Uniform Polychora and Other Four Dimensional Shapes

The Polytope Discord Team has been busy, finding all sorts of new scaliform polychora using a tool they made called Miratope which can now search regiments out finding even lower symmetric finds. They also found many scaliform alterprisms in 5-D, 3 new uniform polytera, a new category J in 6-D, and Vel noticed that there are two types of polytwister - soft and hard - hard ones are on my site. There's also two new uniform polychora in category 20 bringing the count to 2191. I am now offering polychoron collector cards for sell! The first two series are out now, each with 36 cards which are 4" by 6". These cards have a bit more info on them then the ones on my site including long name, conjugate, and Coxeter Dynkin diagrams when possible. Duals are provided for regulars and reverts for certain regiments. They also have nice backs and feel like they're made of plastic. Collect all sets! I also created a polycards webpage to tell more of these cards.

Here is a top ten of the largest regiments for uniform polychora.

10. Afdec - 53

8,9. Stut Phiddix and Getit Xethi - 79 tie

6,7. Padohi and Gidipthi - 81 tie

5. Affixthi - 99

4. Rissidtixhi - 157

2,3. Sadros Daskydox and Gadros Daskydox - 177 tie

1. Siggissido - 333

Most of the graphics was done using Pov-Ray and animations are done with Stella4D. Last updated June 15, 2023.

2023 June 15 - New fissary uniforms in cat 20 - the hidhi fissaries and two new uniform polychora in the ondip regiment - sasdap and gasdap bringing the count to 2191. New uniform hosiap members has been found as well as a fissary alterprism in 5-D, new category J in 6-D. Several new polycards added. Last but not least, I'm selling professionally printed polychoron cards!!!! They look awesome and feel like plastic and they have backs, each series will have 36 cards. New scaliforms have also been found such as a new ico scaliform. The downloads page was been updated as well as the Polytopes of Various Dimensions page due to new metrix prefixes. I also numbered some of the scaliforms and there's a note on 'soft polytwisters' on the polytwister page. Also new content in the news section.

2022 October 27 - "This update will be like herding cats, ferrets, badgers, spiders, zebras, slime molds, tapeworms, rotifers and cthulhus all at the same time" - PlanetN9Ne. This phrase seems to be true. Category S3 had to be revamped, it is a lot larger now. S4 was renamed to Ike and Gishi Scaliforms and the sishi ones moved to S10. I've added more verfs in and separated the 4-D symmetries into several pages to avoid too many videos on one page - Main symmetries, Prismattic Symmetries, Swirldoics, Swirltettics, and Swirlcubics (not done yet). Videos of the swirldoics and swirltettics have been added (these may take a while to load). I also added in the isogonal tessics. The new uniform polychoron is named Tesapdid, giving us 2189 now. Tesapdid is in cat. 20 and is the first snub found with iodecaic symmetry. It was found by Galoomba by accident when he was trying to generate the verf of siggissido using Stella4D. The 'Polytope Discord Team' has been active discovering more scaliforms using Miratope which now has a faceting feature that searches them out giving us odd lower symmetric ones as well. I had to adjust the scaliform categories, two were too small and moved into S3, a few more got added. There's also a new uniform category in 5-D, an infinite one - category E and a couple more 5-D uniforms in the misc category. I also redone many of the 'polychoron cards' which I had many of a low resolution version linked to the polychoron names I made back in the early 2000's. They are now remade with high resolution and more info including symmetry, number of vertices and edges, category and regiment, and orientability and nature. Counts of the remaining Wythoffian 6-D regiments has been done using Miratope leading us to more than 40,000 uniform polypeta so far, there's also a new category I. I also added a top ten (nine for 5-D) regiments for dimensions 4-6. The Miratope Discovery Wave took place. I also added several orbiform polyhedra to the 3-D section of the site. The downloads section has been updated. Some new graphics and a few verfs are now added to cat. 30. Also new content in the news section.

2021 May 10 - New content on Making Polychora pages with piece graphs and several new polychora to build. Massive Discord Discovery Wave announcement which dwarfs the Pedisna Discovery Wave of 2019, six new uniform polychora have been found in 2020. Then in 2021, 333 new uniform polychora in a brand new category 30 - the idtessids, bringing the total to 2188. I later found pseudo uniforms - rissid swirlprisms. We also found comboblends of them which lead me to add a new section to the site with four pseudo-uniform categories. Scaliforms have been revamped entirely with sixteen new categories including an infinite one (29 categories all together now). These new finds were the result of Mecejide and _Geometer searching in the cracks and finding all sorts of subsymmetric polychora in the known regiments as well as new regiments. New symmetry pages that describe symmetries and isogonal polytopes from 2-4 dimensions, including the odd ones like ixitic and ixoic symmetries. Four dimensional symmetry page is here. New regiment finds in dimensions 6-8 by Mecejide. Added new links to the Polytope Discord and Polytope Wiki (in the links under picture below). Also updated some files in the downloads and fixed some counts and numbers in the polyteron pages. Also new content in the news section. This update will have 'aftershock' updates in the near future, stay tuned.

Prievious announcements.

ANNOUNCEMENT 1!!! Six new uniform polychora were discovered in 2020! The uniform polychoron count jumped to 1855 plus many fissaries. These six discovered are sidditsphit, gidditsphit, setut, getut, pecuexidfap, and pecuexdap - four in the sishi regiment, and the last two in a new regiment with the vertices of the grand antiprism - all in category 20. Four were found by _Geometer and I found setut and getut. These were the first uniform polychora found in 14 years. New scaliforms were also found resulting in many new scaliform categories.

ANNOUNCEMENT 2!!! And that's not all - at the end of January 2021, _Geometer found the first 12 members of Category 30 - Yes you heard that right, I looked at it closer and seen that there were 277 new uniform polychora in this regiment, bringing the total count to 2132! I coined the name 'Idtessids' as a play on idcossids for this regiment. But it didn't stop! On Feb. 9, _Geometer cracked open another hidden chamber in the same regiment, leading to another 62 uniform polychora as well as uniform fissaries and compounds. The count went to 2194 and after four more finds in March, it went to 2198. Later ten were found to crash. The uniform polychoron count made it to 2188! This lead to another scaliform category also.

ANNOUNCEMENT 3!!! The scaliform categories has doubled since last update with three more in Feb 2021 and one more in March on top of the ones in 2020! There's now 28 scaliform categories plus an infinite one. This latest discovery wave dwarfs 2019's discovery of seven uniform fissaries and the four new scaliform categories. Another development is the search for isogonal polychora by Grand Antiprism and myself resulting in some seriously wierd shapes with congruent vertices.

ANNOUNCEMENT 4!!! On April 21, 2021 _Geometer found a few more scaliforms and then I made a discovery unrelated to them, Category 31 - Rissid Swirlprisms, but I later noticed that they are pseudo uniform or shall we say 'troll uniforms'. These were formed by blending a distadedip verf with a rissid containing dips or stiddips as long as the result didn't form a 4-at-a ridge case. There were 28 troll uniform polychora here with 2 fissaries and a compound. The uniform count tempararily went to 2216, but went back to 2188. Originally I was looking for rissid scaliforms and found these instead, there should also be rissid pseudo-scaliforms. We later blended these in wierd ways to generate more troll uniforms that are more complicated than the idcossids. I added a new section called Pseudo-Uniform Categories

2020 February 29 - New content on the polyteron and polypeton pages where I reveal the first category of dimension 6 - with graphics and a much larger hax regiment than originally thought. Verfpages are now available which reveal the verfs of many polytera. New updates on regiments page and fixed a few errors on the website. Also new content on home page, Another Reality page, and the polygons page. Massive 'Pedisna Discovery Wave' announcement which includes several regiments of compounds in 4-D, loads of new scaliforms with 4 new scaliform categories discovered, and 7 new fissary uniforms for category 20. Also made an attempt at polychoron compound categories which lists 61 categories, most due to the discovery wave. Added four new 'making polychora' as well as cross section animations to them. I also got some new content on category 26 - the blends. These updates were like herding cats, so there may be others not mensioned. Also check out the added content in the news section.

Previous Updates

2018 June 9 - Finally all uniform polychoron categories are now in. New additions include the rissidtixhi and affixthi regiments which are categories 23 and 16 respectively. I also added the dircospid verfs and some cool browser icons for several web pages including those for the 29 categories. I also added an update on the polypeton site due to the discovery of chosidap. There's also new content on the home page and new content for this latest update.

2016 September 29 - Made version two of the stellachoron nets zipped file under downloads, there are more star polychora including several with gaquatids amongst their cells. Added Making Polychora webpage which links to several pages revealing the instructions and animated GIFs on how to build various star polychoron models including sisp, padohi, getit xethi, and the monsterous gidhiquit paddy and several others with more in the pipeline - assuming you can get to four dimensional space first. Also new content about this latest update.

2016 February 1 - Updated counts in the polypeton page and regiments page. Added Downloads page which has many polytope spreadsheets and a zipped file full of Stella4D files revealing the pieces and nets of over 80 star polychora such as quit gishi, getit xethi, and quiproh. Added link to George Olshevsky's GoFundMe site where he hopes to raise funds for a polychoron book and where he has several interesting posts to read there. Also new content about this latest update. Also new updates on the home page.

2014 April 22 - Added new definitions in glossary, finished fixing transparancy issues in the PNG files. Added Pizza Graphs of the Jewel Regiments page. Added Regiment Maps and Verf Structures page. Added several 7-D regiments in the Regiments web page along with some updated counts and many new short names. New short names added to polygon page. Added Categories 15 and 25 (afdec regiment and getit xethi regiment). Added quicref page displaying sections of polychora with lower symmetries. Lots of new content on uniform polyteron site. Added polypeton site. Also new content about this huge update.

