Quickfur's renders

[quickfur]

Just posting the polytope of the month from my website here, per tradition:



This month I extended my website scripts to be able to generate cross-eyed stereo pairs, so new renders from now on will probably all be cross-eyed stereo pairs. I find that they help a lot to convey the 3D-ness of the projections, much more than monkeying around with the 3D viewpoint / ridge colorings.

Anyway, this is the cantitruncated 24-cell (aka x3x4x3o). With this, coverage of the 24-cell family (convex) uniform polychora on my website is now complete. That leaves just 2 more tesseract family uniforms and 6 120-cell family uniforms. So in about another 8 months or so, uniform polychoron coverage will be complete! After that, I'll probably start on some CRFs... that could last the rest of my life, given the sheer number of them. Either that, or I'll start foraying into 5D polytopes (they will require 4D glasses though ).

[quickfur]

This month's Polytope of the Month is the cantellated 120-cell:



This is a pretty polychoron with 120 rhombicosidodecahedra, 600 octahedra, and 1200 triangular prisms. It can be made by a Stott expansion on the octahedral cells of the rectified 600-cell. I would say it's the closest analogue to the rhombicosidodecahedron itself, and as such exhibits the possibility of gyrations and diminishings. In particular, one can cut decagonal_prism||pentagon segmentochora from it to produce various diminishings, as well as gluing said segmentochoron back the "wrong way", i.e., gyrated, to form a large number of CRF gyrated cantellated 120-cells. The octahedral cells split into square pyramids. The number of possible diminishings/gyrations is the same as the number of non-adjacent pentagons on the 120-cell. I don't know how to accurately count that number, but it has to be quite a big number!

There is (at least) another kind of diminishing possible, that is to cut off rhombicuboctahedron||great_rhombicosidodecahedron.segmentochora. This introduces great rhombicuboctahedron cells to the result, which is still CRF, and so adds another layer of possible CRF diminishings. This type of diminishing can be done in the positions isomorphic to non-adjacent cells on the 120-cell.

All diminishings will produce various diminished rhombicosidodecahedral cells, along with some combination of gyrations -- I haven't proved it, but you should be able to get all of the Johnson solids derived from the rhombicosidodecahedron as cells, by the appropriate truncations/gyrations.

So this little pretty thing is a veritable gold mine of CRFs, even more so than the cantellated tesseract!

On that note, it seems that cantellated polytopes in general have this property of having CRF diminishings and gyrations -- at least AFAICT. The cantellated 24-cell is another example that have yet to be fully explored: it shows a lot of promise of generating a large number of CRFs (though not as many as this one generates).

[Klitzing]

This month's Polytope of the Month is the cantellated 120-cell:

Great pic (as usual )

This is a pretty polychoron with 120 rhombicosidodecahedra, 600 octahedra, and 1200 triangular prisms. It can be made by a Stott expansion on the octahedral cells of the rectified 600-cell. I would say it's the closest analogue to the rhombicosidodecahedron itself, and as such exhibits the possibility of gyrations and diminishings. In particular, one can cut decagonal_prism||pentagon segmentochora from it to produce various diminishings, as well as gluing said segmentochoron back the "wrong way", i.e., gyrated, to form a large number of CRF gyrated cantellated 120-cells. The octahedral cells split into square pyramids. The number of possible diminishings/gyrations is the same as the number of non-adjacent pentagons on the 120-cell. I don't know how to accurately count that number, but it has to be quite a big number!

There is (at least) another kind of diminishing possible, that is to cut off rhombicuboctahedron||great_rhombicosidodecahedron.segmentochora. This introduces great rhombicuboctahedron cells to the result, which is still CRF, and so adds another layer of possible CRF diminishings. This type of diminishing can be done in the positions isomorphic to non-adjacent cells on the 120-cell.

This is neither a "great_rhombicosidodecahedron" nor a "great rhombicuboctahedron"! It is a truncated dodecahedron. (Cf. below.)

All diminishings will produce various diminished rhombicosidodecahedral cells, along with some combination of gyrations -- I haven't proved it, but you should be able to get all of the Johnson solids derived from the rhombicosidodecahedron as cells, by the appropriate truncations/gyrations.

So this little pretty thing is a veritable gold mine of CRFs, even more so than the cantellated tesseract!

On that note, it seems that cantellated polytopes in general have this property of having CRF diminishings and gyrations -- at least AFAICT. The cantellated 24-cell is another example that have yet to be fully explored: it shows a lot of promise of generating a large number of CRFs (though not as many as this one generates).

