IncMats Website Update

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Re: IncMats Website Update

polychoronlover wrote:I noticed that the IncMats website says that o3o3x3x3/2 *a and o3x3x3o3/2 *a lead to 2 superimposed copies of the facetorectified tesseract, firt. However, due to the possibility of demitesseractic symmetry, and the fact that the vertex figure can reduce its symmetry from trigonal prismatic to trigonal pyramidal (see sto and gotto), it appears that o3o3x3x3/2 *a and o3x3x3o3/2 *a form firt, not 2firt.

Confer https://bendwavy.org/klitzing/explain/p ... #xpxqorosa :
Let P=n/d, Q=m/b<>2, R=k/c<>2, S=s/e<>2 and Q<S, then we have as general incidence matrix for xPxQoRoS*a :

Code: Select all
`.... |    G/2k |      k      k |      2k       k       k |      k      k      1      1-----+---------+---------------+-------------------------+----------------------------x... |       2 |    G/4      * |       2       2       0 |      1      2      1      0.x.. |       2 |      *    G/4 |       2       0       2 |      2      1      0      1-----+---------+---------------+-------------------------+----------------------------xx.. |      2n |      n      n |    G/2n       *       * |      1      1      0      0x..o |       s |      s      0 |       *    G/2s       * |      0      1      1      0.xo. |       m |      0      m |       *       *    G/2m |      1      0      0      1-----+---------+---------------+-------------------------+----------------------------xxo. |  g(3)/2 | g(3)/4 g(3)/2 | g(3)/2n       0 g(3)/2m | G/g(3)      *      *      *xx.o |  g(2)/2 | g(2)/2 g(2)/4 | g(2)/2n g(2)/2s       0 |      * G/g(2)      *      *x.oo | g(1)/2k | g(1)/4      0 |       0 g(1)/2s       0 |      *      * G/g(1)      *.xoo | g(0)/2k |      0 g(0)/4 |       0       0 g(0)/2m |      *      *      * G/g(0)`

and the vertex figure would be : x(Q)oRox(S)&#x(2P) = x(Q)Ro || oRx(S)
(where x(Q) means an edge, the size of which is the chord length of the Q-gon; etc.)

P=3 Q=3/2 R=3 S=3 -> n=3 m=3 k=3 s=3 g(0)=24 µ(0)=3 g(1)=24 µ(1)=1 g(2)=24 µ(2)=1 g(3)=24 µ(3)=3 G=384 µ=8
cells are tet and tut.

And as firt itself has for incidence matrix :
Code: Select all
`32 |  6 |  6  6 |  2  6---+----+-------+------ 2 | 96 |  2  2 |  1  3---+----+-------+------ 3 |  3 | 64  * |  1  1 6 |  6 |  * 32 |  0  2---+----+-------+------ 4 |  6 |  4  0 | 16  *12 | 18 |  4  4 |  * 16`

it becomes obvious that x3x3/2o3o3*a ought be its double cover.

Same for the symmetrical case, cf. https://bendwavy.org/klitzing/explain/p ... #xpxqoroqa ,
i.e. with P=n/d, Q=m/b<>2, R=k/c<>2 we get for xPxQoRoQ*a :

Code: Select all
`....   |    G/2k |      2k |      2k      2k |      2k       2-------+---------+---------+-----------------+----------------x... & |       2 |     G/2 |       2       2 |       3       1-------+---------+---------+-----------------+----------------xx..   |      2n |      2n |    G/2n       * |       2       0.xo. & |       m |       m |       *     G/m |       1       1-------+---------+---------+-----------------+----------------xxo. & |  g(2)/2 | 3g(2)/4 | g(2)/2n g(2)/2m | 2G/g(2)       *.xoo & | g(0)/2k |  g(0)/4 |       0 g(0)/2m |       * 2G/g(0)`

where g(0)=g(1) and g(2)=g(3). And the verf becomes x(Q)oRox(Q)&#x(2P) = x(Q)Ro || oRx(Q)

P=3 Q=3 R=3/2 -> n=3 m=3 k=3 g(0)=24 µ(0)=3 g(2)=24 µ(2)=1 G=384 µ=8
but also
P=3 Q=3/2 R=3/2 -> n=3 m=3 k=3 g(0)=24 µ(0)=5 g(2)=24 µ(2)=2 G=384 µ=40
would produce the same one. I.e. x3x3o3/2o3*a = x3x3/2o3/2o3/2*a, both again the double cover of firt.

--- rk
Klitzing
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Re: IncMats Website Update

I'm confused. I was trying to make the case that the family o3o3o3o3/2 *a has demitesseractic symmetry and the polychora (with tesseractic symmetry) that you call a3b3c3d3/2 *a are actually compounds of 2 forms with demitesseractic symmetry.

Your general-case incidence matrix seems to succeed in producing firt's matrix (under demitesseractic symmetry) if the value of G (which seems to be the order of symmetry, correct me if I'm wrong) is set to 192, the order of demitesseractic symmetry. Rit, the head of firt's regiment, is still uniform under demitesseractic symmetry. It has two types of edges and acts as a podiumvert. It is clear then that firt is also uniform under demitesseractic symmetry. Its vertex figure then acts like a crossed antipodium instead of a crossed antiprism, which suggests that it is the other Whythoffian member of the regiment. Since firt can have two types of edges and still be uniform, I see no reason that o3o3x3x3/2 *a, which has two edge types and the exact same vertex figure, needs to wrap around twice.
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polychoronlover
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Re: IncMats Website Update

I see. You might have a point here. This is because o3o3o3o3/2*a symmetry can be obtained as a 4-fold cover of o3o3o *b3o symmetry (group order 192) AND as an 8-fold cover of o3o3o4o symmetry (group order 384). - This kind of is just as in 3D: there o3o3o3/2*a symmetry can be obtained as a 2-fold cover of o3o3o symmetry AND as a 4-fold cover of o3o4o symmetry.

