Uniform Polychora and Other Four Dimensional Shapes


Welcome to my new and improved polychoron web site. Uniform polychoron count still stands at 1849 plus many fissaries, last four discovered are ondip, gondip, sidtindip, and gidtindip. Most of the graphics was done using Pov-Ray. Last updated August 12, 2012.

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!

Previous Updates

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 . . . Glossary . . . On to the Fifth Dimension


Uniform Polychora and Other Four Dimensional Shapes

Polychoron categories . . . My other web pages . . . George Olshevsky's polychoron site . . . Alkaline's site and forum on the 4th dimension . . . Robert Webb's Stella software website . . . Wikichoron website and Mason Green's scaliform site seems to have disappeared . . . Dr. Richard Klitzing's website . . . Wendy Krieger's Polygloss


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 no element in common (well technically they share the -1 dimensional nulloid in common which is sometimes called the empty polytope) - so therefore they are not rigidly locked like the ike-gad case.


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 Bowers Style Acronyms by Richard Klitzing). These short names were the result of writing the polyhedra in an abbreviated form, where I later mentally pronounced the abbreviation, this lead me to change the long name abbreviation into a pronouncible short name. An example is the quasitruncated small stellated dodecahedron - abbreviated to QTSSD, mentally pronounced as quit SIS sid, later spelled as quit sissid. I now have short names for all of the 1849 known uniform as well as the additional scaliform polychora with the exception of many idcossids and dircospids. Also all uniform polyhedra and many of the uniform polytera (5 dimensional) have short names.

It wasn't until 1997, that I contacted other polyhedronist, starting with the legendary Magnus Wenninger, after that contact, I got a letter from another polyhedronist Vincent Matsko who got wind of my discoveries, this letter came immediately after I discovered the massive idcossid and dircospid regiments (as well as their lesser counter parts, the sidtaps and gidtaps) which is one of the biggest discoveries so far. Not long after that, I searched the web for 4-D polytopes and found George Olshevsky's web site and later contacted him. He also got wind of my discoveries before hand, so we compared info and teamed up to search for more polychora. In 1998 I discovered the blends (also known as the sabbadipady regiment), George later found Sto and Gotto, two members of the rit (rectified tesseract) regiment that have demitessic symmetry. George and I later allowed for coinciding cases which brought the polychoron count into the 8000s. In 1999 George found the most unusual polychora to date, the swirlprisms and later I started to investigate another unusual type of 4-D figure which I call polytwisters which are related to Hopf fibration, there are now 27 known regular polytwister plus an infinite group of regular dyadic twisters (also called "dysters"). Not long after this I created my first website. Also during this time I began my polychoron sectioning with POV-Ray, and eventually rendered sections of over 1000 polychora. John Cranmer has volunteered to make scores of polychoron section movies using my POV code, which he has placed on CDs, these were spectacular - however each file is several megabytes, so I can't fit them on my site.

In 2002 I found Iquipadah and it's conjugate Gaquipadah, and later realized that all non-prime dimensions have iquipadah like figures. The discovery of iquipadah was somewhat unusual and mysterious. It began in 2001 while I was at church, the word "iquipadah" clearly popped into my mind, it had a sort of polychoron short name ring to it, although no polychoron at that time had the name. I then broke it down to two possible long names - either 24quasiprismatodis16-choron or inverted quasiprismatodis16-choron. I mentally referred to this undiscovered polychoron as the mysterious iquipadah assuming the possibility that God himself might of revealed the name of an undiscovered polychoron, I also considered the possiblity of a conjugate and coined the name gaquipadah for great quasiprismatodis16. I started to search amongst tessic figures, but found no new polychoron. It wasn't until a year later, when I was looking at sidpith, that I noticed that there was a uniform compound of two sodips (square-octagon duoprisms) inside and it was blendable with sidpith and lead to a new uniform polychoron, of course it also had a conjugate in the gittith regiment. I started to wonder if this could be the mysterious iquipadah, I considered it's cells - 16 tets (arranged in a hex fashion), 16 cubes (arranged in a completely different hex fashion), and 32 trips. There were two groups of 16 cells, there were also prisms amongst the cells, the verf looks like a trigonal antipodium with a huge indention in it. So the name inverted quasiprismatodis16 seemed to fit (the quasi part could represent how this is not a traditional prismato case) - iquipadah was discovered.