You question what lace prism would be obtained as the rhombi-figure first caps of a small rhombated polychoron (i.e. cantellated one). The answer can be given in a closed form. I just have to introduce some secant lengths first:

let x(n, k) be the length of the secant of the regular n-gon, which connects some vertex with the k-th follower. So, for example we have generally x(n, 0) = 0 and x(n, 1) = 1. Likewise x(n, n-1) = 1 and x(n, n) = 0. More generally we have x(n, k) = sin(k pi/n)/sin(pi/n).
(You even could use rational numbers in the first argument (polygrams), the second clearly remains integral, esp. x(P, numerator(P)) = 0.)

Some special values would be:

Code: Select all

P   | x(P,2) x(P,3)
----+--------------
3   | 1      0
4   | sqrt2  1
5   | tau    tau
5/2 | 1/tau  -1/tau


That desired cap of any o-P-x-Q-o-R-x at the x-Q-o-R-x first direction then would be: x-Q-o-R-x || x(P,3)-Q-x(Q,2)-R-x.
Esp. if you ask the bottom figure to be unit edged too, both x(P,3) and x(Q,2) should be either of length 1 or 0. Therefore Q is forced to be 3, and P might be 3 or 4.

You would be just left with:

Code: Select all

o-3-x-3-o-P-x   has caps of the form   x-3-o-P-x || o-3-x-P-x   (i.e. xo3oxPxx&#x)
o-4-x-3-o-3-x   has caps of the form   x-3-o-3-x || x-3-x-3-x   (i.e. xx3ox3xx&#x)


--- rk

[quickfur]

This month's Polytope of the Month is the cantellated 120-cell:

Great pic (as usual )

Thanks!

[...]
There is (at least) another kind of diminishing possible, that is to cut off rhombicuboctahedron||great_rhombicosidodecahedron.segmentochora. This introduces great rhombicuboctahedron cells to the result, which is still CRF, and so adds another layer of possible CRF diminishings. This type of diminishing can be done in the positions isomorphic to non-adjacent cells on the 120-cell.

This is neither a "great_rhombicosidodecahedron" nor a "great rhombicuboctahedron"! It is a truncated dodecahedron. (Cf. below.)

You're right! The piece that's cut off is rhombicosidodecahedron||truncated_dodecahedron. My mistake.

[...]
On that note, it seems that cantellated polytopes in general have this property of having CRF diminishings and gyrations -- at least AFAICT. The cantellated 24-cell is another example that have yet to be fully explored: it shows a lot of promise of generating a large number of CRFs (though not as many as this one generates).

You question what lace prism would be obtained as the rhombi-figure first caps of a small rhombated polychoron (i.e. cantellated one). The answer can be given in a closed form. I just have to introduce some secant lengths first:

let x(n, k) be the length of the secant of the regular n-gon, which connects some vertex with the k-th follower. So, for example we have generally x(n, 0) = 0 and x(n, 1) = 1. Likewise x(n, n-1) = 1 and x(n, n) = 0. More generally we have x(n, k) = sin(k pi/n)/sin(pi/n).
(You even could use rational numbers in the first argument (polygrams), the second clearly remains integral, esp. x(P, numerator(P)) = 0.)

Some special values would be:

Code: Select all

P   | x(P,2) x(P,3)
----+--------------
3   | 1      0
4   | sqrt2  1
5   | tau    tau
5/2 | 1/tau  -1/tau


That desired cap of any o-P-x-Q-o-R-x at the x-Q-o-R-x first direction then would be: x-Q-o-R-x || x(P,3)-Q-x(Q,2)-R-x.
Esp. if you ask the bottom figure to be unit edged too, both x(P,3) and x(Q,2) should be either of length 1 or 0. Therefore Q is forced to be 3, and P might be 3 or 4.

You would be just left with:

Code: Select all

o-3-x-3-o-P-x   has caps of the form   x-3-o-P-x || o-3-x-P-x   (i.e. xo3oxPxx&#x)
o-4-x-3-o-3-x   has caps of the form   x-3-o-3-x || x-3-x-3-x   (i.e. xx3ox3xx&#x)


--- rk

Is it true that a similar cutting is possible in higher dimensions too? I.e., polytopes of the form xPo3x3o3o...3o can have pieces cut off, rotated, and glued back on. Or is it peculiar to 4D?