2013 September 12 - New polyteron of the day site.

2012 August 12 - Small corrections to Category B and Scaliform sector. Over the following weeks, I'll be fixing many of the PNG images due to the transparency issue on newer browsers (they rendered many of the black parts (such as the names) as transparent which wasn't intended).

2012 June 10 - I got a bit carried away, this is by far the largest update to this site ever!!! Added categories 6,10,14,21,22,24,28, and 29. Added vertex figures to the category pages. Added high res graphics and tables to categories 7-9. Added section graphics on the main pages of categories 5,11-13,17,18, and 26. Polytwister page is completely redone and full of graphics. Added a polyhedron website as well as lower dimensional counterparts. Remodeled this page with some graphics also. Added scaliform section to this page with several categories. Added basic figures, 4D dice, and polytwister sections to this page. Small modifications to the polyteron page. Also added a glossary and a page displaying all known polychoric dice up to twenty sides. Also new content about this huge update. - Enjoy!

2010 October 25 - New high res pics for categories 1-4 with clickable picture tables. Also new content about the latest additions.

2009 February 8 - Added Polyteron website with three categories (1,2, and 3) to start it off. Also new content about filling methods as well as pics of powertopes, 13-sided dice, and coiloids (screwballs).

2008 August 23 - Added some polychoron pics in 3D, also added categories A, B, 13 and 19.

2008 March 16 - Added categories 8,9,18,26, and 27. Also started adding piece counts, number of types of pieces, and LOCs thanks to Michael Roedel's WikiChoron site, and lastly I've added the "nature" of the polychora.

2007 August 7 - Added Dice of the Dimensions and Regiments, changed the look of this page, also new content under Polychoron search. Vastly upgraded polytwister page.

Back to the Third Dimension . . . Home Page . . . Polycards Page . . . Glossary . . . On to the Fifth Dimension

Uniform Polychora and Other Four Dimensional Shapes

Polychoron categories . . . My other web pages . . . George Olshevsky's polychoron site . . . Polytope Discord . . . Alkaline's site and forum on the 4th dimension . . . Robert Webb's Stella software website . . . Polytope Wiki . . . Mason Green's scaliform site seems to have disappeared . . . Dr. Richard Klitzing's website . . . Wendy Krieger's Polygloss . . . Quicfur's Polytope website . . . Eric's Honeycombs Page . . . Planet N9ne's List of Hyperbolic Honeycombs . . . List of off files for Uniform and Scaliform Polychora . . . Miratope

Definition of Uniform Polychoron

A polychoron is uniform if its vertices are congruent 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 can't 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 its 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 only two vertices in common - so therefore they are not rigidly locked like the ike-gad case. I decided to relax this requirement to allow pure compound combofaces that are rigidly locked like the compound of three tesseracts (gico) - found in 5-D polytopes and above, but not allow inter-regimental compounds as combofacets.

Polychoron Search

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 Official 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 2191 known uniform as well as the additional scaliform polychora with the exception of many idcossids, dircospids and other large regiments. Also all uniform polyhedra, all known uniform polytera (5 dimensional), and many of the uniform polypeta (6-D) have short names aka OBSAs (pronounced OB sah).

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, several polychoronist and I started to study the scaliform polychora. We found many strange polychora amongst them. The primary polychoronist that searched the scaliforms are Richard Klitzing, Mason Green, George Olshevsky, Andrew Weimholt, and myself. So far there were 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.

Also around this time, 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. Here are a couple of his idcossid sections: Sadros Daskydox and "Darth Vader". 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.

On my old AOL site (last updated in 2002 - and its now gone), I mentioned that there were 8190 uniform polychora, much has happened since that time, although no new uniform polychora have been found between 2002 and 2005, there has been a revision to its definition to keep it more tidy. Because of this, the number of uniform polychora has been greatly reduced to a more managable 1845 (unless one wishes to include the fissary cases which could bring it closer to 3000). But in 2006 4 new uniform polychora have been found!, bringing the total to 1849. These are the first to be found in 4 years - see category 20 for details.

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.

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. Green 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 bloatorectified 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 retroditetragon - 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. Green has also found many new scaliform polychora as well as entire scaliform regiments.

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 - when you buy it, check out the polychoron sidtindip - it looks really interesting when moving.

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 and a few missed ones), 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.

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.

Added March 12, 2008. Last year Andrew Usher suggested to restrict the definition of polychoron even more so only the well behaved ones will be counted, thus cleaning up the set of uniforms - he called his criterion the "no-three-in-a-line" rule which simply means that you can never have a case in the polytope where there are three or more points in a line (this could be on an edge, pseudoedge, vertex figure, face, cell, edge-figure, or anywhere) - this would of cut the polychora down to the 600's and the idcossids would be cut down to one member only - it would eliminate all of the intercepted cases as well as their conjugates. However, instead of cutting the set of polychora - I suggested to group them into three "natures" - tame, wild, and feral. A tame polychoron would fit Usher's rules - to test it, simply look at the edge figures of the polychoron and make sure that no line would have more than two vertices. The wild ones will have edge figures where there are three of more points on a line with a segment on the line (all intercepted cases fit here) - when you have more than two points on a line with a segment on it, it will always lead to ridges intercepting on the cell realms (ex. cuboctahedra with the diagonal hexagons acting as a ridge between two other cells). In between would be the feral cases, they would have edge figures that have three or more points in a line, but never with a segment on the line - the prismasauri fit here - many of these have exotic pseudo-cells. The tame polychora are the best behaved ones. All of the polychora of categories A, B, 1, 2, 7, 8, 9, 10, and 19 are tame - so no need to mention this in those categories.

Added February 8, 2009. The polyteron site has now begun, but there's still a lot to do, like weeding the fissaries out of the larger regiments and putting them in their own class. I've also rendered several polytera, but may need to take a second look due to filling issues - it turns out that the filling method used in Stella4D may be the correct method of filling - I'll stop here and describe a bit about filling methods. When the uniform polyhedra were first displayed, all of the polygons were filled in solid - this turns out to be the way to fill in a regular polygon. But for polyhedra, things aren't as simple. When I first started sectioning polychora, I noticed that groh should have hollow spots inside, simply because they were "missed" by the solid part - I came up with a filling method that I called "core and wedges filling method", lets call it "caw" filling for short. Basically what caw filling would do, was to fill in all the odd density parts as well as all the parts directly touching the outside region - from this I would try to turn the polyhedron into a combination of parts, there was usually always a core in the center and sometimes there were "wedges" or some other polygonal chunks that dangled off the core's edges. I would fill in the core and fill in the wedges - all else was hollow. Groh had a rhombic dodecahedron core and six octagram chunks for the wedges - and so that is how I filled in all of my sections. But caw filling appears to have problems, one thing it is very complicated to figure out for polychora, another thing - not all things work with caw filling. The filling method I would like to use would be the "true" filling method - it fills in all parts that can't be opened to the outside with any type of continuous morph - all odd density regions are filled, but for even density - each region needs to be investigated to see if it can be connected to the exterior by some morphing of the polytope, this is very hard to do sometimes. All of the holes in the caw method can be connected to the outside, so they are true holes.

There are a few other filling methods going around, Norman Johnson prefers the solid filling, no holes aloud - this is quite simple to do, but seems unnatural in some cases where holes would be obvious. A better method is the winding method - it basically fills in everything except density 0 regions, but it only works for orientable polytopes, the caw filling actually filled in a few 0 density regions (in querco for example) - I took a close look at the winding method, and every 0 density region I tried could be connected to the outside, I could even prove this for querco by looking at sections - so the caw method is out - that means, time to rerender some pics. Another filling method is the binary filling method, it fills in only odd density regions, this would cause a pentagram to have a hole in its center. Robert Webb done some study on filling in order to render the cells of the polychora correctly, and noticed that sidhei (the polyhedron with stars and central hexagons) had a cross section of five intersecting rectangles, that when morphed would cause the center part to connect outside - that center point was inside the area under the star's center section - this means that the parts of sidhei under the star centers are completely hollow - now the question is, do we leave the star's center there - if we do, it would be a membrane. For Stella software, he decided to fill all orientables with the winding method and non-orientables with the binary method - I now suspect that this may be the true method - to prevent membranes, it would also be best to fill in all of the elements of a non-orientable in a binary method as well as the polytope itself. For compound facets, it would be best to allow for 0 density regions (even density for non-orientables) to cancel out. So what all of this means is that not only will several polychora and polytera need to be rerendered, but also some of the uniform polyhedra themselves! Groh should have the density 2 regions of the octagrams removed, also sidhei, gidhid, geihid, giddy and gird will also change a bit - last but not least, gidrid would change drastically - it is now hollow!

Also recently I started to render, first with POV-Ray and then with Stella4D, various step tegums (which make dice polytopes) as well as various powertopes. I also rendered with POV-Ray some of the screwballs (which I also call coiloids, these are mensioned on my dice page). Here are a few of those renders:

Tridecachoron unfolded - this polychoron would make a 13 sided die.

Mobius Tridecachoron unfolded - this polychoron is another 13 sided die.

Ocavoc - the octagon of an octagon - one of the powertopes unfolded. Ocavoc's projection.

