Pedisna Discovery Wave

Pedisna Verf12-Sisp Verf12-Gisp Verf12 Sidtaxhi Verf24-Tat Scaliform Verf12 Grand Tapady Blend Verf

Above are the verfs of pedisna (the 12-ex compound), the 12-sisp compound, the 12-gisp compound, the 12-sidtaxhi compound, the 24-tat scaliform blend, and a scaliform blend of 12 grand tapadies. They are not to scale. Below are slice animations of the 12-gisp, 6-ex-6-sishi mixed compound, 6-gogishi, and 6-tico. These animations were made with Stella4D.

Slices of 12-GispSlices of 6-Ex-6-SispSlices of 6-GogishiSlices of 6-Tico

What started as an e-mail from Richard Klitzing about a potential 12-ex compound in rox and it's chiral 6-ex version turned into a massive wave of discoveries of various compounds, scaliforms, and even a few fissary uniforms. I will describe this incredible wave of discovery in the order that it happened.

August 30, 2019 - I received an e-mail from Richard Klitzing asking about a particular compound of 12 exes that he found hidden in the rox army along with a chiral 6-ex compound. He noticed it hidden in what is called a lace city of rox. A lace city is similar to looking at poke sections, but in discriptive form instead of pictures. He shown where the compound was hidden. I then looked at the verf of rox and seen where it would be inside it and then used Stella4D to render the verf. The verf of 12-ex is rendered above and resembles a sort of circus tent ornament, a compound of two ikes meeting at opposite vertices. I tried to render the 4-D compound from the verf using Stella, but it didn't render it. Stella seems to have trouble rendering many 4-D compounds from verfs at this time - a sort of hit and miss situation. So I tried different facetings of the 12-ex including a 12-sisp compound which rendered and was uniform. A surprise was revealed that the 12-sisp was not chiral at all, it had 6 left handed and 6 right handed sisps in it. Both George Olshevsky and I thought it was impossible to get a uniform compound of left and right swirlprisms - not anymore. What I believe happens is that a 6-sisp doesn't have swirl symmetry, but chiral hyic symmetry which can be made achiral when added to its mirror image. I also got the mixed compound 6-ex-6-sisp to render and then removed the sisps to get the 6-ex compound and removed exes to get a 6-sisp compound. I also realized that you could make compounds of 6 A's and 6 B's where A and B are any faceting of ex with at least pentagonal symmetry such as ex, fix, gohi, gahi, sisp, and the scaliform spysp. If they are the same, then you get a 12 A compound. If B was nothing, then you get a 6-A compound.

Not only that, but there would be a 6-gishi and 12-gishi. I was able to get a 12-gisp to render in Stella and it had a really cool looking verf, the sections were amazing - both shown above. I also got the 6-gisp-6-gax to render which lead me to the 6-gax and 6-gisp. This regiment is similar to the 12-ex compound. The real action happens with the 12-sishi where each component A or B could be any sishi faceting with pentagonal symmetry. We can also allow the compound, scaliform, and fissary facetings of sishi to join in on the fun - this leads to an enormous regiment of possible compounds where most are mixed. I just had to try it, the 12-paphacki - Stella couldn't render it though. I mixed paphacki with a swirlprism faceting of sishi and it worked, so I was able to render the 6-paphacki which was excruciatingly intricate and chiral to boot.

At first I thought there would be several ways to render the 12-sisp, where both were right handed or left handed, and two different ways to do a right-left which would aim the X like configuration in the verf in a different spot. I would later prove that this didn't work, but only one 12-sisp was possible.