Later Hironori Sakamoto found four new affixthi regiment members, which lead me to two new afdec members - bringing the uniform polychoron count up to 8190 (which can now be considered as the uniform polychoroid count). In the following years, the polychoron definition was revised to weed out the more degenerate looking figures. I also started to search out the uniform polytera (polyteron in singular form - 5-D polytope) as well as section a few of them - example. I also give short names for all the regiment heads (colonels) of 6 and 7 dimensions. Wendy Krieger started to coin words to represent higher dimensional polytopes, they are as follows: polyteron (5-D), polypeton (6-D), polyecton (7-D), polyzetton (8-D), and polyyotton (9-D). I later extended this to go up to a decillion dimensions!! - an example is polyictron for a 24-D polytope (ic from 1cosi, and tr from tri - so ictri is a short form of icositri = 23, the polyictron has many 23-D facets). In 2005, 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!


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 (||).


Powertopes

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 1849 known.

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.

Below is a slice of an example polychoron from each of the 29 categories. Category Examples

Here is a list of the 29 categories plus the two infinite categories of the uniform polychora.

Category A: Duoprisms - This is the infinite set of duoprisms (also called double prisms). For every two polygons A and B, there is the duoprism AxB. Their verfs are disphenoids.

Category B: Antiduoprisms - This is the infinite set of antiduoprisms (also called antiprism prisms). Each antiprism in 3-D has a prism in 4-D. Their verfs are trapyrs (trapezoid pyramids) or crossed trapyrs.

Category 1: Regulars - (Polychora 1 - 17) These are the 16 regular polychora plus the only faceting of hex - "tho" - there are 17 polychora here. Verfs are regular polyhedra, and in tho's case the verf is a thah.

Category 2: Truncates - (Polychora 18 - 38) These are the truncated and quasitruncated polychora, there are also three ditrigonary truncates. Verfs are pyramids of regular polygons or semiregular polygons.

Category 3: Triangular Rectates - (Polychora 39 - 59) These are the rectified pen, tes, ico, hi, sishi, gaghi, and gogishi and their two primary facetings. There are 7 regiments represented here with three members each (rit and rico has more regiment members mentioned in cat. 12 and cat. 6 respectively). There verfs are triangle prisms along with their facetings.

Category 4: Ico Regiment - (Polychora 60 - 72) These are the facetings of ico, one of them, ihi, has pyrito-ico symmetry, 6 have tessic symmetry, while the other 6 have demitessic symmetry. Verfs are facetings of the cube. There's also a prominent compound called "Gico".

Category 5: Pentagonal Rectates - New graphics on main page (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 - New, June 2012 (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 hi-res pics (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 hi-res pics (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 hi-res pics (Polychora 330 - 351) These 22 polychora are also known as the maximized polychora. Their verfs are irregular tets.

Category 10: Prismatorhombates - New, June 2012 (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 graphics on main page (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 graphics on main page (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 - New graphics on main page (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, June 2012 (Polychora 552 - 611) These 60 polychora are split into 4 regiments of 15, their verfs are skewed wedges and facetings. Many of these are very intricate. The regiments are skiviphado (tessic), gik vixathi, sik vipathi, and skiv datapixady (last three are hyic).

Category 15: Afdec Regiment - (Polychora 612 - 664) The afdec regiment has 53 members plus one fissary member called affic which has 48 cotcoes for cells. Afdec has 48 coes and 48 goccoes for cells, its verf is rectangle trapezoprism (which I first called an antifrustrum).

Category 16: Affixthi Regiment - (Polychora 665 - 763) The affixthi regiment has 99 members plus one fissary member (affidhi). Affixthi's cells are 600 octs, 120 dids, 120 gidditdids, and 120 gaddids. Its verf is similar to afdec's except that the bases have different shaped rectangles (an oct verf and a did verf).