This might tell us, that the double cuver 2firt not truely is a somehow connected one-shielded doubly wound polychoron, but rather decomposes into 2 coincident copies. Or, in other words, the lower order base symmetry group would be the one which is guiding the according Wythoff construction in these cases of coincidence. Esp. I should change that disputed G=384 into G=192 only (and according µ).

But more generally: I will have to rework https://bendwavy.org/klitzing/explain/goursat.htm and https://bendwavy.org/klitzing/explain/pqrstu.htm in that view. (Perhaps kind of similar to the note already contained in https://bendwavy.org/klitzing/explain/schwarz.htm :
For this symmetry should be noted additionally, that as rationals 4/2 and 2 surely are the same. But a spot on the sphere having 4fold symmetry and one having only 2fold symmetry does never interchange within the same symmetry setup. Therefore submultiples 4/2 and 2 have to be distinguished. Esp. no 4/d can be added to 2. – On the other hand, triangular mirror configurations with such an angle of 2π/4 look like those having π/2 instead. Accordingly octahedral symmetry gets reduced to some lesser subsymmetry. I.e. any such potential Schwarz triangle will get the attribute semi. – But also any other in here obtained Schwarz triangle, already being counted within above symmetries surely is semi.
) ...

--- rk
Klitzing
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Re: IncMats Website Update

In fact this not only implied x3x3o3o3/2*a = x3x3o3/2o3*a = x3x3/2o3/2o3/2*a to change from "2firt" to just "firt",
but likewise x3o3o3o3/2*a = x3o3o3/2o3*a = x3o3/2o3/2o3/2*a = x3/2o3o3/2o3/2*a to change from "2tho" to just "tho".
And therefore within 5D we then get x3o3o3o3/2o3*b to change from "2hehad" to just "hehad"
and likewise o3x3o3o3/2o3*b to change from "2rhohid" to just "rhohid" ...
--- rk
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Re: IncMats Website Update

Klitzing wrote:In fact this not only implied x3x3o3o3/2*a = x3x3o3/2o3*a = x3x3/2o3/2o3/2*a to change from "2firt" to just "firt",
but likewise x3o3o3o3/2*a = x3o3o3/2o3*a = x3o3/2o3/2o3/2*a = x3/2o3o3/2o3/2*a to change from "2tho" to just "tho".
And therefore within 5D we then get x3o3o3o3/2o3*b to change from "2hehad" to just "hehad"
and likewise o3x3o3o3/2o3*b to change from "2rhohid" to just "rhohid" ...
--- rk

Really? I'd think x3o3o3/2o3*a would still be 2tho, as the vertex figure is already 2thah, but the 2 thoes would be completely coincidic so as to retain the demitesseractic symmetry. Every element would be doubled -- except for the vertices, which would just have twice the number of elements meeting at them. Such a figure matches what one would expect from the Dynkin diagram; the demitesseractic symmetry implies that each type of cell is to be used 8 times. For x3o3o3/2o3*a, this means the cells are 8 copies of x3o3o, 8 copies of x3o3/2o, and 8 copies of o3x3o. This makes 16 tets and 8 octs, the same cells that would occur in 2 copies of tho. A similar argument can be made about x3o3o3o3/2*a and the others.

Along a similar line, x3o3o3o3/2o3*b, having 2tho as its verf, would have 2 overlapping sets of hehad's terons, cells, and faces; only the edges and vertices wouldn't be in pairs.
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polychoronlover
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Re: IncMats Website Update

Hmm, I agree. Found my error of this morning.
So we just change 2firt into firt, which under demitessic symmetry is produced directly and under tessic one would be produced as a true coincident compound of 2 Independent sheets.
But 2tho remains to be 2tho, not only because the 2thah verf, but also because of the general incidence matrix being applicable here, i.e. xPoQoRoS*a with P=n/d≠2, Q=m/b≠2, R=k/c≠2, S=s/e≠2, P≤S, (P=S ⇒ Q<R):
Code: Select all
`.... |  G/g(0) | g(0)/2 |  g(0)/2  g(0)/2 | g(0)/2m g(0)/4 g(0)/2k-----+---------+--------+-----------------+-----------------------x... |       2 |    G/4 |       2       2 |       1      2       1-----+---------+--------+-----------------+-----------------------xo.. |       n |      n |    G/2n       * |       1      1       0x..o |       s |      s |       *    G/2s |       0      1       1-----+---------+--------+-----------------+-----------------------xoo. | g(3)/2m | g(3)/4 | g(3)/2n       0 |  G/g(3)      *       *xo.o |  g(2)/4 | g(2)/2 | g(2)/2n g(2)/2s |       * G/g(2)       *x.oo | g(1)/2k | g(1)/4 |       0 g(1)/2s |       *      *  G/g(1)`

which for P=R=S=n=m=k=s=3, Q=3/2, g(0)=g(1)=g(2)=g(3)=24, G=192 indeed provides 2tho.
And then too 2hehad remains 2hehad, as its verf is just 2tho (and not simply tho, while hehad has tho for verf). And similarily 2rhohid remains 2rhohid, as it uses 2tho for facet (and not simply tho).
--- rk
Klitzing
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Re: IncMats Website Update

Hi Klitzing, I noticed some errors on your list of polychora.