Also, I was wondering how to count the number of diminishings these polytopes have, esp. for the 120-cell family. I admit I'm not very good at counting things. The number of CRF 600-cell diminishings, for example, still eludes me. I'm currently using the definition of diminishing as the convex hull of a subset of the vertices such that (1) the hull is CRF and (2) has the same edge length as the original polytope.

One interesting direction of research is to find the maximally-diminished polytopes of a given uniform polytope, that is, given some uniform polytope U, what are the smallest subsets of its vertices that are CRF and have the same edge length as U? (Smallest as in, you cannot remove any more vertices from it without making it non-CRF or altering its edge length.) Each U may have more than one maximal diminishing. These may be considered the "fundamental" forms of the diminished uniform polytopes, from which the larger subsets can be rebuilt by gluing on (probably CRF) pieces. I've already enumerated the maximal diminishings for the 5-cell family and tesseract families (but I don't know if the list is complete), and currently I'm looking into the 24-cell family diminishings. Many of the recent CRFs I posted in the other thread were discovered while trying to find maximal diminishings, so this is quite a fruitful way of finding more CRFs. Many of the 5-cell family polytopes have maximal diminishings that are segmentochora; for example, the bidiminished rectified 5-cell yields line||orthogonal_3prism (4.8.2 in your list), which gives us a nice intuitive rationalization of the latter, which otherwise may seem to be an arbitrary shape that just happens to be CRF!

[Klitzing]

Yes similar cuttings are possible in any dimension. Those are just a property of how the net will be built: Consider some facet polytope (or alternatively some lower dimensional element) and all its (full dimensional) neighbouring facets. If all those neighbours are either monostratic themselves (in that specific orientation, for sure) or do have themself the possibility for segmentotopal caps, then this property of the used facets will be lifted one dimension up!

Gyration, i.e. cutting off some cap and glueing it back within a different orientation, OTOH looks like being more an effect of 3D. Here the relevant caps are cupolae. The base has alternating edges which connect to lacing triangles resp. lacing squares. So some rotation (gyration) can interchange those edges. In order to have the same effect for 4D figures, you would need sections, i.e. base polyhedra of the relevant cap, which has a higher symmetry than the cap itself. (Else any symmetry applied to the base, being required for glueing back, would result in an identity operation of the whole cap!) This looks to me rather restrictive in higher dimensions...

--- rk

[Klitzing]

Gyration, i.e. cutting off some cap and glueing it back within a different orientation, OTOH looks like being more an effect of 3D. Here the relevant caps are cupolae. The base has alternating edges which connect to lacing triangles resp. lacing squares. So some rotation (gyration) can interchange those edges. In order to have the same effect for 4D figures, you would need sections, i.e. base polyhedra of the relevant cap, which has a higher symmetry than the cap itself. (Else any symmetry applied to the base, being required for glueing back, would result in an identity operation of the whole cap!) This looks to me rather restrictive in higher dimensions...

4D caps with one subsymmetrical base are:

• tet || oct (= tet-cap of rap), but in fact = rap itself --> so no gyration possible
• tet || co (= tet-cap of spid); spid = tet || co || dual tet --> ortho bi-(tet || co) possible
• oct || tut (= oct-cap of srip); wrong way round --> no gyration possible
• co || tut (= co-cap of srip); wrong way round --> no gyration possible
• tut || toe (= tut-cap of prip) --> gyration possible

--- rk

[quickfur]

Gyration, i.e. cutting off some cap and glueing it back within a different orientation, OTOH looks like being more an effect of 3D. Here the relevant caps are cupolae. The base has alternating edges which connect to lacing triangles resp. lacing squares. So some rotation (gyration) can interchange those edges. In order to have the same effect for 4D figures, you would need sections, i.e. base polyhedra of the relevant cap, which has a higher symmetry than the cap itself. (Else any symmetry applied to the base, being required for glueing back, would result in an identity operation of the whole cap!) This looks to me rather restrictive in higher dimensions...

4D caps with one subsymmetrical base are:

• tet || oct (= tet-cap of rap), but in fact = rap itself --> so no gyration possible
• tet || co (= tet-cap of spid); spid = tet || co || dual tet --> ortho bi-(tet || co) possible
• oct || tut (= oct-cap of srip); wrong way round --> no gyration possible
• co || tut (= co-cap of srip); wrong way round --> no gyration possible
• tut || toe (= tut-cap of prip) --> gyration possible

--- rk

You're right, it only works for a few select cases.