Duocavoc - the dual of ocavoc unfolded - this is also a powertope and a 128 sided die. Duocavoc's projection

Bicoiloid - a one sided curved die, one of the coiloids. Four views (2-fold symmetry on, 2-fold 90 degree rotation, orthogonal view, orthogonal 90 degree rotation).

Tricoiloid and Bitricoiloid - two distinct one sided curved die, two more coiloids. Four views (3-fold equator on, 3-fold 90 degree rotation, orthogonal view, orthogonal 90 degree rotation).

Added October 25, 2010. Lately I've been rendering polychora with high res cross sections. There are twelve cross sections on each picture instead of four or five, they will be in PNG format for better quality. In the category pages, starting with 1-4 with more to follow, there will be a nicely rendered clickable table - just click on the polychoron section images to bring up the high res picture files. I've also been rewriting my POV-Ray code to render the polychora with the density filling for orientables and binary filling for the non-orientables - this seems to be the most consistent filling method, so some polychora will look different (this will be noticed more in category 11 and beyond). Due to this filling method, even some uniform polyhedra will look different; eg. groh, gidditdid, sidhei, and gidrid (which is now hollow).

I've also investigated a potential new type of figure in 8 dimensions - the "polyswirlers" (at this time, I'm 95% sure that they work - but need an answer to two questions in order to be 100% sure). If they work, they will be like polytwisters, except they can roll like a glome (4-D sphere) on many sides. Their symmetries will match the polyteron symmetries - for example, there should be a demipenteract swirler. There should also be 16 polyswirlers for each regular polyteron (there's 4 polytwisters for each regular polyhedron). Polyswirlers are related to quarternion Hopf fibration in the way polytwisters are related to complex Hopf fibration (which is the best known Hopf fibration). There may also be octonion Hopf fibration counterparts in 16 dimensions - if these work, lets call them polywhirlers. I'll need to look more into this though. If anyone is quite familiar with quarternion Hopf fibration, please contact me - I'd like to be 100% that these work.

Added June 10, 2012. This latest website update started in Dec, 2011 and was worked on until June, 2012 - it is by far the largest update to this website ever. Lots of new pages and sections have been added including the lower dimensional hubs. Graphics have been added on many of the category pages, some revealing slices of most or all of the polychora in the category fused together - I call these "fusion graphics". Many new categories have been added in and vertex figures have now been placed on the pages (I have yet to add verfs to the dircospid and prism pages). Since my last update, I found new polytwisters, there are now 222 of them, and rendered sections of each one. These can be seen on the revamped polytwister page. I've also looked at various 4-D dice in detail and now have those up to twenty sides revealed on a webpage with more to come in the future. I used Stella4D to render the dice and POV-Ray to write the OFF files for the dice. This lead to many crazy looking 4-D shapes. I've also built models of many of the "dice cells". Another fun project was the "semi-uniform" polyhedra, more info on these can be seen on the new polyhedron webpages. I've also finally wrote POV-Ray code to render idcossid and dircospid sections, so I've added those two categories to this update and finally given them all names. One new page that I added is the glossary which defines many of the terms seen on this site. So in summary - new content is all over the place - explore!

Added April 22, 2014. My original intent was to have an update done to the site last year, but due to the "call of the wild" and a couple monitor hick ups, the update was delayed until now. Back in late 2012, I started playing around with six dimensional polytopes and even rendered a few verfs - it was around this time that my monitor blown out a capacitor. Due to a very tight income, I wasn't able to repair it until four months later - so I went back to pen and paper and started drawing several "regiment maps" of uniform polytera and polypeta. This was quite a fun project and I even searched out the regiments of a few polypeta. Once my monitor got repaired, I started working on some updates, but then the "call of the wild" hit me - a fierce yearning to render fields of cross sections of the uniform polytera. I kept at it for a while and started to break a few milestones - including hitting every non-prismatic Wythoffian regiment, and rendering all known uniform polytera with hinnic and hixic symmetry! When I finally started to slow down, I created a Polyteron of the Day website, but due to some pending unfinished website updates, I didn't link it here yet, but mentioned it on Alkaline's 4D forum. I also discovered a new regiment in 5-D, called Hosiap - it is a blend of 6 demipenteracts. At the end of 2013, another capacitor fried in my monitor, it only took two months before I could get it fixed. During the monitor-down time, I searched out more six dimensional regiments using the regiment maps. I now have the final count for all typical uniform polypeta with hoppic (6-simplex) symmetry - 923, assuming nothing got missed. Once the monitor got fixed, I wrote some code with POV-Ray to search out regiments and weed out inter regimental compounds. I would only need to manually remove the fissaries (a process I call "defissing") and check for pure compounds. One of the regiments I've checked listed 1924 polypeta! (it's not defissed yet). I also rechecked the nit regiment, there were no new ones. I then got back to working on my site, mainly in the 5-D area. The new updates includes two of the remaining four polychoron categories, several new polyteron categories, fields of sections of many polytera revealed on the category pages along with their verfs, the polypeton hub is now up, and a few extra pages (pizza graphs, regiment maps, and knotboxes), a change on the home page, and new content in the glossary. Enjoy this latest update! - lets just hope no more capacitors fry!

Added February 1, 2016. Good news, no more capacitors fried - bad news my computer fried, the hard drive is unfixable. Good news, I had my most important files on a flash drive or hidden on my web space - bad news, it was a few months behind, so I lost some content I was hoping to add to this update and of course the crash delayed this update. Before the crash I continued some more polypeton regiment counts, then I got on an exciting project to figure what the nets of many star polychora looked like. I used Rob Webb's Stella4D software to excavate the pieces of many "stellachora". Stella4D can find the nets of star polyhedra and convex polychora, but it can't reveal the nets of star polychora yet - so I decided to develop them. Some took a while and were quite challenging, like wavdatixady, some were easy like gofix. I also excavated one of the pieces of gaquapac, it looked like a sea urchin. I also worked on webpages once in a while, like the 'spaces' one for my array notation site. I also coded the rissid verfs on POV-Ray to prepare for the Cat. 23 webpage. I then made a spreadsheet of over 10,000 short names to help me avoid duplicate names. It was around this time that my computer crashed. I went to a Fed Ex shop to look at the flash drive to see what I had left, there were my POV-Ray files and my website up to the last update - but no spaces page, no Stellachoron nets (NOOOO!!!), no short name lists, and I lost the Brag regiment listing which I recently redone. My POV-Ray files were updated to a few months before the crash, so there went the rissid verfs I coded. So from May to October, I had no computer - only a Playstation 3 to get on the net and check e-mails. During the crash time, George Olshevsky started something interesting - a GoFundMe site where he was trying to raise funds for a polychoron book. He also wrote several articles there. Sadly he only raised $210 as of this date, so I added a link to my site today hoping to help out. Also during this timeout I thought out a possible proof that all Wythoffian regiments are now known in all dimensions - I have yet to write it in detail. They come in the following forms above 4-D: simplex - oooo....oooo, cubics - oooo....ooo'o, demicubics oooo...oo6o, stellicubics oooo...oox"x, and goccoics oooo...oooG, oooo...ooGo, oooo...oGoo, oooo...Gooo, ... all the way to ..., Gooo...oooo where the G is the gocco loop (o'x"x) and where x nodes are marked and o nodes are either unmarked or marked. There are also the E6, E7, and E8 symmetries.

I later decided to see if I could boot up my mom's laptop for it got to a point it was impossible to boot up. I found out the problem, it had bugs in it - and I really do mean BUGS - cockroach skins everywhere inside, thankfully no live ones. The laptop booted up fine after I removed the bugs. Soon afterwards, I looked at my webspace hoping to find any lost files I may have put there - and there they were - the Stellachoron Nets - YES!!!, they were not lost after all. So I made a zip file out of them and created a Downloads page for my website to put them and several of my spreadsheets so they won't get lost anytime in the future and for others to view as well - so I hope ya'll enjoy. P.S. I plan on buying a new computer in a couple months, for the laptop is quite old and is making those hard drive noises, I don't know how long it'll last.

Added September 29, 2016. I'm back in the game. I got a new computer back in March and later got Stella4D again. After that I got back into those stellachoron nets and made several more including gaquatid containing ones. One of the pages that I lost during the crash was a page called "Making Padohi" which revealed instructions and rotating GIFs to explain how you would build an actual model of padohi if you could get to a four dimensional place. It was a webpage version of one of my stellachoron files with added instructions. Well I remade that page along with many others which can be viewed at Making Polychora.

Added June 9, 2018. A little over a month after being back in the game, I was knocked out of the game due to becoming septic after a bout of cellulitis in my left foot. I had to be admitted into the hospital for two months (Nov and Dec, 2016). I also found out that I was diabetic. While I was there I sketched Another Reality cartoons to be added later to Another Reality books 8 and 9. I also took a look at the polyteron hosiap and discovered new regiments related to it from dimensions 6 on. They are all formed by blending six rectified simplexes. I also found an infinite series of swirl compounds of jak and mo. After returning, I got back into element collecting again after finding gallium for a low price. I also finished Another Reality 8 (I have yet to scan these cartoons in). Recently I finished the rissidtixhi and affixthi pages to get this web page to a more finished state along with dircospid verfs and browser icons. On the home page, I added a strange glossary which you may find amusing and an elements page which has a bit of work to do.