The polychoron you have listed as "qrahi" should be "qrigogishi", at least that's what Hedrondude has on his site. Similarly, "querfix" should be "qrigfix", "qraghi" should be "qqragishi", and "paqraghi" should be "paqrigshi".

Since x3o3o3o3/2*a = 2tho and x3o3x3o3/2*a = 2ratho, I would think x3x3x3o3/2*a = 2titho (titho = truncated tho).

Also, there are a few Wythoffian polychora not listed. These are the ones you are missing:

x4o3x3x3/2*b = kaviphdit (skewverted prismato16distesseract, in skiviphado regiment)

x3o5x3x5/3*b = kavipathi (skewverted prismatotris120, in sik vipathi regiment)

x3x3o3x3*a3/2*c = rawvhitto = o4x3x3o3/2*b (this is just a half-symmetry representation of it)

x3x3o4x4*a3/2*c *b4/3*d = kavahto (skewverted 16trisoctachoron, in skiviphado regiment)

x3x3o5x5*a3/2*c *b5/3*d = skiv datixathi (small skewverted ditrigonary 600tris120, in skiv datapixady regiment)

x3x3o5/3x5/3*a3/2*c *b5*d = gikkiv datixathi (great skewverted ditrigonary 600tris120, in gik vixathi regiment)

x5x5o5/3x5/3*a3/2*c *b3*d = kevuthi (skewverted tetris120, in sik vipathi regiment)

I think that's all of the Wythoffian polychora your missing I might make a list of Wythoffian polytera sometime based on Hedrondude's spreadsheet of polytera.
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Re: IncMats Website Update

username5243 wrote:Hi Klitzing, I noticed some errors on your list of polychora.

Well, "errors" is a harsh word here.

The polychoron you have listed as "qrahi" should be "qrigogishi", at least that's what Hedrondude has on his site. Similarly, "querfix" should be "qrigfix", "qraghi" should be "qqragishi", and "paqraghi" should be "paqrigshi".

Way back in 2001 those truely were named like that. Hedrondude must have changed their namings since silently...

Since x3o3o3o3/2*a = 2tho and x3o3x3o3/2*a = 2ratho, I would think x3x3x3o3/2*a = 2titho (titho = truncated tho).

Well, my page just states there that it would contain "2thah". But you are right, this Grünbaumian polychoron could be considered a 2titho.

Also, there are a few Wythoffian polychora not listed. These are the ones you are missing:

x4o3x3x3/2*b = kaviphdit (skewverted prismato16distesseract, in skiviphado regiment)
x3o5x3x5/3*b = kavipathi (skewverted prismatotris120, in sik vipathi regiment)
x3x3o3x3*a3/2*c = rawvhitto = o4x3x3o3/2*b (this is just a half-symmetry representation of it)
x3x3o4x4*a3/2*c *b4/3*d = kavahto (skewverted 16trisoctachoron, in skiviphado regiment)
x3x3o5x5*a3/2*c *b5/3*d = skiv datixathi (small skewverted ditrigonary 600tris120, in skiv datapixady regiment)
x3x3o5/3x5/3*a3/2*c *b5*d = gikkiv datixathi (great skewverted ditrigonary 600tris120, in gik vixathi regiment)
x5x5o5/3x5/3*a3/2*c *b3*d = kevuthi (skewverted tetris120, in sik vipathi regiment).

I never said that all the listed Wythoffian polychora in my Webpage were already fully complete. Those you mentioned at least all are already listed, just missing the according referenciation to their OBSAs ("official Bowers style acronyms").

I think that's all of the Wythoffian polychora your missing I might make a list of Wythoffian polytera sometime based on Hedrondude's spreadsheet of polytera

While pasting in your additions (so far only in my offline copy) - thanks for fiddling these out! - I saw that several more Wythoffian polychora (symbols) do still lack according referenciation to their OBSAs (and their to be worked out structure) ...

--- rk
Klitzing
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Re: IncMats Website Update

username5243 wrote:x3x3o3x3*a3/2*c = rawvhitto = o4x3x3o3/2*b (this is just a half-symmetry representation of it)

No it isn't. The symmetry of the verf can reduce, but not to the required triangular pyramidal symmetry. Instead, the verf can only reduce from a faceted triangular prism (o3o4o3o symmetry, order 1152) to a faceted wedge (o3o3o4o symmetry, order 384) or a faceted skewed wedge (o3o3o *3o symmetry, order 192). None of these symmetries

Oops, I just realized I was thinking of frico, not rawvhitto. Never mind, you're right.

I made a similar list back in 2015 and I'll look at it soon to do a more thorough investigation. But one thing I remember on my list that yours is missing is three hemi-Whytoffian polychora in the rissidtixhi regiment. The polychora are sidditdy (o3x5/2o3/2x5/3 *a3*c *b5*d / 2), ridditdy (x5o5/2x5o5/2 *b3/2*d / 2), and gidditdy (o3x5/4o3/2x5*a3*c *b5/3*d / 2). Or are you not listing hemi-Whytoffians?

I also noticed that x6o6/5x~*a is listed as hoha on the tessellations page when it is actually 2hoha.
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polychoronlover
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Re: IncMats Website Update

That list was only for true Wythoffians

@polychoronlover: Very interesting. I think it would be cool to see exactly how many uniform oylchora can be given as variants of Wythoffians (either with faces removed, or multi-covers of them etc.) In 3D it seems all polyhedra can be represented this way, but I don't think so in 4D.

Here is the first part of my list of Wythoffian (and some hemi-wythoffian) polytera. This part will cover all those with hexateral (hixic) symmetry. Most of the names are taken from Hedrondude's spreadsheet.