Makes me wonder, though, whether it might be possible in 5D if the base of a cut-off segmentoteron is in the shape of a duoprism; then gyration in theory should be possible in two directions. I don't know if there are any actual instances of this among the 5D uniforms, though.

[Klitzing]

You're right, it only works for a few select cases.

Makes me wonder, though, whether it might be possible in 5D if the base of a cut-off segmentoteron is in the shape of a duoprism; then gyration in theory should be possible in two directions. I don't know if there are any actual instances of this among the 5D uniforms, though.

The question is not the symmetry of the section alone.
It rather is that the larger base polytope of the cap shall have a larger symmetry than the cap as a whole (around its orthogonal axis)!

--- rk

[quickfur]

You're right, it only works for a few select cases.

Makes me wonder, though, whether it might be possible in 5D if the base of a cut-off segmentoteron is in the shape of a duoprism; then gyration in theory should be possible in two directions. I don't know if there are any actual instances of this among the 5D uniforms, though.

The question is not the symmetry of the section alone.
It rather is that the larger base polytope of the cap shall have a larger symmetry than the cap as a whole (around its orthogonal axis)!

--- rk

Ah, right, otherwise rotating it will not produce a distinct shape. Got it.

[quickfur]

It's been awfully quiet in here for the last while. Where did everyone go? Anyway, let's remedy that with a pretty little uniform polychoron:



This is the runcitruncated tesseract (C1101), or, in Wendy's notation, x4x3o3x. It can be constructed by the Stott expansion of the truncated cubes in the truncated tesseract (C1100 / x4x3o3o).

Here's a mind-boggling animation to go with it:



Don't you just love Clifford double rotations?

This is the first animation I did with visibility clipping turned on (and also the first 3D stereographic animation I made!), so you may notice cells popping in and out (although from this particular viewpoint it's not too bad, and the "shadowy" outline of the cuboctahedra right before they emerge on the near side is a happy coincidence - they are caused by the ridges in the cuboctahedra being assigned the color of the cell, so when they lie on the limb of the polytope, they show up as a "shell" or outline of the actual cell which is actually hidden from view).

[Nick]

Quickfur, how does one even begin to envision these rotations in one's head? I can kinda do it with a hypercube, but these things are ridiculously complex!

[quickfur]

Quickfur, how does one even begin to envision these rotations in one's head? I can kinda do it with a hypercube, but these things are ridiculously complex!

One can understand them via dimensional analogy. And yeah, they are ridiculously complex. They're lots of fun, though!

(And this particular polychoron is just a member of the relatively simple tesseract family... try this 120-cell family polychoron to blow your mind with:

)

[quickfur]

I missed last month's polytope of the month, and indeed, I haven't been on this forum for a while. But anyway, here's the polytope of the month for January:



This is the truncated 120-cell. Something that's conceptually rather simple, but nevertheless beautifully intricate, just like the rest of the 120-cell family of polychora.

I'm not too fond of the coloring scheme used here, but it does allow one to see into the projection at least as far as the central truncated dodecahedron, especially when one is viewing cross-eyed.

I only have about a handful of uniform polychora left to render on my website; I'm thinking to go into the CRFs next. What do you people think? Should I start doing CRFs or should I start doing 5D projections?

(And of course, I'm saving the omnitruncated 120-cell for the last; it will be the grand-daddy of them all to cap off the uniform polychora projections. )

[quickfur]

It's that time of the month again, and this month, we take a detailed look at one of the gyrated polychora that Klitzing & myself have been talking about last month. Here's the paratetragyrated cantellated tesseract:



It is made by gyrating four 8prism||square segments of the cantellated tesseract x4o3x3o, locating on two orthogonal rings, and antipodal to each other. This gyration turns the 8 rhombicuboctahedra into J37's (gyrate rhombicubocahtedra), and splits 12 of the 16 octahedra into 24 square pyramids. This CRF polychoron is interesting because all of the rhombicuboctahedra become J37's.

[Klitzing]

In order to provide references:

• Quickfur came up with this shape on 2012 / 08 /06 in this post. Then he called it "pseudo cantellated tesseract", which in hedrondudes terminology would read "pseudo (small) rhombitesseract", because of its seeming analogy to the 3D "pseudo (small) rhombicuboctahedron".
   