Added February 29, 2020. This period was a bit intense when it comes to polytopes. In 2018 after the last update, I got into a frenzy to render graphics of polyteron vertex figures, not just small pics, but high res graphics that are labelled as well. I spent quite some time to render these 'verfpages' as I called them. I completed all with hixic symmetry and even finished the nit regiment along with it's fissaries - this was a massive undertaking. I started slowing down before reaching verfpage 120. Due to the size of them, I wasn't sure I would have space on this website, so I currently have them on my Google Drive space for viewing. After these I decided it was time to work on the first category in 6 dimensions, the primaries. I started to make the page, but there were no graphics. So I started to develop a POV-Ray code to start rendering cross sections of six dimensional shapes, yes you heard that right. I got several of them coded. But how in the world will I display them - I found a method :) There is an example on the primary6 webpage, it involved playing around with Javascript. I hope to add more of these later. I also coined names for several of them. Last year I even coined names for a few 'uniform polyhedra' with Klein Quartic symmetry, that 24 heptagoned 'polyhedron' with 56 vertices. Here are a few of them: Quart {7,3}, Klein {3,7}, Sisqua (small stellated quart) {7/2,7}, Gaqua (great Q) {7,7/2}, Gisqua {7/3,4}, Fa {4,7/3}, there were truncates and quasitruncates there also like Tikle and Quit Sisqua. I have yet to add these to the site. I also considered starting a you-tube channel, but right now there's no content yet. I hope to render animations of 4-D shapes for it as well as some googology content and some humorous stuff.

Something big loomed on the horizon, it started when Richard Klitzing found a compound of 12 exes in the rox army and sent me an e-mail about it, I named the compound Pedisna for short. This lead me to find a pandora's box of strange uniform compounds in 4-D, it didn't stop at compounds, but four new scaliform categories with a few more additions in existing categories. I also found seven fissary uniform polychora. It may be a while before I add all of them to this site. After this Pedisna Discovery Wave, I decided it was time to update this site, so I continued with the primary6 page and this time to render verfs of these polypeta. I coded the hax regiment in and noticed that the hax regiment would have subsymmetric compounds and polytopes. I then investigated it and this led to an intense search there, hax no longer has 10 members, but 145 members not counting fissary and compound members. I also made a spread sheet on polytope short names and it currently has over 10,000 shapes there. I once did it before but lost it in the computer crash a few years ago. I also started Another Reality 9 and added the latest cartoons to this website. I also made an attempt to list polychoron compound categories which is now on the website, but only as a list with descriptions. There's a few other things I hope to add later in a follow up update like the new scaliform categories and mo regiment verfs which I have yet to render (did get jak regiment verfs in). I did finally work on the spaces page for the large numbers site (googology) and got to X^^(X2) spaces, I'll do a follow up to continue this. This update was like hearding cats, I hope I didn't miss anything.

Added May 10, 2021. This last update was intense, it was like hurding cats, ferrets, and badgers at the same time. This was due to the creation of the polytope Discord and many discoveries being made by various members there. My original plan was to add several 'making polychora' and let that be it - but plans were about to change - BIG TIME. Various members of the discord like Mecejide, GAP, and _Geometer got involved with finding all sorts of uniform, scaliform, and isogonal shapes from 4-D to 8-D. Some including subsymmetric naq facetings, the Eldritch Horror Snubs, and even the doubling of the scaliform categories and to top it all off several new uniform polychora bringing the count to 2188. This includes the new Category 30 - the Idtessids. Many pseudo-uniforms were also found and we thought we had categories 31 and 32. This discovery wave was like a massive storm of discoveries, the largest since the 90's. Updating the site with these new finds will require several future updates (which I'll refer to as 'aftershocks'). I also investigated 4-D symmetries and therefore added a symmetry section to the site. The Discord Discovery Wave is still continuing and may so for some time to come. I still have many things to add to the site, like verfs of the new finds, new scaliform category pages, as well as future isogonal pages - so stay tuned for the future aftershocks which I hope to bring in monthly.

Added October 27, 2022 My original plan was to make several small aftershock updates, but plans changed. I started by analyzing some of the symmetries in kingdoms 4 and 5 - ah what - Kingdoms? Well the first kingdom are the primary symmetries such as pennic, tessic, icoic, and hyic as well as their families including ionic, pyritic, demis, and ixoic etc. The second kingdom are the polyhedral prism symmetries and similar. The third are the swirlchoron symmetries, ie skitettic, swirldoic, askicubic, spyritic etc. The fourth are the duoprism symmetries and similar, they also include duoantiprisms, prism pyramids, and even bilateral etc. The fifth are the swirlprism symmetries (swirls based on prisms instead of the original definition as found in sisp) This includes swirlantiprism and swirlpyramid symmetries. The sixth kingdom are the gyochoric and various bigyric symmetries. This took some time and was quite tricky, I have yet to add these to my website. Various polytope explorers from the discord were still busy with new finds including the discovery of new uniform regiments in 6-D, gyropentic polyteron scaliforms, two new uniform polytera, new sirsid and seside containing scaliforms now in cat. S3, thatoth and thaquitoth blends, and to top it all off a brand new uniform polychoron - tesapdid. No doubt this would make the next update take longer than expected - then . . .

Disaster strikes! On Christmas, I found George Olshevsky's obituary online, he died a couple weeks earlier. Although my mom and I enjoyed a nice Christmas with both my brother's families, I started to feel ill afterwards, mom did also. I had cought Covid. I wasn't aware that mom had it too, she had no fever but was getting weaker. On Jan 7, I found her dead and I had to go to the hospital for ten days and put on oxygen. I did enjoy the food there, but was ready to go home. My oxygen level was at 80 percent sometimes lower, I had the severe form. My brother Jeff also had to go to the hospital, he caught it too and it was like a punch to the gut, but he got over it faster than me. Justin, my other brother, had it earlier, but he didn't need to go to the hospital. After being released, I had to be on oxygen for two months and hyperventilated a few times. I started getting my strength back and got better. I also lost my last two remaining uncles before summer - three funerals this year. I didn't have access to my computer until March and had to access discord with a new cellphone, and noticed a new wave of discoveries while I was sick.

The Miratope discovery wave officially began. They added a faceting feature to Miratope making if possible to search regiments, lots of new finds came and many zip files containing raw data of the sishi regiment, pecuexdap regiment, idtessids, and all sorts of new members of cat S3, like the ipe blends. There was a flood of zip files - too many to search them out in detail. Many regiments were searched with lower symmetries, this lead to scaliform ex and gishi members with 5-skitettic symmetries. Sishi was overwelmed with scaliforms, also several fissary uniforms were found, the idtessids may have over 10K scaliforms. Also a new polyteron category was found - an infinite one. I couldn't wait to get back on Stella to check out all of these finds. Once I got back on my computer, I started back on the website updates, but there was so much to add - I took it easy and mainly stuck with 4-D finds including cat S3. I also got back to the polychoron cards, something I made over a decade ago, and made far better ones with higher resolution, some are included on the site now. More recently, a few members of the discord - PlanetN9ne, Creeperman 7002, Username 5243, and H A M used Miratoe to count the larger uniform polyteron regiments, mainly by faceting the verfs - the top ten had over 1000 members each. Even today, I got a second count on the shopjak regiment which was a bit different. After this update, I'll work on some of the polycards and if nothing crazy happens, I hope to have a small polycard update near the end of November.

Added June 15, 2023 What started as a plan for a small update with a few new polycards turned into a massive endeavor to not only add tons of updated polycards to this site, but to also make them available for sale as professionally printed cards with backs. I updated 72 cards into 'phase 2' and put these in sets 1 and 2 and sent them in to MPC ('Make Playing Cards') and ordered them for myself - they're stunning. I then created a store where others can buy them. I hope to one day get these in a store that can buy them in bulk to help reduce the price from $40 to something like $20 or even $10. So far I only got one sale and earned $7 - great start :P, well hopefully this gets better now that the collector cards are finally mentioned on my website. Meanwhile, various Discord polytopist found several new polytopes, a new category J in 6-D, new uniforms in the hosiap regiment in 5-D, more 5-D scaliforms. Tons of scaliforms in 6-D and up, which I haven't kept up with well, some new 6-D recounts - turns out that several 6-D iquipadah like uniforms are scaliform instead. There's also two new uniforms in 4-D, bringing us to 2191 - I even made polycards of them for this site. Other updates may have been added but not mentioned due to my polycard focus.

I also looked into skewed polygons and symmetries between n-block and n-cube symmetries for a fun small project. I even considered skew polytopes that exists on the elements of an n-cube, such as a skew polyhedron made of squares bending around the squares on a hexeract. Imagine if this was a floor plan of a palace in some videogame. Worse would be a skew polychoron made with cubes bending around all over some 9 dimensional cube, if this was the 'space' a game was designed into, it would be like Escher on steroids, even non-orientable skew polytopes could exist - these n-cube faced skew polytopes were called 'Hotels' by George Olshevsky. Now imagine a videogame using a skew-polyteron with tesseract rooms bending all over some 12 dimensional cube. A game built here could have space-time rooms where space and time could shift orientation Escher style and would loop on itself. Time could go backwards, you could become mirror imaged, you could find yourself walking on walls, the ceiling, or the future or past - what sort of labyrinths would be possible - yikes!

I then got back into drawing more Another Reality cartoons, ten (cartoons 811-820) were yet to be scanned in and were done previously, I drew 30 more getting to the half point of AR9 - so there are 40 new AR cartoons on the Another Reality page and many are quite crazy. I even designed the front of AR 10. I hope to finish AR9 within the next few months and start up book 10 where all of its cartoons are now currently planned as well as the 'Don't You Hate it When' book. Story based cartoons such as Rusty, Sergeant's Elite, Zok and Dorp, and others have story lines planned up to AR 13, so I plan on speeding up with these instead of taking multiyear breaks from cartooning (AR7 started in the 1990's and was finished in 2011 to give you an idea of what multiyear breaks does) - hopefully I'll get to AR 13 before I'm 70.