Code: Select all
`x3o3o3o3o = hix (hexateron)o3x3o3o3o = rix (rectified hexateron)x3x3o3o3o = tix (truncated hexateron)o3o3x3o3o = dot (dodecateron)x3o3x3o3o = sarx (small rhombated hexateron)o3x3x3o3o = bittix (bitruncated hexateron)x3x3x3o3o = garx (great rhombated hexateron)x3o3o3x3o = spix (small prismated hexateron)o3x3o3x3o = sibrid (small birhombidodecateron)x3x3o3x3o = pattix (prismatotruncated hexateron)x3o3x3x3o = pirx (prismatorhombated hexateron)o3x3x3x3o = gibrid (great birhombidodecateron)x3x3x3x3o = gippix (great prismated hexateron)x3o3o3o3x = scad (small cellidodecateron)x3x3o3o3x = cappix (celliprismated hexateron)x3o3x3o3x = card (cellirhombidodecateron)x3x3x3o3x = cograx (celligreatorhombated hexateron)x3x3o3x3x = captid (celliprismatotruncated dodecateron)x3x3x3x3x = gocad (great cellidodecateron)x3o3o3o3/2x = 2firx (firx = facetorectified hexateron, in rix regiment)o3o3x3x3o3/2*c = rawx (retrosphenary hexateron, in sarx regiment)x3o3x3x3o3/2*c = sircrax (small retrocellirhombated hexateron, in card regiment)o3x3x3x3o3/2*c = rapirx (retroprismatorhombated hexateron, in pirx regiment)x3x3x3x3o3/2*c = rocgrax (retrocelligreatorhombated hexateron, in cograx regiment)o3o3o3x3x3/2*c = dehad (dodecahemidodecateron, in scad regiment)o3o3x3o *b3/2x3*c = rippix (retroprismated hexateron, in spix regiment)o3o3x3x *b3/2x3*c = racpix (retrocelliprismated hexateron, in cappix regiment)o3x3x3o *b3/2o3*c = rabird (retrobirhombidodecateron, in sibrid regiment)x3x3x3o *b3/2o3*c = roptix (retroprismatotruncated hexateron, in pattix regiment)x3x3x3x *b3/2o3*c = recaptid (retrocelliprismatotruncated dodecateron, in captid regiment)x3x3o3/2*a3o3/2x3*a = recard (retrocellirhombidodecateron, in card regiment)`

As far as I know, this list is complete, at least for Wythoffian and hemi-Wythoffian hixics. I'll probably do the pentics (only full symmetric ones) next.
Last edited by username5243 on Sun Apr 23, 2017 11:54 am, edited 2 times in total.
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Re: IncMats Website Update

What are ye meaning by hemi-wythoffian?
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the dream we dream together is reality.

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wendy
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Re: IncMats Website Update

wendy wrote:What are ye meaning by hemi-wythoffian?

If I understand it, he means polytopes that aren't Wyhoffian by itself, but a degenerate double cover is.

A simple 3D example is thah (tetrahemihexahedron, in oct regiment). Thah isn't Wythoffian, but its double cover is represented by the symbol x3/2o3x.
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Re: IncMats Website Update

One of my favorite families of uniform polytera is o3o4o4/3o3o3/2*b3*d *c4*e. The diagram is shaped like a "tetrahedron with tail" and acts like a 5-D version of kavahto's family. It contains a member of the skatbacadint regiment -- raktatant (o3o4x4/3x3x3/2*b3*d *c4*e) -- and members of the skivbadant and sibacadint regiment too -- skevatant (o3x4x4/3x3o3/2*b3*d *c4*e) and scatnit (o3x4x4/3x3o3/2*b3*d *c4*e) respectively. Its conjugate family also has raktatant, as well as gakevatant and gactanet, the conjugates of skevatant and scatnit.

Another interesting thing is that the family with 2hehad, o3o3o3o3o3/2*b, also contains members of the sirhin, siphin, and pirhin regiments. This kind of occurrence is what finally made me realize that even finding all the Whythoffian families with a particular symmetry was a lot less trivial than I thought.
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polychoronlover
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Re: IncMats Website Update

polychoronlover wrote:... It contains a member of the skatbacadint regiment -- raktatant (o3o4x4/3x3x3/2*b3*d *c4*e) -- and members of the skivbadant and sibacadint regiment too -- skevatant (o3x4x4/3x3o3/2*b3*d *c4*e) and scatnit (o3x4x4/3x3o3/2*b3*d *c4*e) respectively. Its conjugate family also has raktatant, as well as gakevatant and gactanet, the conjugates of skevatant and scatnit. ...

• Twice the same symbol for skevatant and scatnit?
• What are the long names of those acronyms?
• What means "conjugate" wrt. to a simplicial symbol? Should there be switched all 3 <-> 3/2 and all 4 <-> 4/3?
--- rk
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Re: IncMats Website Update

Klitzing wrote:Twice the same symbol for skevatant and scatnit?

Oops, I meant to say x3x4x4/3x3o3/2*b3*d *c4*e for scatnit.

Klitzing wrote:What are the long names of those acronyms?

Looking at the LN abbreviations on the spreadsheet, they appear to be: retroskewtrigonary trispenteractitriacontiditeron for raktatant, small cellitrispenteractitriacontiditeron for scatnit, and small skewverted trispenteractitriacontiditeron for skevatant.

Klitzing wrote:What means "conjugate" wrt. to a simplicial symbol? Should there be switched all 3 <-> 3/2 and all 4 <-> 4/3?