• But already in its direct comment at the next day (at least in its corrected version thereof) quickfur already calls it "para-tetra-gyrated cantellated tesseract".
   
• About 4 months later, at 2013 / 1 / 12, in the context of the srit (small rhombitesseract = cantellated tesseract = o3x3o4x) to odip (octagonal duoprism = x8o x8o) correlation investigation(*), I re-discovered that figure independingly., cf. this post. There I called that very figure a "bi-para-(bi-)gyrated small rhombitesseract" (even proposing an OBSA (official Bowers-style acronym) "bipgy srit"). In that very post also the unwrapped torus surface of the related odip is displayed, including the markings of the to be used augmentations thereof, in order to result in this figure of consideration. Also its incidence matrix is provided there.
   
• In the run of the following discussion, on the next day in this post, I then argued why this later naming would be much more concise:

  Well we should not only use the total count, but use the internal symmetry as well. So cyclo tetra is unique. But not so para tetra. Therefore I used bi para bi instead. And, as para itself refers to 2 opposite ones, the second bi here is obsolete.

  
  In fact, in each ring of 8 ops (octagonal prisms) then every second one will have to be augmented by xx4ox ox&#x lace prisms (i.e. {4}||op segmentochora), but one such opposite pair in some to be chosen orientation, and the other opposite pair in the relatively gyrated one.
  So -para-(bi-)gyrated therefore refers to such a pair of opposite augmentations, which are to be gyrated.
  And the further qualifyer bi- additionally explains, that this has to be done at both (relatively orthogonal) rings of odip.
  (Note that the choice of the relative starts of counting is not relevant, as because of those gyrations any such choice would result in the very same figure here.)
   
• Sure, boiling down to the grand total, quickfur is correct in that 4 (tetra) of the 8 alternatingly augmented ops will have to be gyrated, and those would come in opposite pairs (para). So his "para-tetra" is not completely wrong, just imprecise.
  And this impreciseness of naming then result in an according impreciseness of construction device. As then one just can speak of "some". Or alternatively, if trying to be exact at least there, one would have to refer back to exactly what was omitted in its naming...
   
• Concluding, I'd strongly recommend the bi-para-(bi-)gyration adjective instead of that mere para-tetra-gyration one!

(*) PS: For short explanation - but unrelated to my point of this post -:

Augmenting any second op of odip (in an ungyrated fashion) would result in sirco (small rhombicuboctahedra) cells instead of those J37 (esquigybcu = elongated square gyrobicupola). If those alternations would be started relatively in the right way within those 2 rings of odip, then the outcome will be nothing else as srit (small rhombitesseract = cantellated tesseract = o3x3o4x).

The other way round, the srit could be diminished (by chopping off some squares) in such a way, that the result would be odip. In fact, both figures not only have the same circumradius by chance, odip rather uses a subset of vertices of srit. Thereby reducing tesseractic symmetry to a mere duoprismatic one, for sure. (In fact, the "some", used within the first line of this paragraph, could be given equally precise: being its bi-cyclo-tetra-diminishing.)

--- rk

[Keiji]

Thanks for the excellent renders and structure explanation as always Quickfur, and thanks for the history Klitzing, as I have been away for too long to catch up otherwise!

I shall not join in on the biparabi- vs. paratetra- debate since I am too ignorant to be taking part in that But as Klitzing has been so kind as to provide the imat I shall endeavor to add it (and its cell, J37) to the Polytope Explorer ASAP.

Edit: Here we go! http://teamikaria.com/hddb/explorer/?n=64

Interestingly, the dual of this shape has two kinds of cells... one is the trigonal bipyramid, and the other is the same as the (unique kind of) cell found in the dual of the BXD! This polytope is essentially an irregular triangular prism, with one of the square faces broken into two triangular faces, creating an extra edge.

This latest addition also has the most complex imat I've added so far, the dual's edge kinds taking up over half the alphabet, and very nearly filling my 1920x1080p screen as displayed in the explorer. I hope we can find some interesting polytopes between the 14-kind K4.8 and the whopping 43-kind "4D J37"...

[quickfur]

@klitzing: you're right, I totally forgot to update my draft webpages for this polytope after we had the discussion about the naming scheme. I'll have to rename the page to "biparabigyrated cantellated tesseract" later when I get some free time. Thanks for the reminder!

[quickfur]

[...]
This latest addition also has the most complex imat I've added so far, the dual's edge kinds taking up over half the alphabet, and very nearly filling my 1920x1080p screen as displayed in the explorer. I hope we can find some interesting polytopes between the 14-kind K4.8 and the whopping 43-kind "4D J37"...