Basic Four Dimensional Shapes

These five shapes can be considered as the basic 4-D shapes, they are either flat in each dimension or they join them in a uniform curve. The five shapes are:

Tesseract - which can be generalized as a variety of tetrablocks when the dimensions are of different length. The most symmetric is the tesseract of the dyad, then there are the cube prisms (dyad cubed time dyad), there are the square-square duoprisms, the square - rectangle duoprisms, and finally there are the blocks (dyad times dyad times dyad times dyad). This shape would be the basic building block in four dimensional space. It could be represented by ||||.

Cubinder - This is the cross product of a disk and a square (or rectangle for variants). It has four cylinders as sides and one curved side. There are also a large family of prisminders related to it. It is flat in two dimensions, the other two form a curve and it has rollability of 1. It could be represented as ||().

Duocylinder - This is the square of a circle, or the cross product of two circles. Both sides are curved with rollability of 1. It could be represented as ()().

Spherinder - It has one flat dimension and three curved ones. It has rollability of 2 on curved sides. This is the prism of the sphere and may be the can shape of 4 dimensions. It could roll like a ball. It could be represented as |(|).

Glome - This is the four dimensional sphere, solid versions are sometimes called gongols. It has rollability of 4, for all four dimensions are curved. It could be represented as (||).


I've also been interested in a new type of polytope, which I call a Powertope. A simple one is the dodecahedron triprism - it is like taking the cube of a dodecahedron which is 9-D. It is also possible to take the octahedron of the dodecahedron - which turns out to be the dual of an ike triprism - the result is the dodecahedron tritegum (the word tegum was created by Wendy Krieger to represent the duals of prisms). But as it turns out, you could also find the quitco of a doe if you wished. The a of b always works if a has some sort of brick like symmetry (dyadic sym.(1-D), rectangle sym.(2-D), box sym.(3-D), etc) which includes all those with an n -cube symmetry and their pyritic counterparts. And b has a central inversion symmetry (point (x,y,z,...) implies point (-x,-y,-z,...)), although some cases work if b doesn't have this central inversion depending on what a is. For example the following work: the octagon of the pentagon (4-D), the gike of sirsid (9-D), the quitco of the circle (6-D), the octagram of the sphere (6-D), the iquipadah of gadros daskydox (16-D) - all of these by the way are isogonal.

Polychora in 3-D

Get your 3-D glasses and reach out and grab these polychoron sections (make sure that the cyan lens is over the right eye and the red over the left) snid phiphi, sirhihy, gisp, gidpaxhi, sprapivady, sadtaphix, quiphi, gardatady, gaqrigafix, more to come!

Uniform Polychoron Categories

A polychoron is uniform when all of its vertices are congruent and its cells are uniform polyhedra. There are currently 2191 known, yes you read that right.

Verfs (vertex figures) of the polychora are now viewable on the category pages. Polychoron pics can be viewed by simply clicking on the polychoron's name within each category file. For a few examples, take a look at sidtixhi, gabbathi, quiphi, and gaqrigafix. Hi-Res pics can be seen by clicking on clickable tables for categories 1-4 and 7-9. On some categories, sections are displayed on the category pages.

Here's a quick reference showing a series of sections of all the pennic, tessic, and prismattic (trigon and square based) uniform polychora. These have potential of being facets of five dimensional uniforms.

Below is a slice of an example polychoron from each of the 29 categories, which needs an update due to category 30. Category Examples

Here is a list of the 30 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 - New and improved polycards linked to the names, Oct 2022. (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 - New and improved polycards linked to the names, June 2023. (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 - New and improved polycards linked to the names, Oct 2022. (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 - New and improved polycards linked to the names, Oct 2022. (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 - New and improved polycards linked to the names and updated fusion graphic, Oct 2022. (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 - Updated fusion graphics due to giid and qrid ones coded. New and improved polycards for srip, rico, wavathi, and swav ditathi regiments. (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).

Category 7: Bitruncates - New and improved polycards linked to the names, June 2023. (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 - New and improved polycards linked to the names, June 2023. (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 - New and improved polycards linked to the names, June 2023. (Polychora 330 - 351) These 22 polychora are also known as the maximized polychora. Their verfs are irregular tets.

Category 10: Prismatorhombates - Updated fusion graphics due to giid and qrid ones coded. (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 - New and improved polycards linked to the names, June 2023.(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 - New and improved polycards linked to the names, June 2023. Updated fusion graphic, Oct 2022. (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 square antiprism verf. Each regiment also has 2 fissary members.

Category 14: Skewverts - New and improved polycards linked to the names, June 2023. Updated fusion graphics due to giid and qrid ones coded. (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 - Updated fusion graphics due to giid and qrid ones coded. (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 - New and improved polycards linked to the names, Oct 2022. (Polychora 764 - 777) Sishi is the regular small stellated 120-cell which has a dodecahedron shaped verf, these 14 polychora are its non-regular facetings with hyic symmetry. 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 - New and improved polycards linked to the names, June 2023.(Polychora 778 - 888) These polychora all have 600 vertices, there are three regiments of 37, each regiment also has 4 fissaries, a compound, 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 - Two new uniforms found - sasdap and gasdap! New and improved polycards linked to the names including the gondip regiment. New fissaries found - the hidhi fissaries, June 2023. New polychoron found in October 2021 - tesapdid and tons of new fissary sishis found in 2022 to add to the six previously found uniform polychora in 2020 and the seven fissary swirlprisms found in 2019 due to a goof. (Polychora 963 - 984, 1846 - 1855, 2189) These 33 polychora (plus 60+ fissaries) include iquipadah, gaquipadah, the ondip type, the antiprisms (two new ones), snubs (one new one), and swirlprisms (two new ones), and two entirely new 'ixitic' symmetric ones and all sort of fissary sishis with weird symmetries and the hidhi based ones. The grand antiprism (gap) belongs here. This set contains all sorts of odd shaped polychora.

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). It also has 30 fissaries. 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 also has 30 fissaries. It's verf is a triangle cupola, cells are 600 tets, 120 sidtids, 120 gaddids, and 120 quit gissids.

Category 26: Blends - New additions, 2020 (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 - New and improved polycard - Sadsadox, Oct 2022. (Polychora 1474 - 1491) These 18 polychora are split into two regiments of 9, there were also some exotics here too 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 - New and improved polycard - Sadros Daskydox, Oct 2022. (Polychora 1492 - 1668) The idcossids once had 2749 polychora, but nearly all of them were exotic-celled or coincidic, etc., now only 177 uniform idcossids are left (along with many scaliforms). Even the polychoron that the idcossids were named after was exotic-celled. The idcossids are based off of the 10-padohi compound, where 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 - New and improved polycard - Gadros Daskydox, Oct 2022. (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.

Category 30: Idtessids - Added some verfs and graphics for 2022. New Category in 2021!(Polychora 1856 - 2188) The idtessids are related to rappisdi and disdi and have 96 of the vertices of sishi, these start with a root polychoron, siggissido and then blends it with various ico members and later rappisdi. The verf of the first one looks like a blend of a pentagonal fustrum faceting with a pentagonal antipodium faceting with an antiprismic twist, later other core idtessids were found but they weren't as fruitful. There are five cores that blend with ico compounds to generate these. Two fissary cores are also known as well as four cores that lead to compounds. There are 333 idtessids. This regiment utterly creams the skiviphado regiment which used to be the largest tessic regiment, now this one is - heck it even creams the idcossids. As a matter of fact, this category quadrupled the number of tessics. _Geometer found the first of these on January 2021.

Pseudo-Uniform Polychoron Categories

A polychoron is pseudo-uniform if all of the cells are uniform (or pseudo-uniform) polyhedra and all of its vertices have the same vertex figure, but it is not quite transitive on the vertices. These things may even trick Stella4D into thinking they are uniform and had many of us polychoron researchers thinking we had new uniform polychora. I decided to add these due to the tricky nature of these things and some are quite beautiful and complex.

Here is a list of the 4 known categories of the pseudo-uniforms which could also be called 'troll uniforms'.

Category P1: General Pseudo-Uniforms - New finds in 2020! This is basically the miscellaneous pseudo-uniforms, these include the prisms of the two pseudo-uniform polyhedra and some tetrasidpith and tetragittiths that were found in 2020.

Category P2: Gyroic Sishis - Newly found Category in 2020! Before the idtessids were found, we were onto what we thought were 12 new uniform sishi blends, these were formed by blending various sishis with ico compounds, they turned out to be pseudo-uniform.

Category P3: Rissid Swirlprisms - New Category in 2021! What started as a search for potential scaliforms and were originally thought to be category 31, resulted in 28 new pseudouniforms which are in the rissidtixhi regiment, but with sispic symmetry. These form by blending a dip or stiddip containing rissid with a compound of 12 distadedips which I call 'fasp', one distadedip verf blends in the verf. Just make sure the original rissid being blended doesn't have both decagons and decagrams or the blend crashes. I found these on April 21, 2021, these are actually biform, but have identical vertex figures.

Category P4: Rissid Blend Monstrocities - Newly discovered Category in 2021! _Geometer suggested a way to blend verfs of category P3 together and both Spirit Backup and myself looked into them and found these horrific snub polychora that give the dircospids a run for their money. These are formed by blending 12 rissids together, which can be six of one and 6 of another creating an idcossid like blend. They have two sets of 3600 vertices, 7200 total and have horrific verfs. There are 406 of these that are true polychora, even though their vertices have the same verf, these are actually biform and therefore pseudo-uniform. These were initially dubbed the 'Idrissids'.