Only 4 and 4/3 get switched. Switching 3 and 3/2 can lead to an overlapping figure, but switching 4 and 4/3 never does unless the original shape overlaps as well (actually I'm not entirely sure about this, but it seems to be true in all cases with hypercubic symmetry).
Climbing method and elemental naming scheme are good.
polychoronlover
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Re: IncMats Website Update

Here come the pentic Wythoffians. Again, most of the names are from Hedrondude's spreadsheet, except for two of the skatbacadint members (which don't seem to have listed names - in fact, not even lists of facets are given beyond the first few in the skatbacadint regiment). To denote names I coined I will use a * - you might want to check with Hedrondude as to the official names. Klitzing, I hope you might list some of these on the next update to your polytope site...

Code: Select all
`x3o3o3o4o = tac (triacontiditeron)o3o3o3o4x = pent (penteract)o3x3o3o4o = rat (rectified 32)o3o3o3x4o = rin (rectified penteract)x3x3o3o4o = tot (truncated 32)o3o3o3x4x = tan (truncated penteract)o3o3x3o4o = nit (penteracti32)x3o3x3o4o = sart (small rhombated 32)o3o3x3o4x = sirn (small rhombated penteract)o3x3x3o4o = bittit (bitruncated 32)o3o3x3x4o = bittin (bitruncated penteract)x3x3x3o4o = gart (great rhombated 32)o3o3x3x4x = girn (great rhombated penteract)x3o3o3x4o = spat (small prismated 32)o3x3o3o4x = span (small prismated penteract)o3x3o3x4o = sibrant (small birhombipenteracti32)x3x3o3x4o = pattit (prismatotruncated 32)o3x3o3x4x = pattin (prismatotruncated penteract)x3o3x3x4o = pirt (prismatorhombated 32)o3x3x3o4x = prin (prismatorhombated penteract)o3x3x3x4o = gibrant (great birhombipenteracti32)x3x3x3x4o = gippit (great prismated 32)o3x3x3x4x = gippin (great prismated penteract)x3o3o3o4x = scant (small cellipenteracti32)x3o3o3x4x = capt (celliprismated 32)x3x3o3o4x = cappin (celliprismated penteract)x3o3x3o4x = carnit (cellirhombipenteracti32)x3o3x3x4x = cogrin (celligreatorhombated penteract)x3x3x3o4x = cogart (celligreatorhombated 32)x3x3o3x4x = captint (ceilliprismatotruncated penteracti32)x3x3x3x4x = gacnet (great cellipenteracti32)o3o3o3x4/3x = quittin (quasitruncated penteract)o3o3x3o4/3x = quarn (quasirhombated penteract, in wavinant regiment)o3o3x3x4/3x = gaqrin (great quasirhombated penteract)o3x3o3o4/3x = quappin (quasiprismated penteract, in fawdint regiment)o3x3o3x4/3x = quiptin (quasiprismatotruncated penteract)o3x3x3o4/3x = quiprin (quasiprismatorhombated penteract, in gibtadin regiment)o3x3x3x4/3x = gaquapan (great quasiprismated penteract)x3o3o3o4/3x = quacant (quasicellipenteracti32, in ginnont regiment)x3o3o3x4/3x = quacpot (quasicelliprismated 32)x3o3x3o4/3x = quacrant (quasicellirhombipenteracti32, in wacbinant regiment)x3o3x3x4/3x = quacgarn (quasicelligreatorhombated penteract)x3x3o3o4/3x = caquapin (celliquasiprismated penteract, in getitdin regiment)x3x3o3x4/3x = quicpatint (quasicelliprismatotruncated penteracti32)x3x3x3o4/3x = quicgrat (quasicelligreatorhombated 32, in gibcotdin regiment)x3x3x3x4/3x = gaquacint (great quasicellipenteracti32)o3o3o3x4/3x4*c = ginnont (great penteractipenteracti32)x3o3o3x4/3x4*c = skatbacadint (small skewtrigonary biprismatocellidispenteracti32)o3x3o3x4/3x4*c = skivbadant (small skewverted biprismatodispenteracti32)x3x3o3x4/3x4*c = sibacadint (small biprismatocellidispenteracti32)o3o3x3o4/3x4*c = rawn (retrosphenary penteract, in sirn regiment)x3o3x3o4/3x4*c = rawcbinant (retrosphenary cellibiprismatopenteractipenteracti32, in carnit regiment)o3x3x3o4/3x4*c = sibtadin (small biprismato32dispenteract, in prin regiment)x3x3x3o4/3x4*c = sibcotdin (small biprismatocelli32dispenteract, in cogart regiment)o3o3x3x4/3x4*c = nottant (penteractitruncated penteracti32)o3x3x3x4/3x4*c = nurrant (penteractirhombated penteracti32)x3o3x3x4/3x4*c = niptant (penteractiprismatotruncated penteracti32)x3x3x3x4/3x4*c = nippant (penteractiprismated penteracti32)o3o3x3o4x4/3*c = wavinant (sphenoverted penteractipenteracti32)x3o3x3o4x4/3*c = wacbinant (sphenary cellibiprismatopenteractipenteracti32,)o3x3x3o4x4/3*c = gibtadin (great biprismato32dispenteract, in)x3x3x3o4x4/3*c = gibcotdin (great biprismatocelli32dispenteract)o3o3o3x4x4/3*c = sinnont (small penteractipenteracti32, in scant regiment)x3o3o3x4x4/3*c = giktabacadint* (great skewtrigonary