But then there was the thought of a real nightmare - since there is a 10-rox compound (which lead to the sidtaps), there should also be a 120-ex compound with a verf combining 4-ikes together. Two ikes had a verf which resembled the 12-ex compound. But 2 different ikes looked like the more well known 10-ex compound's verf (the one under hi). The ikes could be thought of as a 2x2 arrangement. You could keep one ike to get a 30-ex compound, two ikes in three different positions to get a 60-ex compound, three ikes to get a 90-ex compound, and all ikes to get the full 120-ex compound. Sadly none of these were 'Stellable' (Stella4D couldn't render it from the verf). Since there are four ikes in the verf, you could mix match 4 different facetings of ex, so you now have a 30-ABCD compound where each of the letters represent some faceting of ex. There will also be a 120-gishi compound that behaves similar. Worst of all is a sort of Pandora's Box - the 120-sishi with a 4-doe verf. You could mix-match any grouping of sishi facetings with pentagonal symmetry (not sure of the lesser symmetries yet) - so you could get some sort of 30-sidtixhi-gaghi-paphacki-spaxhi compound. Imagine the horrific chiral 60-paphacki, there are three of them - but one has scaliform blending and turns into a compound of 6 gastakixes. The 120-paphacki will actually become a 12-gastakix. It also turns out that the 120-ex is a compound of 10 12-ex's as well as 12 10-ex's.

September 2, 2019 - After playing around with these new compounds by drawing their verfs in a spiral, calculating how many regiment members and so on. I decided to look at the 6-ex compound that I had previously rendered and I looked at the dual which would be a 6-hi. I took the convex hull and seen that all of the vertices were alike, therefore the 6-hi as well as the 12-hi was uniform. I was able to figure that the verf of the 12-hi would look like an X disphenoid which is a compound of two tets where two edges form a golden ratio X and the opposite edges would turn the X 90 degrees. Since these compounds work, so will the 6-gogishi and the 12-gogishi. I got the 6-hi and 6-gogishi rendered in Stella, the 6-gogishi is chiral and its sections look like a morphing swarm of tetrahedra which is seen in the animations above. But if these work . . . so . . . will . . . sidtaxhi, dattady, and gadtaxady compounds and all are uniform! They also play the mixmaster game of 6A and 6B compounds, so you could take two different dattady members and combine them. Their verfs look like compounds of two verfs of one of these three regiments. I used a trick in Stella to figure the verf by rendering sections of the 6-hi's convex hull vertex on and highlighting the vertices, looking for an arrangement of highlighted vertices that looked like 2 tuts combined. This gave me the vertices of the verf of 12-sidtaxhi. I also noticed that some of the vertices of the verf combined together which lead me to a shocker - some of the sidtaxhi 'compounds' blends together and will be actual polychora, scaliform polychora that is! The same will happen to the 12-gadtaxady. The 12-dattady will only produce compounds. I decided that these should produce one or even two new scaliform categories. I announced these discoveries to Richard and mentioned the new category S10.

September 3, 2019 - I counted the number of scaliforms in the 12-sidtaxhi regiment, there are 66 plus one fissary - the fissary one is a 12-sitphi blend. This means that there are also the same number of scaliforms in the 12-gadtaxady regiment. So I decided to add two new categories S10 and S11. The scaliform count is up by 132 plus 2 fissaries. I also rendered several of them in Stella and they are quite complex looking, almost idcossid like. Some turned out hollow with intricate interiors. I then coined names for some of those 12-ex compounds. 12-ex was named Pedisna for pental disnubbachoron. 6-ex was named Kepisna for chiropental snubbachoron. Other names include Kadhep (6-hi), Dadhep (12-hi), Sidadhep (12-sishi), Skadhep (6-sishi), Gisdadhep (12-gishi), and Giskadhep (6-gishi). I named a few others as well. If you are wondering, the 6-gisp is called Kiparp.