Scaliform Polychoron Categories

A polychoron is scaliform if all of its vertices are congruent and all of its edges have the same length, which causes its faces to be regular polygons. Their cells are not limited to uniform polyhedra, but they will be circumscribable in a sphere. New cells include pyramids, cupolas, and a wide range of strange blends. All uniforms are also scaliform. In 3-D, the scaliforms and uniforms are the same set of polyhedra. To generalize to tilings (Euclidean or hyperbolic), we need to add another stipulation: all n-D elements are circumscribable in an n-D sphere.

Near the dawn of the millennium, during some polychoron discussion online, Dr. Richard Klitzing suggested a looser definition for uniform polychoron (the scaliform definition above). I thought it would be best to keep the current uniform definition, but use the looser one for a new class of polychoron. I later suggested the word "scaliform" as a fusion of the words "scale" and "uniform". The scale part refers to the fact that the edges have the same length (scale). Many polychoronist got together to search for many of these and so far there are atleast 855 (not counting uncounted categories) plus many fissaries, many are idcossids or dircospids. I suspect there are still many undiscovered ones which a whole herd of them were found in 2019 as part of the Pedisna Discovery Wave and a massive explosion of finds in 2020 and 2021 as part of the Discord Discovery Wave! The count is currently unknown and I suspect the sishi regiment has hundreds if not over a thousand! Last year there was 13 categories here, now there's 29 - one is infinite and called category SA. I also changed the order a bit.

Here is a list of the 28 categories plus one infinite category of known scaliform polychora.

Category SA: Hemiantiprisms - Newly discovered in 2020 - _Geometer found the first one, and then I realized there's an infinite series of these strange polychora. They have a duoantiprism like nature, but with hemicells shaped like a bowtie tegum.

Category S1: Simple Scaliforms - New and improved polycards linked to the names, Oct 2022. - These are the more simpler scaliforms. Included are four prismattic cases and three more members of the hex regiment.

Category S2: Podary Scaliforms - New and improved polycards linked to the names, June 2023. - These 12 polychora are in or "almost in" the sidpith and the gittith regiments.

Category S3: Special Scaliforms - New finds as well as more polycards June 2023. Lots and I mean LOTS of new finds and sub-categories here in 2022, shoot most of them are new now! Previously four new discoveries in 2019 added. Also new polycards linked to the newly found ico scaliforms, Oct 2022. - These 20 (two are fissary) from several (17 now) small categories that all get grouped here - oops the count is 80 now instead of 20. Included is the 4-D version of sirsid! And there are two new sirsid containing ones now also along with their seside counterparts.

Category S4: Ex and Gishi Scaliforms - New polycards linked to the names, June 2023. New name for 2022 along with many new shapes! - These 28 are scaliforms found in the ex and gishi regiments and many more are expected - they have various swirlprism symmetries: 5-swirldoic, 5-skidoic, 3-swirldoic, 2-swirldoic, 5-spyritic, 5-skitettic, and many are expected to have 3-skidoic, 2-skidoic = 1-swirldoic, and 1-skidoic. Previously called Scaliform Swirlprisms, I moved the spidrox pair to S3 and the sishi ones to S10.

Category S5: Ondip Family - Six new finds, June 2023. - Ondip and gondip seem to have a tirade of strange relatives, 20 known so far plus 6 fissaries.

Category S6: Hexadecagonal Scaliforms - New polycards linked to the names, June 2023. These eight are made by blending octagon-hexadecagon duoprisms together as well as their star versions, they have contic symmetry.

Category S7: Scaliform Sidtaps and Gidtaps - Four new fissary discoveries by Mecejide in 2019 - We're not done with the baby monster snubs, there are a dozen scaliforms plus two fissaries in each regiment.

Category S8: Typical Scaliform Idcossids - These are the scaliforms within the idcossids formed by blending the verfs of two uniform padohis together, but have non-uniform cells, there are 310 of these monsters, plus a large number of fissaries.

Category S9: Typical Scaliform Dircospids - These are the scaliforms within the dircospids, formed by blending the verfs of two uniform padohis together, but have non-uniform cells, there are 310 of these monsters, plus a large number of fissaries.

Category S10: Sishi Scaliforms - More thorough searches done in 2022 for 5-skidoic and even 3-skidoic symmetries, previously - newly discovered in 2020 (only two are old ones) - This category began as _Geometer searched for sishi facetings with various subsymmetries. The count is currently unknown and may have hundreds of shapes, if not thousands. Sishi has utterly exploded on us. This category usually has cells that are bizarre edge facetings of ike, sissid, and sidtid. I moved the S4 members here that were sishi scaliforms.

Category S11: Sidtaxhi Swirlprisms - 12 more fissary finds in 2022, newly discovered in 2020 - These 11 plus 54 fissaries and two compounds are in the sidtaxhi regiment, but have either sisp symmetry or 'kisispic' symmetry (also called 5-skidoic symmetry). _Geometer found the first of this category and I searched out the others, then Username5243 found some new ones from the listing provided by H A M using Miratope.

Category S12: Dattady Swirlprisms - 6 new ones found in 2022, newly discovered in 2020 - These 17 plus 51 fissaries are in the dattady regiment and have sisp or gyrosisp symmetry. I found these after suspecting that they must also exist due to categories S11 and S13, The last six were found by Username5243 using a list made by Miratope by H A M.

Category S13: Gadtaxady Swirlprisms - 12 more fissary finds in 2022, newly discovered in 2020 - These 11 plus 54 fissaries and two compounds are in the gadtaxady regiment, but have sisp or gyrosisp symmetry. _Geometer found the first of these and Username5243 found the most recent ones using H A M's list made with Miratope.

Category S14: Sidtaxhi Comboblends - Newly discovered in 2019, new finds in 2020 - These 66 plus 20 fissaries form due to blending in the uniform compound of 12 sidtaxhis. There's likely many more of these now.

Category S15: Gadtaxady Comboblends - Newly discovered in 2019, new finds in 2020 - These 66 plus 20 fissaries form due to blending in the uniform compound of 12 gadtaxadies. There's likely more of these now.

Category S16: Tetrasidpiths - Newly discovered in 2019 - These form from blending in the 24-sidpith compound, their verfs fuse four sidpith verfs together. The count is currently unknown, but suspected to be high.

Category S17: Tetragittiths - Newly discovered in 2019 - These form from blending in the 24-gittith compound, their verfs fuse four gittith verfs together. The count is currently unknown, but suspected to be high.

Category S18: Ixitic Padohis - Newly discovered in 2020 - These 18 plus 59 fissaries form by blending a compound of 5 prissis with a padohi member as well as related figures, these have ixitic symmetry. Mecejide found several of these and _Geometer found some during the Pedisna wave. I searched them out with a regiment map.

Category S19: Ixitic Gidipthis - Newly discovered in 2020 - These 18 plus 59 fissaries form due to blending a compound of 5 prarsis with a gidipthi member and the related figures. These have ixitic symmetry.

Category S20: Prissi Blended Idcossids - Newly discovered in 2020 - These idcossids include the blending of prissis as well as ixitic padohi members, count unknown.

Category S21: Prarsi Blended Dircospids - Newly discovered in 2020 - These dircospids include the blending or prarsis as well as ixitic gidipthis, count unknown.

Category S22: Pecuexdap Regiment - Renamed in 2022, Miratope seach has been done and their appears to be thousands of members! Newly discovered in 2020 - This category was added after pindap was found by _Geometer, this regiment includes the two new uniforms pecuexdap and pecuexidfap and are a derivative of the sishis, but removed twenty vertices and having the same vertices of gap. The pecuexdap regiment (also called pindap regiment) appears to have thousands of members, most are fissary, and the count isn't finallized yet. There's even one with a convex verf, but it has fewer edges than pindap. Previously called 'Scaliform Duoantiprisms'.

Category S23: Disdi Super Regiment - New Category for 2022 - These are the scaliform members of the disdi super regiment, some of these have fewer edges. Rappisdi would fit here. Disdi and Mesdi belongs here, but will be 'grandfathered' in S3 and the others here. We will exclude the idtessids here which must have cells parallel to the cells found in the ico compound hidden within. The count is unknown, but suspected to be very high.

Category S24: Scaliform Idtessids - Newly discovered in 2021 - These are the scaliform members of the idtessid sort. The count is unknown, but suspected to be very high.

Category S25: Decasishi Scaliforms - Newly discovered in 2021 - These scaliforms are in the same regiment as the 10 sishi compound in the hi army excluding the two known ones from earlier. These have some chaotic looking verfs.

Category S26: Bhidhi Army - Newly discovered in 2021 - These are also derived from the sishi compounds, but have 240 less vertices giving them the same vertices as the bi-hecatonicosidiminished hecatonicosachorn (bhidhi). They have either 2-swirldoic or 2-skidoic symmetry.

Category S27: Octicoes and Hexadekicoes - Hexadekicoes found in 2022, Newly discovered in 2021 - These are blends of eight or sixteen ico members and can have two types. The octicoes have either demitessic or kidemitessic symmetry.

Category S28: Monster Ipe Blends - Newly discovered in March 2022 - These are formed by blending members of the greater ipe blends or ixitic ipe blends from S3 in an idcossid sort of way. These were found by H A M, a list of these has yet to be provided, there should be several, but likely less than 100. Their verfs are horrific looking.