biprismatocellidispenteracti32, in skatbacadint regiment)o3x3o3x4x4/3*c = gakvebidant (great skewverted biprismatodispenteracti32, in gikvacadint regiment)x3x3o3x4x4/3*c = gibacadint (great biprismatocellidispenteracti32, in gidacadint regiment)o3o3x3x4x4/3*c = naquitant (penteractiquasitruncated penteracti32)o3x3x3x4x4/3*c = noqrant (penteractiquasirhombated penteracti32)x3o3x3x4x4/3*c = naquipptant (penteractiquasiprismatotruncated penteracti32)x3x3x3x4x4/3*c = noquapant (penteractiquasiprismated penteracti32)o3o3x3o *b4x4/3*c = fawdint (frustisphenary dispenteracti32)o3o3x3x *b4x4/3*c = getitdin (great tesseracti32dispenteract)x3o3x3o *b4x4/3*c = gikvacadint (great skewverted cellidispenteracti32)x3o3x3x *b4x4/3*c = gidacadint (great discellidispenteracti32)o3x3o3o *b4x4/3*c = sirpin (small retroprismated penteract, in span regiment)x3x3o3o *b4x4/3*c = setitdin (small tesseracti32dispenteract, in cappin regiment)o3x3o3x *b4x4/3*c = sikvacadint (small skewverted cellidispenteracti32, in skivbadant regiment)x3x3o3x *b4x4/3*c = sidacadint (small  discellidispenteracti32, in sibacadint regiment)o3x3x3o *b4x4/3*c = danbitot (dispenteractibitruncated 32)o3x3x3x *b4x4/3*c = sadinnert (small dispenteractirhombated 32)x3x3x3o *b4x4/3*c = gadinnert (great dispenteractirhombated 32)x3x3x3x *b4x4/3*c = danpit (dispenteractiprismated 32)o4o3x3x3o3/2*c = rawt (retrosphenary 32, in sart regiment)x4o3x3x3o3/2*c = sircarn (small retrocellirhombated penteract, in carnit regiment)o4x3x3x3o3/2*c = repirt (retroprismatorhombated 32, in pirt regiment)x4x3x3x3o3/2*c = srocgrin (small retrocelligreatorhombated penteract, in cogrin regiment)o4o3o3x3x3/2*c = 2rinhit (rinhit = retropenteractihemi32, in rat regiment)x4o3o3x3x3/2*c = katacbadint* (skewtrigonary cellibiprismatodispenteracti32, in skatbacadint regiment)x4/3o3x3x3o3/2*c = qracorn (quasiretrocellirhombated penteract, in wacbinant regiment)x4/3x3x3x3o3/2*c = gorcgrin (great retrocelligreatorhombated penteract, in quacgarn regiment)o3o3x4o *b3/2x3*c = ript (retoprismated 32, in spat regiment)o3o3x4x *b3/2x3*c = sorcpit (small retrocelliprismated 32, in capt regiment)o3x3x4o *b3/2o3*c = ribrant (retrobirhombipenteracti32, in sibrant regiment)x3x3x4o *b3/2o3*c = roptit (retroprismatotruncated 32, in pattit regiment)o3x3x4x *b3/2o3*c = sroptin (small retroprismatotruncated penteract, in pattin regiment)x3x3x4x *b3/2o3*c = sircaptint (small retrocelliprismatotruncated penteracti32, in captint regiment)o3x3o4x *b3x3/2*c = skovactaden (small skewverted celli32dispenteract, in skivbadant regiment)x3x3o4x *b3x3/2*c = scadnicat (small cellidispenteracticelli32, in sibacadint regiment)o3o3x4/3x *b3/2x3*c = gorcpit (great retrocelliprismated 32, in quacpot regiment)o3x3x4/3x *b3/2o3*c = groptin (great retroprismatotruncated penteract, in quiptin regiment)x3x3x4/3x *b3/2o3*c = gircaptint (great retrocelliprismatotruncated penteracti32, in quicpatint regiment)o3x3o4/3x *b3x3/2*c = gokvactaden (great skewverted celli32dispenteract, in gikvacadint regiment)x3x3o4/3x *b3x3/2*c = gacdincat (great cellidispenteracticelli32, in gidacadint regiment)x3o3/2x3*a4x4/3o3*a = recarnit (retrocellirhombipenteracti32, in carnit regiment)x3o3/2x3*a4x4/3x3*a = narptint (penteractiretroprismatotruncated penteracti32, in niptant regiment)x3o3/2x3*a4/3x4o3*a = garcornit (great retrocellirhombipenteracti32, in wacbinant regiment)x3o3/2x3*a4/3x4x3*a = noqraptant (penteractiquasiretroprismatotruncated penteracti32, in naquiptant regiment)o3o4x4/3x3x3/2*b3*d *c4*e = raktatant (retroskewtrigonary trispenteracti32, in skatbacadint regiment)o3x4x4/3x3o3/2*b3*d *c4*e = skevatant (small skewverted trispenteracti32, in skivbadant regiment)x3x4x4/3x3o3/2*b3*d *c4*e = scatnit (small cellitrispenteracti32, in sibacadint regiment)o3x4/3x4x3o3/2*b3*d *c4/3*e = gakevatant (great skewverted trispenteracti32, in gikvacadant regiment)x3x4/3x4x3o3/2*b3*d *c4/3*e = gactanet (great cellitrispenteracti32, in gidacadintregiment)`

Again, I'm fairly sure that's all of them... Next one will contain the hinnics (and some members of pentic regiments with hinnic symmetry).
Trionian

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Re: IncMats Website Update

wendy wrote:What are ye meaning by hemi-wythoffian?

If I understand it, he means polytopes that aren't Wyhoffian by itself, but a degenerate double cover is.

