusual type of 4-D figure which I call polytwisters which are related to Hopf fibration, there are now 27 known regular polytwister plus an infinite group of regular dyadic twisters (also called "dysters"). Not long after this I created my first website. Also during this time I began my polychoron sectioning with POV-Ray, and eventually rendered sections of over 1000 polychora. John Cranmer has volunteered to make scores of polychoron section movies using my POV code, which he has placed on CDs, these were spectacular - however each file is several megabytes, so I can't fit them on my site.
In 2002 I found Iquipadah and it's conjugate Gaquipadah, and later realized that all non-prime dimensions have iquipadah like figures. The discovery of iquipadah was somewhat unusual and mysterious. It began in 2001 while I was at church, the word "iquipadah" clearly popped into my mind, it had a sort of polychoron short name ring to it, although no polychoron at that time had the name. I then broke it down to two possible long names - either 24quasiprismatodis16-choron or inverted quasiprismatodis16-choron. I mentally referred to this undiscovered polychoron as the mysterious iquipadah assuming the possibility that God himself might of revealed the name of an undiscovered polychoron, I also considered the possiblity of a conjugate and coined the name gaquipadah for great quasiprismatodis16. I started to search amongst tessic figures, but found no new polychoron. It wasn't until a year later, when I was looking at sidpith, that I noticed that there was a uniform compound of two sodips (square-octagon duoprisms) inside and it was blendable with sidpith and lead to a new uniform polychoron, of course it also had a conjugate in the gittith regiment. I started to wonder if this could be the mysterious iquipadah, I considered it's cells - 16 tets (arranged in a hex fashion), 16 cubes (arranged in a completely different hex fashion), and 32 trips. There were two groups of 16 cells, there were also prisms amongst the cells, the verf looks like a trigonal antipodium with a huge indention in it. So the name inverted quasiprismatodis16 seemed to fit (the quasi part could represent how this is not a traditional prismato case) - iquipadah was discovered.
Later Hironori Sakamoto found four new affixthi regiment members, which lead me to two new afdec members - bringing the uniform polychoron count up to 8190 (which can now be considered as the uniform polychoroid count). In the following years, the polychoron definition was revised to weed out the more degenerate looking figures. I also started to search out the uniform polytera (polyteron in singular form - 5-D polytope) as well as section a few of them - example. I also give short names for all the regiment heads (colonels) of 6 and 7 dimensions. Wendy Krieger started to coin words to represent higher dimensional polytopes, they are as follows: polyteron (5-D), polypeton (6-D), polyecton (7-D), polyzetton (8-D), and polyyotton (9-D). I later extended this to go up to a decillion dimensions!! - an example is polyictron for a 24-D polytope (ic from 1cosi, and tr from tri - so ictri is a short form of icositri = 23, the polyictron has many 23-D facets). In 2005, I started to study scaliforms, and found many with help from a few other polychoronist such as Richard Klitzing, Mason Green, George Olshevsky, and Andrew Weimholt. So far there are 60 non-uniform scaliforms (not counting the potential hundreds of idcossidic ones) plus two fissary cases. I also studied polytwisters in more detail, finding that there are 27 regular ones and an infinite number of dysters. I also rendered sections of them.
NEW - Added May 28, 2006. Recently Mason Green found 9 more regular polytwisters, bringing the total to 36 - the new ones, which are called the inverted polytwisters, are copycats of the bloated cases - which is why they escaped my notice, these polytwisters are of the (3/2, 3/2) caliber. Mason also found variations of the co twister as well as other rectified twisters, some of which I've known about - this caused me to investigate the uniform polytwisters a bit closer and I have found many unusual and unexpected surprises. There are the various redysters (rectified dysters and quasirectified dysters). Also many uniform polytwisters doesn't have uniform polyhedron counterparts, but may have a degenerate polyhedron counterpart - for instance Sacroter's ring figure is a tetrapod - with 4 triangle twisters and 4 square twisters meeting at each ring - sacroter is short for small cubiretrooctatwister, it's polyhedron counterpart is an oct with diagonal squares (which is degenerate, but sacroter is not degenerate). Also the degenrate polyhedron cid (complexicosidodecahedron) which has a complete pentagon verf - doesn't have a degenerate polytwister counterpart - instead there are two true polytwister counterparts, one with an antitruncated pentagon (pentagon with triangles dangling off corners) rinf, the other with a semiuniform decagram rinf. There's also sheaved polytwisters - which have dyad twisters between the polygonal twisters - an example is the sheaved icosatwister - it's rinf is a decagon with 5 dyad twisters and 5 triangle twisters joining at the rings - so the set of uniform polytwisters will be far more interesting than originally thought. Mason has also found many new scaliform polychora as well as entire scaliform regiments.