Category S29: Hexasidpiths - New for 2022 - Their verfs have six sidpiths meeting there. I have yet to prove these work and not sure how large this category is, it was found by the Discord Team.

Category S30: Hexagittiths - New for 2022 - Their verfs have six gittiths meeting there. I have yet to prove these work and not sure how large this category is, it was found by the Discord Team.

Category S31: Sidpixhi Sishi Blends - Found November, 2022 - These things are terrifying, a never ending supply of fissaries and many legit ones found by the p o kekodjek in November 2022 using Miratope. They have the convex hull of sidpixhi and are based on a 40 sishi compound that exists there.

Compound Polychoron Categories

A compound polychoron is uniform when its components are uniform and all of its vertices are congruent. It is scaliform if its conponents are scaliform (which includes uniform). Sometimes several components meet at a vertex and they can be different types, if this is the case, they are called mixed compounds. When several vertices meet at the same place, we can call these 'combovertices'. Mixed compounds are not technically uniform since vertices are of different types, however they can be 'combo-uniform' since their combovertices are congruent or combo-scaliform. These will be included in this list. If any component is fissary, it is called a fissary compound. Here is my second attempt to put compounds in categories, and as bad as this sounds, it is likely not complete. The Pedisna Discovery Wave is responsible for categories C7-C10, C18-C21, C32-C60, and partially for C26-C29 - yikes! There will be new categories here, but have yet to be listed. There's even mixed compounds that have rapsady in it.

Here is a list of the categories of known uniform and scaliform compound polychora.

Category CA: Duoprism Compounds - Cross product of AxB where A and B are compound polygons, or A is a compound polygon and B is a regular polygon.

Category CB: Antiduoprism Compounds - Compounds of antiduoprisms with duoprism symmetries.

Category CC: Duoantiprism Compounds - Compounds of hex or gudap that have duoprism and related symmetries.

Category CD: Gap and Padiap Prismattic Compounds - Compounds of gaps or padiaps that have duoprism or related symmetries.

Category C1: Regular Compounds - Included here are the 32 compounds considered regular or the extras that are partially regular (5-ex compound) by HSM Coxeter. I'll also add any compound that is vertex, edge, face, and cell-transitive such as the 2-pen compound. Many are related to the 10-ex compound which I call 'sodi'.

Category C2: Unregular Compounds - These are those wierd extra 'regular' compounds with regular convex hulls with regular components that are alike, but the compounds themselves are not so regular.

Category C3: Regular Regiments - Any other member of the regiments found in category C1 or C2, like the 10-sishi regiment.

Category C4: Tupens and Thretesses - Compounds derivable by 2-pen and 3-tes (gico), includes 2-hex ones also.

Category C5: Tuicoes and Sixtesses - Compounds derivable by 2-ico and 6-tes.

Category C6: Truncations of Regulars - Compounds derived by various truncations of the regulars excluding those in previous categories or in giant regiments.

Category C7: Pedisna and Friends - Compounds in the 12-ex (pedisna) and 12-gishi (gisdadhep) regiments which are in the rox army. Also included are the 6 and 12 hi and gogishi compounds which are uniform. Pedisna was found by Richard Klitzing, it was this compound that began the giant 'Pedisna Discovery Wave' or PDW for short. All members in this category were found in this discovery wave.

Category C8: Sidadhep Regiment - Compounds in the massive 12-sishi regiment, 702 are currently known and most are mixed. These were found due to the PDW.

Category C9: Ditusna and Friends - Compounds in the 20-ex (ditusna) and 20-gishi regiments which are in the rahi army. I checked the 20-hi, it wasn't uniform so is not included. These were also a result of the PDW.

Category C10: 20-Sishi Regiment - Compounds in the 20-sishi regiment, there's a lot of them here.

Category C11: Decarox Compounds - Compounds based off of 10-rox which form the sidtaps, 10-rigfix which form the gidtaps, and the related 10-righi and 10-rasishi regiments. Includes 5-rox compounds and friends.

Category C12: Compound Idcossids - Compounds found amongst the idcossids, based on 10-padohi, includes 5-padohi compounds.

Category C13: Compound Dircospids - Compounds found amongst the dircospids, based on 10-gidipthi, includes 5-gidipthi compounds.

Category C14: Compound Prisms - Prisms of the uniform compound polyhedra.

Category C15: Compound Antiprisms - Antiprisms of tet containing compound polyhedra, these then will be hex compounds as well as their two distinct tho compound counterparts for each 3-D tet compound.

Category C16: Main Prismattics - Compounds of prisms of uniform polyhedra or uniform compound polyhedra along major symmetry axises.

Category C17: Main Antiprismattics - Compounds of compound antiprisms along major symmetry axises.

Category C18: Tetrasidpith and Tetragittith Compounds - Compounds of regiment members of sidpiths, gittiths, ondips, or gondips found within the 24-sidpith and 24-gittith regiments. These usually are compounds of 6 or blending occurs. Related compound of tats, or quitits are here also. Brought to you by PDW.

Category C19: Decasidtaxhi Regiment - Compounds found in the 12-sidtaxhi regiment. Found during the PDW.

Category C20: Decadattady Regiment - Compounds in the 12-dattady regiment. Found during the PDW.

Category C21: Decagadtaxady Regiment - Compounds found in the 12-gadtaxady regiment. Found during the PDW.

Category C22: Swirl Twister Compounds - Swirl symmetric compounds of polychora that are uniform/scaliform in a swirl-symmetry (this includes ico, ex, sishi) which approach polytwister symmetries. There are infinite groups of these on each polychoron, some which can be phased in different ways.

Category C23: Gyro Compounds - Gyro symmetric compounds of polychora which are uniform under some gyrochoron symmetry (such as pen, some duoprisms, and perhaps deca). These form infinite groups approaching coiloid symmetries.

Category C24: Hyped Square Compounds - Compounds related to powertopes based on A2 symmetries and their sub-symmetries.

Category C25: Other Compounds of Pennic Polychora - Any compound of pennic polychora that doesn't fit in any other category.

Category C26: Other Hex and Tes Compounds - Any compound of tes, hex, or tho that doesn't fit in any other category.

Category C27: Other Compounds of Tessic Polychora - Any compound of tessic polychora that doesn't fit in any other category.

Category C28: Other Ico Compounds - Any compound of ico or regiment members that doesn't fit in any other category.

Category C29: Other Compounds of Icoics - Any compound of icoic polychora that doesn't fit in any other category.

Category C30: Other Compounds of Prisms - Any compound of polyhedral prisms or antiprisms that doesn't fit in any other category.

Category C31: Other Compounds of Duoprisms - Any compound of duoprisms or related that doesn't fit in any other category.

Category C32: 120-Ex Regiment - Compounds based on the 120-ex compound - remember pedisna, the 12-ex in rox army, well there's a 10-rox compound - mix these concepts and you get this horrific regiment of monsters, many are mixed. This can also be thought of as a compound of 10 pedisnas, or alternatively 12 sodis. Found during the PDW.

Category C33: 120-Gishi Regiment - Compounds based on the 120-gishi compound - I bet the 120-gax and the chiral 60-gax will look horrificly complex and could put the dircospids to shame. Found during the PDW.

Category C34: Pandora's Box - Compounds in the horrificly massive 120-sishi regiment found during the PDW. There are more than 10,000 members in this regiment! The vertex figure of the 120-sishi is a compound of 4 dodecahedra. This compound has 3600 combovertices. Now - imagine - the - horrific - 120 - paphacki - AAAAAAAAA!!!! It blends though and should be a scaliform compound, but still - AAAAAAAAAA! Even the 6-paphacki from category C8, which I seen sections of, rival the dircospids in complexity.

Category C35: 24-Ex and 24-Gishi Compounds - Compounds related to the 24-ex compound and their conjugates under tex. These have two exes/gishis per vertex just like all other ex compound categories below this one unless mentioned different.

Category C36: 24-Sishi Regiment - Compound of 24 sishis and its regiment, has same vertices as tex. These have two sishis per vertex just like all other sishi compound categories below unless mentioned different.

Category C37: 40-Ex and 40-Gishi Compounds - Compounds related to the 40-ex compound and their conjugates under thi.

Category C38: 40-Sishi Regiment - Compound of 40 sishis and its regiment, has same vertices as thi.

Category C39: Sidpixhic 40-Ex and 40-Gishi Compounds - Compounds related to the 40-ex compound and their congugates under sidpixhi. Some of these may have sadis/rasdis added to the mix.

Category C40: Sidpixhic 40-Sishi Regiment - Compound of 40 sishis and its regiment, has same vertices as sidpixhi. Some of these may have sirhapsippadi regiment mixed in.

Category C41: Xhiic 60-Ex and 60-Gishi Compounds - Compounds related to the 60-ex compound and their congugates under xhi.

Category C42: Xhiic 60-Sishi Regiment - Compound of 60 sishis and its regiment, has same vertices as xhi.

Category C43: Srahic 60-Ex and 60-Gishi Compounds - Compounds related to the 60-ex compound and their congugates under srahi.

Category C44: Srahic 60-Sishi Regiment - Compound of 60 sishis and its regiment, has same vertices as srahi.

Category C45: Srixic 60-Ex and 60-Gishi Compounds - Compounds related to the 60-ex compound and their congugates under srix.

Category C46: Srixic 60-Sishi Regiment - Compound of 60 sishis and its regiment, has same vertices as srix.

Category C47: Grixic 120-Ex and 120-Gishi Compounds - Compounds related to the 120-ex compound and their congugates under grix.