There's another "pseudo cantellated tesseract" (or J37 analogue) candidate, which was discovered by Klitzing (I had actually considered its construction before, but I wrongly rejected it by mistakenly assuming that it was the same as the cantellated tesseract itself). This one is the octagyrated x4o3x3o, which has 8 rhombicuboctahedra just like the x4o3x3o, but they are in two rings that are "misaligned" with each other, so that no octahedral cells are formed, but there's a toroidal net of square pyramids that surround two orthogonal rings of rhombicuboctahedra. I'm planning to do that one at some point, too.

[Klitzing]

[...]
There's another "pseudo cantellated tesseract" (or J37 analogue) candidate, which was discovered by Klitzing (I had actually considered its construction before, but I wrongly rejected it by mistakenly assuming that it was the same as the cantellated tesseract itself). This one is the octagyrated x4o3x3o, which has 8 rhombicuboctahedra just like the x4o3x3o, but they are in two rings that are "misaligned" with each other, so that no octahedral cells are formed, but there's a toroidal net of square pyramids that surround two orthogonal rings of rhombicuboctahedra. I'm planning to do that one at some point, too.

In fact there are 2 such!

Both were contained in that post of mine, providing their "patterns" (showing the according augmentations to the unwrapped toroidal surface of squares of the underlying 8,8-duoprism) and their incidence matrices:

• the cyclotetragyrated small rhombitesseract (just gyrating all {4}||ops of one ring - with respect to the uniform small rhombitesseract), and
• the bicyclotetragyrated small rhombitesseract (gyrating all {4}||ops of both rings)

Both having exactly the same net-count of cells: 8 sircoes + 32 trips + 32 squippies!

PS: srit = small rhombitesseract = cantellated tesseract = x4o3x3o.

--- rk

[quickfur]

[...]
There's another "pseudo cantellated tesseract" (or J37 analogue) candidate, which was discovered by Klitzing (I had actually considered its construction before, but I wrongly rejected it by mistakenly assuming that it was the same as the cantellated tesseract itself). This one is the octagyrated x4o3x3o, which has 8 rhombicuboctahedra just like the x4o3x3o, but they are in two rings that are "misaligned" with each other, so that no octahedral cells are formed, but there's a toroidal net of square pyramids that surround two orthogonal rings of rhombicuboctahedra. I'm planning to do that one at some point, too.

In fact there are 2 such!

Both were contained in that post of mine, providing their "patterns" (showing the according augmentations to the unwrapped toroidal surface of squares of the underlying 8,8-duoprism) and their incidence matrices:

• the cyclotetragyrated small rhombitesseract (just gyrating all {4}||ops of one ring - with respect to the uniform small rhombitesseract), and
• the bicyclotetragyrated small rhombitesseract (gyrating all {4}||ops of both rings)

Both having exactly the same net-count of cells: 8 sircoes + 32 trips + 32 squippies!
[...]

Yeah, I was referring to the second variant (bicyclotetragyrated -- more accurate than octagyrated I guess). The first variant actually I've already known of when I first studied the diminishings of the srit (x4o3x3o). But I wrongly assumed that performing the gyration in both rings produces the srit again, so I didn't actually study it any further. It was only until your post that I realized that actually it produces something new.

[quickfur]

I had some free time over the weekend so I decided to (finally!) do the proper structural renders of the square magnabicupolic ring (aka 4cup||8g, aka octagonal_prism||square):



Including projections from less common POV that shows why Klitzing classified it as a "wedge":

[Klitzing]

Thanks for the excellent renders and structure explanation as always Quickfur, and thanks for the history Klitzing, as I have been away for too long to catch up otherwise!

I shall not join in on the biparabi- vs. paratetra- debate since I am too ignorant to be taking part in that But as Klitzing has been so kind as to provide the imat I shall endeavor to add it (and its cell, J37) to the Polytope Explorer ASAP.

Edit: Here we go! http://teamikaria.com/hddb/explorer/?n=64

Interestingly, the dual of this shape has two kinds of cells... one is the trigonal bipyramid, and the other is the same as the (unique kind of) cell found in the dual of the BXD! This polytope is essentially an irregular triangular prism, with one of the square faces broken into two triangular faces, creating an extra edge.