NEW - Added June 14, 2007. Robert Webb, the designer of Stella software, has recently created Stella4D, which shows sections of all of the uniform polychora, although his software uses a different "filling method" than I use in my section pictures (Stella4D renders may have more holes and tunnels in them). This is some really great software, not only do you get still images, but also interactive movies, you can also check out the duals - if anyone buys it, check out the polychoron sidtindap - it looks really interesting when moving.
NEW - Added June 14, 2007. For a few years now Michael Roedel has been creating a polychoron Wiki called WikiChoron, which contains sections of all of the uniform polychora (except prisms), it also reveals how the cells are chopped up as well as the number of pieces of the polychoron. Also it reveals the LOC (level of complexity) of each polychoron - LOC is equal to the value of the polytope divided by its half order, where the value of a polytope is equal to the sum of the complete values of each piece (excluding hidden cavities) of the polytope, and complete value is the sum of the complete values of each piece (including cavities) of the polytope. The value of a segment is 1. I came up with LOC and value several years ago. The WikiChoron site does have a few errors and some archaic short names (made a few name changes in recent years), but the content is great.
NEW - Added June 14, 2007. Several months ago I discovered some new uniform compound polychora, not just some, but many infinite families of swirl compounds. For example there's a uniform compound with 43 gogishis, and one with 74983 gadtaxadies - what happens is that any polychoron with the same vertices as pen, tes, hex, ico, hi, or ex has atleast one infinite family of swirl compounds that approach polytwisters in the same way that there are infinite compound families of any polygon that approach a circle (for example the uniform compound of 43 pentagons has 215 corners and looks like a rippled circle - imagine creating this by taking a strobe light and putting it next to a spinning pentagon - this would make the rotating pentagon look like many pentagons - if the strobe flashed 43 times per rotation you'll get the 43 pentagon compound) - the swirl compounds do the same thing by taking a polychoron and swirling it next to a strobe light, these compounds approach polytwisters, you can swirl ex in 3 different ways to get three distinct families (one with 12 rings, one with 20, and one with 30), hi can spin to give 60 ring swirl compounds. Take a look at 12-Swirlico for an example.
Here is a list of the 29 categories plus the two infinite categories of the uniform polychora.
Category A: Duoprisms - This is the infinite set of duoprisms (also called double prisms). For every two polygons A and B, there is the duoprism AxB. Their verfs are disphenoids.
Category B: Antiduoprisms - This is the infinite set of antiduoprisms (also called antiprism prisms). Each antiprism in 3-D has a prism in 4-D. Their verfs are trapyrs (trapezoid pyramids) or crossed trapyrs.
Category 1: Regulars - (Polychora 1 - 17) These are the 16 regular polychora plus the only faceting of hex - "tho" - there are 17 polychora here. Verfs are regular polyhedra, and in tho's case the verf is a thah.
Category 2: Truncates - (Polychora 18 - 38) These are the truncated and quasitruncated polychora, there are also three ditrigonary truncates. Verfs are pyramids of regular polygons or semiregular polygons.
Category 3: Triangular Rectates - (Polychora 39 - 59) These are the rectified pen, tes, ico, hi, sishi, gaghi, and gogishi and their two primary facetings. There are 7 regiments represented here with three members each (rit and rico has more regiment members mentioned in cat. 12 and cat. 6 respectively). There verfs are triangle prisms along with their facetings.
Category 4: Ico Regiment - (Polychora 60 - 72) These are the facetings of ico, one of them, ihi, has pyrito-ico symmetry, 6 have tessic symmetry, while the other 6 have demitessic symmetry. Verfs are facetings of the cube. There's also a prominent compound called "Gico".
Category 5: Pentagonal Rectates - (Polychora 73 - 132) These are the polychora that belong to the rox army, there are four regiments here, the rox, righi, ragishi, and rigfix regiments, each having 15 members, there are also two coinciding members and five exotic members in each regiment, which are no longer counted as polychora. The verfs are varient facetings of varient pentagon prisms.
Category 6: Sphenoverts - (Polychora 133 - 297) These are the cantellates (also called small rhombates) of the polychora along with others with similar verfs. Verfs are wedges and their facetings, each of the 24 regiments have 7 members (rico has had 3 members already counted in cat. 3). Sirgax belongs to this group.