A simple 3D example is thah (tetrahemihexahedron, in oct regiment). Thah isn't Wythoffian, but its double cover is represented by the symbol x3/2o3x.

Well, we have to be careful when we say "double cover", because (A) elements lower than ridges don't need to be doubled, they can just have twice the number of higher-dimensional elements incident to them and (B) instead of having a pair of overlapping facets, you can have one facet which is a double cover itself and therefore has density 2. The definition I use allows for all of these to happen; it lets each facet of the single covering correspond to either a pair of overlapping facets or a single facet which is a double cover itself, under the same definition.
Climbing method and elemental naming scheme are good.
polychoronlover
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Re: IncMats Website Update

If you can write it in my notation like x3/2o3x, it's wythoffian. The grand antiprism in 4d is not wythoffian.
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wendy
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Re: IncMats Website Update

Polychoronlover is right. Any mixture of multiple completely coincident elements or single elements, which are multicovers themselves, can occur. Even overlays of subdimensional elements which are coincident wraps of multicovered elements with several single covered ones might occur. Esp. when you look in higher covering numbers. E.g. x5/2x5/2x5/2*a = 6doe, here every pentagon of doe has 3 layers of doubly wound decagons!

Wendy, yes, you are right, x3o3/2x = 2thah is Wythoffian, but thah itself not. And this was the initial point here. Username5243 was calling polytopes "hemi-Wytoffian" whenever a double cover of it can be given to be Wythoffian... Admitted, neither a really clear, nor constructive definition.

But on the other hand, Wythoffian itself becomes a bit more difficult, whenever it is Grünbaumian. This is why often Grünbaumian polytopes are rejected completely. OTOH several one-sided uniform polytopes appear just as an ironed Wythoffian double cover, where double coveres (of any type) just become replaced by the single appearing shape...

--- rk
Klitzing
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Re: IncMats Website Update

And finally, here are all the hinnic (demipenteractic) Wythoffians. Several are repeats of the pentics; i've just included them for completeness.

Code: Select all
`x3o3o *b3o3o = hin (demipenteract)x3x3o *b3o3o = thin (truncated demipenteract)x3o3o *b3x3o = sirhin (small rhombated demipenteract)x3x3o *b3x3o = girhin (great rhombated demipenteract)x3o3o *b3o3x = siphin (small prismated demipentereact)x3x3o *b3o3x = pithin (prismatotruncated demipenteract(x3o3o *b3x3x = pirhin (prismatorhombated demipenteract)x3x3o *b3x3x = giphin (great prismated demipenteract)o3o3o *b3o3x = taco3o3o *b3x3o = rato3o3o *b3x3x = toto3x3o *b3o3o = nito3x3o *b3o3x = sarto3x3o *b3x3o = bittito3x3o *b3x3x = gartx3o3x *b3o3o = rinx3o3x *b3o3x = spatx3o3x *b3x3o = sibrantx3o3x *b3x3x = pattitx3x3x *b3o3o = bittinx3x3x *b3o3x = pirtx3x3x *b3x3o = gibrantx3x3x *b3x3x = gippitx3o3o3o3/2o3*b = 2hehad (hehad = hexadecahemidecateron, in tac regiment)o3x3o3o3/2o3*b = 2rhohid (rhohid = rectified hehad, in rat regiment)x3x3o3o3/2o3*b = 2thehid (thehid = truncated hehad, in tot regiment)o3o3x3o3/2x3*b = 2brohahd (brohahd = birectified hehad, in nit regiment)x3o3x3o3/2x3*b = 2howoh (howoh = hexadecasphenohexadecateron = small rhombated hehad, in sart regiment)o3x3x3o3/2x3*b = 2bathehad (bathehad = bitruncated hehad, in bittit regiment)x3x3x3o3/2x3*b = 2grahehad (grahehad = great rhombated hehad, in gart regiment)o3x3x3o3/2o3*b = quafdidoh (quasifacetodispenteractidishexadecateron, in sirhin regiment)x3x3x3o3/2o3*b = ripperhin (retroprismatorhombated demipenteract, in pirhim regiment)o3o3o3x3x3/2*b = fedandoh (facetodispenteractidishexadecateron, in siphin regiment)x3o3o3o3o3/2*a = 2dah (dah = dishexadecateron, in hin regiment)x3o3x3o3o3/2*a = 2hiquah (hiquah = hexadecaquasihexadecateron = rectified dah, in nit regiment)x3x3x3o3o3/2*a = 2tadah (tadah = truncated dah, in thin regiment)x3x3o3o3o3/2*a = firn (facetorectified penteract, in rin regiment)o3x3x3/2o3*b3o = rawto3x3x3/2o3*b3x = ripthin (retroprismatotruncated demipenteract, in pithin regiment)x3x3x3/2o3*b3x = repirto3x3x3x3o3*b *c3/2*e = brewahen (biretrosphenary demipenteract, in sibrant regiment)x3x3x3x3o3*b *c3/2*e = ropith (retroprismatotruncated hexadecateron, in pattit regiment)o3o3/2x3o3/2x3*a *c3*e = 2harhan (harhan = hexadecaretrohemipenteract, in nit regiment)x3o3o3x3x3*a *c3/2*e = rorpdah (retroretroprismated dishexadecateron, in spat regiment)`

That's all the hinnics. (I know several more pentics have half symmetry symbols, but I haven't bothered to figure those out yet...)
Trionian

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Re: IncMats Website Update

You missed 2rawroh, the existence of whose Dynkin symbol only came to me a few days ago. Its symbol is x3/2o3x3o3/2x3*a3*d *c3*e, which puts it in a new family. Rawroh, the retrosphenary retrohexateron, has a verf which is a bowtie-wedge like Howoh's, but crossed. Its terons are 16 rawvtips, 10 tithoes, 10 rathoes, and 5 rawvhittoes.