Category 17: Sishi Regiment - New graphics on main page (Polychora 764 - 777) Sishi is the regular small stellated 120-cell which has a dodecahedron shaped verf, these 14 polychora are its non-regular, non-swirlprism facetings. There are also 2 fissaries and several exotic-celled members. Three of these have verfs shaped like the three ditrigonary polyhedra. Paphicki and paphacki (the small and great prismasauri) are also here.

Category 18: Ditetrahedrals - New graphics on main page (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 - New graphics on main page (Polychora 963 - 984, 1846 - 1849) These 22 polychora include iquipadah, gaquipadah, the newly discovered ondip type, the antiprisms, snubs, and swirlprisms. The grand antiprism (gap) belongs here. This set contains all sorts of odd shaped polychora. Several scaliforms would fit amongst these, since many are swirlprisms.

Category 21: Padohi Regiment - New, June 2012 (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 - New, June 2012 (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 - New, June 2012 (Polychora 1304 - 1382) The stut phiddix regiment now has 79 members (once it had 238). Its verf is a triangle cupola, cells are 600 tets, 120 sidtids, 600 coes, and 720 stips. There are some beautiful polychora amongst the stuts.

Category 25: Getit Xethi Regiment - (Polychora 1383 - 1461) The getit xethi regiment also has 79 members (once it had 238). It's verf is a triangle cupola, cells are 600 tets, 120 sidtids, 120 gaddids, and 120 quit gissids.

Category 26: Blends - New graphics on main page (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 graphics on main page (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, June 2012 (Polychora 1492 - 1668) The idcossids once had 2749 polychora, but nearly all of them were exotic-celled or coincidic, etc., now only 177 are left (however there are scaliforms here). Even the polychoron that the idcossids were named after was exotic-celled. The idcossids are based off of the 10-padohi compound, 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, June 2012 (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.


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 719 plus many fissaries, most are idcossids or dircospids. I suspect there are still many undiscovered ones.

Here is a list of the 9 categories of known scaliform polychora.

Category S1: Simple Scaliforms - (New!, June 2012) These are the more simpler scaliforms. Included are four prismattic cases and three more members of the hex regiment.

Category S2: Podary Scaliforms - (New!, June 2012) These 12 polychora are in or "almost in" the sidpith and the gittith regiments.

Category S3: Special Scaliforms - (New!, June 2012) These 16 (two are fissary) form several small categories that all get grouped here. Included is the 4-D version of sirsid!

Category S4: Scaliform Swirlprisms - (New!, June 2012) These 14 are the funky cousins of sisp and gisp and a few others with larger vertex counts.

Category S5: Ondip Family - (New!, June 2012) Ondip and gondip seem to have a tirade of strange relatives, 20 known so far plus 6 fissaries.

Category S6: Hexadecagonal Scaliforms - (New!, June 2012) 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 - (New!, June 2012) We're not done with the baby monster snubs, there are a dozen scaliforms in each regiment.

Category S8: Scaliform Idcossids - There are 310 of these monsters.

Category S9: Scaliform Dircospids - There are 310 of these monsters.


Polytwisters

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 - Complete Overhaul, Loaded with Graphics! (June, 2012) - This page lists the 194 222 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 - NEW! 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 - This is my home page, it has links to some of my non-polytope pages such as Array Notation, Infinity Scrapers, Elements, Existence of God, etc.

Glossary - New, June 2012 - What does all of those wierd terms mean? Check here to find out.

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

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

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 - Coming Soon - This site will describe the 6 dimensional uniform polytopes which will be divided into 39 categories plus 8 prismattic categories.

Uniform Polytera and Other Five Dimensional Shapes - Added Feb. 8, 2009 - This web site is the 5-D version of the polychoron web site, it will list the uniform polytera in 19 categories plus 4 extra prismattic categories, there are pics!

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

Uniform Polygons and Other Two Dimensional Shapes - New, June 2012 - 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 store my polychoron pics. Without the domain there wouldn't of been enough room on my site to store all the pics.


Polyhedron Dude
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e-mail = hedrondude at suddenlink dot net