Category 7: Bitruncates - (Polychora 298 - 306) These nine polychora (deca, tah, cont, xhi, shihi, dahi, gixhi, gic, and ghihi) are the bitruncates, they all have disphenoid verfs. Cont, gic, and deca have only one type of cell. There are also two fissary cases sitphi and gitphi which have only one type of cell, their verfs are compounds of three disphenoids.
Category 8: Grombates - Coming Soon (Polychora 307 - 329) These 23 polychora are also known as the great rhombates and their kin. There verfs are scalenoids (a scalene like disphenoid).
Category 9: Omnitruncates - Coming Soon (Polychora 330 - 351) These 22 polychora are also known as the maximized polychora. Their verfs are irregular tets.
Category 10: Prismatorhombates - (Polychora 352 - 441) These 90 polychora are grouped into 30 regiments of three, they seem to be quite attractive. Their verfs are trapyrs and facetings. One of my favorites is giphihix.
Category 11: Antipodiumverts - (Polychora 442 - 481) These 40 polychora are grouped into 5 regiments of 7 and one regiment of five. They have triangle antipodium shaped verfs along with facetings. The small prismates, like sidpith, belong here. There are some scaliforms in the sidpith regiment also.
Category 12: Podiumverts - (Polychora 482 - 511) These 30 polychora are grouped into 4 regiments of 7 and the extra two members of the rit regiment (sto and gotto). Their verfs are triangle podiums and their facetings, sixhidy belongs here. Previously known as frustrumverts. There are some scaliforms amongst the gittith regiment.
Category 13: Spic and Giddic Regiments - (Polychora 512 - 551) These 40 polychora are split into two regiments of 20. Spic has 48 octs and 96 trips as cells, Giddic has 48 octs and 48 quiths as cells. They both have a sort of square antiprism verf. Each regiment also has 2 fissary members.
Category 14: Skewverts - (Polychora 552 - 611) These 60 polychora are split into 4 regiments of 15, their verfs are skewed wedges and facetings. Many of these are very intricate. The regiments are skiviphado (tessic), gik vixathi, sik vipathi, and skiv datapixady (last three are hyic).
Category 15: Afdec Regiment - (Polychora 612 - 664) The afdec regiment has 53 members plus one fissary member called affic which has 48 cotcoes for cells. Afdec has 48 coes and 48 goccoes for cells, its verf is rectangle trapezoprism (which I first called an antifrustrum).
Category 16: Affixthi Regiment - (Polychora 665 - 763) The affixthi regiment has 99 members plus one fissary member (affidhi). Affixthi's cells are 600 octs, 120 dids, 120 gidditdids, and 120 gaddids. Its verf is similar to afdec's except that the bases have different shaped rectangles (an oct verf and a did verf).
Category 17: Sishi Regiment - (Polychora 764 - 777) Sishi is the regular small stellated 120-cell which has a dodecahedron shaped verf, these 14 polychora are its non-regular, non-swirlprism facetings. There are also 2 fissaries and several exotic-celled members. Three of these have verfs shaped like the three ditrigonary polyhedra. Paphicki and paphacki (the small and great prismasauri) are also here.
Category 18: Ditetrahedrals - (Polychora 778 - 888) These polychora all have 600 vertices, there are three regiments of 37, each regiment also has 4 fissaries, 20 exotic-celled cases, and 11 coincidic cases. The three regiments are the sidtaxhi, dattady, and gadtaxady regiments. Sidtaxhi's cells are 600 tets and 120 sidtids, verf is tut like. Dattady's cells are 120 gissids and 120 sidtids, verf is also tut like. Gadtaxady's cells are 120 gissids, 600 tets, and 120 gids, verf is a golden cuboctahedron (looks like a co, but squares are turned to golden rectangles). Sitphi and Gitphi can also go here as well as a similar compound which shows up in the dattady regiment.
Category 19: Prisms - (Polychora 889 - 962) These 74 polychora are the prisms of 74 of the 75 uniform polyhedra (we excluded the cube since the cube prism is the tesseract). Verfs are pyramids of the polyhedron verfs.
Category 20: Miscellaneous - (Polychora 963 - 984, 1846 - 1849) These 22 polychora include iquipadah, gaquipadah, the newly discovered ondip type, the antiprisms, snubs, and swirlprisms. The grand antiprism (gap) belongs here. This set contains all sorts of odd shaped polychora. Several scaliforms would fit amongst these, since many are swirlprisms.
Category 21: Padohi Regiment - (Polychora 985 - 1065) The padohi regiment now has 81 members (it once had 354, where most of them were exotic-celled or coincidic). If we added the fissaries back in, the padohi regiment would double in size. Padohi's verf is a pentagonal antipodium. It's cells are 120 sissids, 120 ikes, 720 stips, and 1200 trips.