The pentic Wythoffians and hemi-Wythoffians under demipentic symmetry you missed are:

Code: Select all
`o3o3x3x3x3*b3/2*d = ripto3x3x3o3x3*b3/2*d = ribrantx3x3x3o3x3*b3/2*d = roptit`

I think this is all of them unless I've missed any.

By the way: Giktabacadint, one of the names you coined for an shape unnamed in the spreadsheet, is already on Klitzing's site, listed as the conjugate of Skatbacadint.
Climbing method and elemental naming scheme are good.
polychoronlover
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Re: IncMats Website Update

I'm experimenting with a notation for this, but i should need to do some modeling. The current model is to derive the hemi-wythoffs from the wythoff by an additional rune, something in line with Klitzing's * notation. In essence, the wythoff notation is a construction, and the hemiate is a modification to it, eg

x5o3o = o5x5/2x *+ is a d3 reduction of the latter to the former. We can't use letters after *, because they are all reserved here.
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wendy
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Re: IncMats Website Update

Just realized that the last two on my list can have simpler linearizations...

rorpdah can be represented more simply as o3x3x3x3o3*a3/2*c.

"2harhan" can be represented more simply as x3o3/2x3o3o3/2*a3*c.

These Dynkin diagrams would look like a square+triangle 2-loop.
Trionian

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Re: IncMats Website Update

Did you ever consider to do a similar research as for the addition of Schwarz triangles or addition of Goursat tetrahedra for the 4D orthoschemes of 5D symmetries? - Then you could get sure to have produced all valid Dynkin diagrams, which thereafter could be decorated. You even would get the respective multiplicities for free.
--- rk
Klitzing
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Re: IncMats Website Update

Wait, did you just update your incmats site? It seems the following text has appeared on the version history page:

Klitzing, on his history page wrote:2017 / 5 / 19

- outline on rectify vs. ambo
esp. its application to the Catalans oPmQoRo, resulting in the Gévay polychora
- research in area of C4nRF,
esp. those occuring within Partial Stott Expansions when allowing additionally for the "diagonally expanded square"
- expanded section on non-convex Wythoffian polytera
- expanded section on "non-convex" Wythoffian honeycombs (i.e. including non-convex cells, retrograde cells, or 2D tilings)
- elaboration of the main hexacombs and their relation to lattices
- more than 100 new incmats files, more than 200 new incidence matices
Trionian

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Re: IncMats Website Update

username5243 wrote:Wait, did you just update your incmats site? It seems the following text has appeared on the version history page: ...

yes indeed: I herewith want to inform you all about the next IncMats Website update:

Version Changes
2017 / 5 / 19

- outline on rectify vs. ambo
esp. its application to the Catalans oPmQoRo, resulting in the Gévay polychora
- research in area of C4nRF,
esp. those occuring within Partial Stott Expansions when allowing additionally for the "diagonally expanded square"
- expanded section on non-convex Wythoffian polytera
- expanded section on "non-convex" Wythoffian honeycombs (i.e. including non-convex cells, retrograde cells, or 2D tilings)
- elaboration of the main hexacombs and their relation to lattices
- more than 100 new incmats files, more than 200 new incidence matices

--- rk
Klitzing
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Re: IncMats Website Update

The next update of my IncMats website finally is up now:

Version Changes
2017 / 9 / 19

• new page on Catalan solids and their 4D counterparts
• ursatopes and tutsatopes added to axials page
• found one further (ico-wise) 24-diminishing and added also several (spid-wise) 20-diminished CRFs
• finally got around to evaluate the huge incidence matrix of gotanq,
which is the omnitruncated form of the 7D Gosset polytope 32,1
• cross-refs to several known diminishings of the various EKFs
• more than 80 new incmats files, more than 130 new incidence matrices
--- rk
Klitzing
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Re: IncMats Website Update

I've to report the next IncMats website update now finally.

Main topics of work of the last few month got its preciptation within the list of Version Changes as follows:

2018 / 7 / 4

--- rk
Klitzing
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Re: IncMats Website Update

The next update of my IncMats website is up!

2019 / 4 / 24
- started to collect 6D - 8D noble polytopes
- found 2 more scaliform polyzetta: codify and brocalbroc
- first lace hyper cities, e.g. 3D displays for naq = 321, laq = 231, lin = 132, hax = 131,
and 4D ones for rek, codify
- some general thoughts on a roundness measure of polytopes
- added a (non-unit-edged) toroidal polyhedron, which allows to tile 3D-space without gaps and overlaps
- about 90 new incmats files and exactly 200 new, mostly quite huge incidence matrices, thus summing up to the following totals:

Total count of actually available polytopes: 3398
Total count of contained incidence matrices: 8111

--- rk
Klitzing
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Re: IncMats Website Update

Just 2 months later the next update of my IncMats website is up already!

2019 / 6 / 22
- added an own page on ridge facetings as well as several according sub-pages
- added an own page on lace simplices, listing all the easiest cases each
- added a section on the Dehn-Summerville equations for simplicial polytopes
- several broken links, esp. on the pages for hyperbolic tilings and to pics of polyterons, now got fixed
- exactly 50 new incmats files as well as nearly 150 additional incidence matrices, many of them for various segmentotera, thus summing up to the following totals:

Total count of actually available polytopes: 3448
Total count of contained incidence matrices: 8256

--- rk
Klitzing
Pentonian

Posts: 1621
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

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