Category C48: Grixic 120-Sishi Regiment - Compound of 120 sishis and its regiment, has same vertices as grix.

Category C49: Grahic 120-Ex and 120-Gishi Compounds - Compounds related to the 120-ex compound and their congugates under grahi.

Category C50: Grahic 120-Sishi Regiment - Compound of 120 sishis and its regiment, has same vertices as grahi.

Category C51: Prixic 120-Ex and 120-Gishi Compounds - Compounds related to the 120-ex compound and their congugates under prix.

Category C52: Prixic 120-Sishi Regiment - Compound of 120 sishis and its regiment, has same vertices as prix.

Category C53: Prahic 120-Ex and 120-Gishi Compounds - Compounds related to the 120-ex compound and their congugates under prahi.

Category C54: Prahic 120-Sishi Regiment - Compound of 120 sishis and its regiment, has same vertices as prahi.

Category C55: Idcossidic 240-Ex and 240-Gishi Compounds - Compounds related to the 240-ex compound and their congugates with the same vertices as the idcossids, four exes/gishis per vertex.

Category C56: Idcossidic 240-Sishi Regiment - Compound of 240 sishis and its regiment, has same vertices as the idcossids, 4 sishis per vertex.

Category C57: Dircospidic 240-Ex and 240-Gishi Compounds - Compounds related to the 240-ex compound and their congugates with the same vertices as the dircospids, four exes/gishis per vertex.

Category C58: Dircospidic 240-Sishi Regiment - Compound of 240 sishis and its regiment, has same vertices as the dircospids, 4 sishis per vertex.

Category C59: 240-Ex and 240-Gishi Compounds - Compounds related to the 240-ex compound and their congugates under gidpixhi. These form continuums.

Category C60: 240-Sishi Regiment - Compound of 240 sishis and its regiment, has same vertices as gidpixhi. These form continuums.

Category C61: Miscellaneous - Compounds or mixed compounds that don't fit anywhere, this includes the mixed compounds found in the blends.


Polytwister Slices

Polytwisters are strange polyhedron like four dimensional objects which have equatorial circles as their simplest element, sample slices are pictured above. The circles (rings) swirl around each other. Their sides, which I call "twisters", look like bloated out polygonal rods that have been twisted 360 degrees and then curved into a ring like a cylinder. They are the "lovechild" of Hopf fibration and polyhedra. If a polytwister was placed on a flat surface in the fourth dimension, it could roll like a cylinder no matter how it was placed. If it was tipped onto another side, it would roll in a different direction.

A polytwister is regular if all of its rings are congruent and all of its twisters are regular and congruent. A twister is regular when all of it's rings are congruent and all of its strips are congruent. There are 36 regular polytwisters plus one infinite group called the dysters.

A polytwister is uniform if all of its rings are congruent and all of its twisters are regular. There are 222 uniform polytwisters plus three infinite groups, this count includes the regulars.

There are many polytwisters that act as fair dice, many are duals to the convex uniform polytwisters.

There are also many semi-uniform polytwisters, likely millions of them, where tens of thousands are petsu.

Polytwisters - updated June 2023, note on soft and hard polytwisters - This page lists the 222 known uniform polytwisters with sample slices of all of them. There are also three infinite sets. Polytwisters are like rollable twisted "polyhedra" in 4 dimensions. It appears that the fourth dimension is only the beginning.

Four Dimensional Dice

In a geometric sense, dice are normally defined as convex polytopes with congruent sides. We could allow curved objects into the mix by changing the definition so that a die is any convex shape with congruent "contact regions". Contact regions can be thought of as the part that can contact a surface when the object is setting on one. The contact region of a tesseract is a cube cell (creases such as faces, edges or vertices don't count as contact regions), the contact region of a duocylinder is a disc, and the contact region of a glome is a point. Four dimensional dice can be put into these groups:

Polychoron Dice

Uniform - These are the six convex regular polychora plus deca and cont from category 7.

Squared Polygons - These are the squares of the polygons, they are all duoprisms.

Regular Gems - These are the tegums of the five regular polyhedra.

Catalan Gems - These are the tegums of the catalan polyhedra.

Duogems - AKA the duotegums, they are duals of the duoprisms, including amongst them are the diamonds.

Crystal Gems - These are the duals of the antiduoprisms.

Duocrystals - These are also called duoantitegums.

Catalan - These are the duals of the uniform polychora.

Snub Duals - These are the duals of isogonal snub polychora that didn't make the cut to be uniform.

Scaliform and Duals - These are the dice found amongst the scaliforms and their duals.

Other Snub Duals - I recently found some dice that didn't fit anywhere on this list, so I added a category here. Their duals usually have snub tetratetrahedra (an alteration of ike) with a spattering of various tets. I found them by investigating uniform compounds and taking the dual of their convex hulls.

Dublets - These 26 dice form by taking the intersection of two dice with pennic or icoic symmetry in opposite orientations.

Duocs - These are powertope dice which are the "duoc" of the polygons. Duoc being an octagon standing on its corner.

Antiduocs - These are formed like the duocs, but where there is a phase shift.

Grand Crystals - These are related to the Grand Antiprism dual.

Grand Gems - These are related to the dual of sidpith.

Phased Gems - These are like the grand gems but with a phase change.

Gyrochora - Also called step tegums. These have a lot of variety and are quite strange looking. Many have a prime number of sides. There are an infinity of these and the more sides they have the more they approach dysters, coiloids, and the duospindle.

Bigyros - Sometimes the intersection of two stretched gyrochora (stretched in an orthogonal directions) lead to new dice. Lots of strange ones here.

Swirlchora - These form several infinite sets of swirlprism dice that approximate polytwisters the more side they have.

Dice up to Twenty Sides - Added June 2012 - This page describes the polychoron dice up to twenty sides in detail.

Curved Dice

Basic Curved Dice - These are basic curved shapes along with their duals.

Polygonal Spindles - This is the axis product of the circle with a polygon.

Regular Polytwisters - These are the five convex regular polytwisters.

Dysters - These are the convex dyadic polytwisters, they form an infinite set.

Catalan Polytwisters - These polytwisters are based off of the Catalan solids.

Gemtwisters - These polytwisters are based off of the polygonal gyms - it is an infinite set.

Crystaltwisters - These polytwsiters are based off of the polygonal crystals - an infinite set.

Coiloids - These shapes curve in a spiraling way.

Dice of the Dimensions - This is my old page that describes the fair dice up to the 4th dimension, including those with curved sides.

Following is a list of some of my other polytope related pages as well as some that are coming soon.

Home Page - New Content, 2020 - This is my home page, it has links to some of my non-polytope pages such as Array Notation, Infinity Scrapers, Another Reality, Elements, Existence of God, etc.

Polyteron of the Day - Added, Sept. 2013 - Here I reveal verfs and cross sections of selected uniform polytera.

Making Polychora - New Content, 2020 - Find instructions on how to build actual starry polychora complete with animated GIFs.

Polycards - New!, June 15, 2023 - Information on the new 'Shapes of the Fourth Dimension' collector cards that I'm offering for sale.

Verfpages - New for 2020! - Labelled vertex figures of the uniform polytera in hi-res pics, nearly halfway finished.

Downloads - Last Update, June 2023 - Download various polytope spreadsheets and Stella files here.

Glossary - Last Update, April 2014 - What does all of those wierd terms mean? Check here to find out.

Symmetries up to Three Dimensions - New for 2021! - Explore the symmetries of polytopes and shapes up to three dimensions.

Four Dimensional Symmetries - Updated October 2022, New for 2021! - Explore the symmetries of four dimensional shapes. Prismatic Symmetries - Swirldoic Symmetries - Swirltettic Symmetries.

Isogonals with Pennic and Decaic Symmetries - New for 2021! - Check out the isogonal polychora with various pennic and decaic symmetries - full of animations, may take a while to load.

Isogonals with Tessic Symmetries - New for October 2022! - Check out the isogonal polychora with various tessic symmetries.

Polytopes of Various Dimensions - Updated a few names due to new metric prefixes, June 2023 - 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 polyronna.

Regiments - Added a few more names, 2020 - 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 7 at the moment) and hope to get to dimension 8.

Regiment Maps and Verf Structures - Added, April 2014 - This page introduces "regiment maps" and verf structures which are graphs and diagrams showing how lower dimensional regiments connect together to form a higher dimensional regiment.

Pizza Graphs of the Jewel Regiments - Added, April 2014 - This page shows a cool way to represent members of regiments with symbols of the form xooo...oox.

Powertopes - Coming Soon - This page will describe in more detail of what the powertopes actually are - powers of polytopes. There will be some pics.

Uniform Polypeta and Other Six Dimensional Shapes - Updated June 2023 - This site describes the 6 dimensional uniform polytopes which are divided into 38 categories plus 9 prismattic categories.

Uniform Polytera and Other Five Dimensional Shapes - Updated June 2023 - This web site is the 5-D version of the polychoron web site, it will list the uniform polytera in 19 categories plus 5 extra prismattic categories, there are pics!

Uniform Polyhedra and Other Three Dimensional Shapes - Updated October 2022, Added June 2012 - This page describes all sorts of polyhedra: uniform polyhedra, uniform compounds, 3-D dice, semi-uniforms, orbiform - complete with pictures.

Uniform Polygons and Other Two Dimensional Shapes - New Content 2020 - This page describes the polygons, with links to lower dimensions as well.

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.

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

Polyhedron Dude
Page created by Jonathan Bowers, © 2006-2023
e-mail = hedrondude at suddenlink dot net