Category 22: Gidipthi Regiment - (Polychora 1066 - 1146) The gidipthi regiment also has 81 members since it is the conjugate of the padohi regiment. It's verf is a pentagonal podium. It's cells are 120 sissids, 120 ikes, and 120 gaddids. Many of its members are very intricate.
Category 23: Rissidtixhi Regiment - (Polychora 1147 - 1303) The rissidtixhi regiment (sometimes called the rissids) has 157 members (once it had 316) it also has a few fissary cases. It's verf is a ditrigon prism. Cells are 120 sidtids, 600 octs, and 120 gids. Some strange looking verfs show up in this regiment.
Category 24: Stut Phiddix Regiment - (Polychora 1304 - 1382) The stut phiddix regiment now has 79 members (once it had 238). Its verf is a triangle cupola, cells are 600 tets, 120 sidtids, 600 coes, and 720 stips. There are some beautiful polychora amongst the stuts.
Category 25: Getit Xethi Regiment - (Polychora 1383 - 1461) The getit xethi regiment also has 79 members (once it had 238). It's verf is a triangle cupola, cells are 600 tets, 120 sidtids, 120 gaddids, and 120 quit gissids.
Category 26: Blends - Coming Soon (Polychora 1462 - 1473) These 12 polychora belong to the strange sabbadipady regiment which also contains 4 fissaries, its cells are 120 gissids, 720 stips, 720 pips, and 120 quit sissids. The verf looks like a triangle antipodium with a pyramid stuck on it's base. Some of the facetings have some really odd verfs.
Category 27: Sidtaps and Gidtaps - (Polychora 1474 - 1491) These 18 polychora are split into two regiments of 9, there were also some exotics here two as well as scaliforms. The sidtaps (or the sadsadox regiment) are based off of the blended compound of 10 roxes (which is no longer a compound, but a true polychoron). Likewise the gidtaps (gadsadox regiment) is based off of the blended compound of 10 raggixes. These are also known as the baby monster snubs and are related to the idcossids and dircospids. The verfs are facetings of a 2-pip blend
Category 28: Idcossids - (Polychora 1492 - 1668) The idcossids once had 2749 polychora, but nearly all of them were exotic-celled or coincidic, etc., now only 177 are left (however there are scaliforms here). Even the polychoron that the idcossids were named after was exotic-celled. The idcossids are based off of the 10-padohi compound, were the verfs are facetings of a pentagonal antipodium duo-combo. Many of these have millions of pieces. I now consider sadros daskydox as the head of this regiment (the conjugate of gadros daskydox).
Category 29: Dircospids - (Polychora 1669 - 1845) The dircospids are based off of the 10-gidipthi compound, only 177 are left as true polychora plus many scaliforms. The verfs are facetings of a pentagonal podium duo-combo. Gadros daskydox is considered the head. The dircospids are so far the most complex of the uniform polychora.
Something new!, Following is a list of some of my other polytope related pages as well as some that are coming soon!
My Home Page - This is a remake of my old 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 27 regular polytwisters and this time there are pics of these bizarre curvaceous swirling 4-D shapes.
Uniform Polytera - Coming Soon - This web site will be the 5-D version of the polychoron web site, it will list the 1410 known uniform polytera in 19 categories plus 4 extra prismattic categories, there will be pics!
Scaliform Polychora - Coming Soon - This page will describe the various non-uniform scaliform polychora.
Powertopes - Coming Soon - This page will describe in more detail of what the powertopes actually are - powers of polytopes. There will be some pics.
Polytopes of Various Dimensions - New - 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 - New - This page describes the fair dice of variant dimensions, including those with curved sides.
Regiments - New - 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 4 at the moment) and hope to get to dimension 8.
Uniform Polypeta - Planned for the Future - This site will describe the 6 dimensional uniform polytopes which will be divided into 39 categories plus 8 prismattic categories.
Uniform Polyhedra - Coming Soon - This page will describe the uniform polyhedra in some detail, pictures are linked below.
The following picture pages show the uniform polyhedra with their short names: 1, 2, 3, 4, 5, 6, 7.
Uniform Compound Polyhedra - Coming Soon - This page will describe the uniform compound polyhedra in some detail, pictures are linked below.
The following picture pages show the uniform compounds with their short names: 1, 2, 3, 4, 5, 6, 7.
Special thanks to Andrew Weimholt, who has let me use his polytope.net domain to store my polychoron pics. Without the domain there wouldn't of been enough room on my site to store all the pics.
Page created by Jonathan Bowers, 2006 e-mail = hedrondude at suddenlink dot net