September 5-7, 2019 - I decided to look at the 2-ico compound and it's symmetry and found various tes and hex compounds, one particular hex compound had a verf that looked like the 'vertex intensified' oct, it was under cont's vertices. Take an oct and create three more by spinning an oct halfway on each opposite vertex pair. You could also do compounds of 6-hex, by keeping one of the new octs in the vio (vertex intensified oct) or keeping the original, the first compound was chiral, the second in not chiral (core-6-hex). There would be two 12-hex compounds one with the core-6-hex, the other with only side-6-hexes. There's also 2 18-hex compounds, one with the core, and the other without the core and it is achiral. The noncored 18-hex rendered on Stella and I was able to generate the noncored 12-hex and the chiral 6-hex compounds by removing hexes. Of course I looked at the tes compounds which were uniform. Stella can also truncate, rectify, and expand polychora - so I checked to see if thex, tat, quitit, sidpith, ico, dual ico, rit, rico, tico, srit, grit, grico, and srico compounds would work on the 6,12, and 18 compounds. The following is the result: The 6-rit was uniform with a bizarre convex hull with some sort of swirly symmetry with long square antiprisms. Tico, thex, spic, srico, grico, srit, and grit compounds weren't uniform. The 6-ico was uniform and chiral and its convex hull was spic. The dual of 6-ico was the same figure. The 6-rico was uniform with a convex hull having 48 snub cubes with smaller squares and twisted 3APs and irregular tets. The tat, quitit, and sidpith compounds were uniform. The 6-tat's convex hull has 48 cubes, 144 long square antiprisms, 288 tets, and 576 disphenoids and is vertex transitive (a requirement for uniformity). The 6-quitit and 6-sidpith has the same convex hull - it looks similar but with large cubes, flattened squaps, and the tets are pulled out into disphenoid shapes. There will also be a 6-gittith with a convex hull like the 6-tat. The convex hull of the 12-tes looks rather odd, it is vertex transative with 96 disphenoids and 48 strange polyhedra that looks like you started with a tic and sheaved off the triangles where one point hit the adjacent triangle, keeping the shape with a chiral square prism symmetry. Front and back sides are square, and the sides are tilted hexagons. The 12-rit and 12-ico has two types of vertices. The 18-tes is uniform with a cont shaped convex hull. The 18-rit was uniform with a verf looking like two rit verfs combined where two rectangles form a plus shape. The 12-tat and 18-tat work where the vertices double and triple up respectfully, similar is the 12 and 18 quitit. The 18 compounds have triangular pyramid symmetry, where one vertex joins the three tat/quitit verfs at their bases. The most amazing thing was that the 24-quitit and 24-tat had verfs that fused four verfs together into scaliforms. The cored 18 and cored 12 compounds should also fuse into scaliforms, the cored 12 will be compounds of already known scaliforms (otbott and otbaquitit). So that adds four new scaliforms to category S3. The tat and quitit scaliforms were actually found on the 7th.

The biggest surprise were the 24-sidpith and 24-gittith compounds, they fused four verfs together as well and open up the floodgates to potential scaliforms. I rendered the 24-sidpith, 18-sidpith, 24-quidpith, and 18-quidpith scaliform blends. Mixmatched ones didn't render, but they should work. The four verfed blend has one core verf and three side ones. This will lead to all sort of possibilities of combinations and most will be scaliform, some will be compounds of members of the ondip and gondip regiment, that is if they blend a core with one side verf. I have yet to count which ones work, many have a four-facet-meet-at-a-ridge case, so these regiments will need to be checked closely. There should be several members in each, so there are now two more scaliform categories S12 and S13, I called these the tetrasidpith and tetragittith regiments. I found these two on the 5th and analyzed them some on the 6th.

September 8, 2019 - On the 8th, I coined names for a couple hex compounds found by someone on the space forum (50-hex-hesadti and 25-hex-ketspatti), I also looked at various hex compounds as well.

September 10, 2019 - I tried to render a mixed compound of 6 sidtixhis and 6 tipsops by creating the verf in Stella so I could get a 6-sidtixhi compound, it worked - but something looked off. The 'tipsop' verf looked a bit pointy. I looked at it closer and seen what I did wrong, I put in half of the trips and put pips in instead of paps. I then realized I found a new uniform polychoron, but it was fissary. I checked to see if I could blend it with other sishi members and also found the conjugate polychoron. The end result was seven new uniform polychora, all were fissary. One had the symmetry of spittid, the others had the symmetry of sisp. I also coined names for them, they are now put in Category 20. The one I found as a mistake is now called 'sispdohix'.

September 11, 2019 - I investigated all sorts of odd hex compounds which seemed to be all over the place, none rendered in Stella, but their vertex arrangement was found in the vertices. I then looked at the vertex arrangements of rahi and at the depth of .095492, the same depth where ex's verf is, there were active vertices that formed a compound of two ikes. This lead me to find compounds of 10-ex and 20-ex under rahi. The 20-ex verf looks like a compound of two ikes where the second one is formed by twisting the first one on a 3-fold axis halfway. This lead to three new compound regiments, 20-ex, 20-gishi, and 20-sishi. This time, swirlprisms with a 3-fold symmetry should work in the compounds.

September 12-13, 2019 - I rendered verfs of the pedisna regiment, the 12-gishi regiment, 20-ex regiment, and 20-gishi regiment in POV-Ray and wrote macroes for the 12-sishi. I named the 20-ex 'Ditusna' for disicosahedronary tetrasnubbachoron. I also counted the 12-sishi regiment to have 702 members (likely more), and the 20-sishi has at least 347 members, these include scaliform members and fissary.

September 22-23, 2019 - I started to update the site, due to this big discovery wave. I also looked for scaliforms in the sishi regiment due to the sispdohix find earlier, there were 7 scaliforms where all but two are fissary and all but one has sisp symmetry, the other has spittid symmetry. On the next day, I started to write POV code to render the verfs of these new sishi members.

September 25, 2019 - Klitzing e-mailed about compounds he found under rico of 4 and 8 icoes. I realized that under the 2-rico, there would be 8 and 16 ico compounds, as well as triple the numbers to get tes and hex compounds. I decided to look at vertex arrangement of various other hyics using Stella. I noticed that at depth .095492 there was always at least two ikes worth of active vertices, sometimes as many as 4 ikes. So there are more compounds of exes that exist as well as their sishi and gishi counter parts as well as chiral halves. These include: 24-ex under tex, 240-ex under 10-tex (14,400 vertices), 40-ex under thi, 60-ex under srahi, 240-ex under gidpixhi, 240-ex under idcossid army (4-ex verf), a 40-ex-25-sadi compound under sidpixhi - YIKES!! There now appears to be a continuum of 240-ex compounds that meet together at certain points to give smaller number of ex's. The verfs look like two ikes sharing a center where they can be rotated freely of each other - imagine the horrific 240-paphacki compounds now, they should be the most complex looking uniforms in 4-D. After e-mailing suggestions to Robert Webb for Stella updates to make compounds easier to render, I started to write this page.

October 2-7, 2019 - I turned my attention to the 6th dimension to generate vertex figures of the hax regiment and realized that the tratets could be split into many orientations resulting in many new compounds of tetdips instead of just the three that I was aware of (wa, eh, and fu). I first listed out 28 compounds. I also realized that the tetdips could be split in half and connected to other facets leading to a sort of 'mixmaster' of half-tetdip orientations. I then started drawing diagrams to help me figure how many ways these half tetdips could work. I coded POV-Ray to draw the hexagonal graphs and looked through 256 of them to find the result. I found 13 ways that the half-tetdips could be arranged to connect to other facets and also found 15 configurations of facets that required the half-tetdips - 5 of these were fissary. Not long before this, I found six new full symmetric members of the hax regiment which I wasn't aware of, one of these were fissary. After all of this analysis and coding a verf rendering macro for the hax regiment, I have counted out 145 members in the hax regiment, which only 15 are full symmetric. There's also 66 fissaries (1 is full symmetric), and 18 compounds (I made a mistake in my original count). Now, I'm adding the verfs to the first polypeton category page. I'm also looking at the jak regiment to see if the compound 'pok' could split, I'm not sure yet, but if it does, the split would be into four sets of 9 hixips. I later found an overlooked member of the jak regiment, a fissary called Jep.

Below are the verfs of members of the pedisna (12-ex), gisdadhep (12-gishi), ditusna (20-ex), and gedidhop (20-gishi) regiments. Pedisna VerfsGisdadhep VerfsDitusna VerfsGedidhop Verfs

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