This latest addition also has the most complex imat I've added so far, the dual's edge kinds taking up over half the alphabet, and very nearly filling my 1920x1080p screen as displayed in the explorer. I hope we can find some interesting polytopes between the 14-kind K4.8 and the whopping 43-kind "4D J37"...

Just a question, Keiji:
Why are your "imats" in the polytope explorer triangular ones only?
I usually provide square matrices.
Whereas you just use the subdiagonal entries (the count of the elements of the elements) and finally add what I use for diagonal (the count of elements) in a separate row below.

Not that this is wrong. And not even that this display reduces the contained information, as there is a useful equation on IncMats (my usage) entries which allows to (re-)calculate the (in your case) missing superdiagonal entries. In fact, this equation (assuming my IncMats setup) runs:

Code: Select all

I(m,m)*I(m,n) = I(n,m)*I(n,n)

But why do I provide those numbers additionally?
Its because those numbers contain informations about those polytopes readily accessible, which otherwise would be hidden:

• The entries of the superdiagonal parts of any vertex row of the IncMat are exactly the element counts of the corresponding vertex figure.
• The entries of the superdiagonal parts of any edge row of the IncMat are exactly the element counts of the corresponding edge figure.
• etc.

Further, there is an other useful thing for my square matrices, which is missing in your triangular display:
The IncMat of the dual of some polytope clearly is obtained just by a rotation of the matrix (within the paper plane) about 180°.
(Thus you usually have to set up 2 entries in your "polytope explorer", while the complete info would be contained in a single square matrix. - Which nonetheless could be provided in either orientation, if you would like.)

--- rk

[Keiji]

Just a question, Keiji:
Why are your "imats" in the polytope explorer triangular ones only?
I usually provide square matrices.
Whereas you just use the subdiagonal entries (the count of the elements of the elements) and finally add what I use for diagonal (the count of elements) in a separate row below.

I omit the redundant information. According to Wikipedia the full matrix duplicates the diagonal on the top, bottom, left and right so I simply chose to remove the diagonal rather than removing the bottom row. My reason for this is so that I don't have to include all the "*" entries. It also makes more sense, because then this bottom row is included as a row in higher dimensional polytopes that use it.

Besides, omitting the redundant information provides the "step" appearance, making it easy to identify boundaries between ranks, and how many kinds of elements there are in each.

Further, there is an other useful thing for my square matrices, which is missing in your triangular display:
The IncMat of the dual of some polytope clearly is obtained just by a rotation of the matrix (within the paper plane) about 180°.
(Thus you usually have to set up 2 entries in your "polytope explorer", while the complete info would be contained in a single square matrix. - Which nonetheless could be provided in either orientation, if you would like.)

Ah, but when I actually go to add a polytope, it works out and adds the dual automatically - I don't need to calculate it myself. If you want to see the values for the dual, just click the link to it.

Maybe I'll add a "full matrix view" some day, though

[Klitzing]

Removed full quote of last post ~Keiji

" it works out and adds the dual automatically " so there is a piece of software behind? and your imats simply are a matter of display?

I for one, did calculate all the thousands of IncMats, provided at my site, by hand. It is the process of calculation, which is the most interesting to me. (Kind of similar to solving a sudoku. )
And then it often is essential to have that "upper part" included as well, making lot of use of that cited equation. Often about 3/4 of the matrix are more or less obvious. But it amounts in solving for the remainder, which shows whether one was right or wrong so far!

--- rk

[Keiji]

Yes, it's a PHP script. If I entered a matrix that doesn't make sense, it will reject it and tell me something is wrong so I can go calculate the elements again.

Although calculating the elements can be fun, not all of us have time to do this. I'm also someone who makes a lot of silly mistakes in their arithmetic so I prefer to have a computer check it over for me. However when I add a 4D shape that I don't know the dual of, I do go calculate the full matrix by hand first, because I need to know what the dual cells are before I can add it to the system (as it requires all lower dimensional elements of the polytope and its dual to be added first). At some point, I hope to have the system calculate the dual imat by itself and find the cells itself, only prompting on non existing cells (and then giving me their imats) to make this process faster.

[Klitzing]

would you like to outline the ideas / algorithms of your script?

--- rk

[Keiji]

No polytope this month?

Klitzing: there isn't really that much to explain. Most of the work is probably just in checking that I've entered a valid matrix - it checks that the number of rows in each group match with the number of columns in each group. It calculates the upper triangle using the formula you mentioned, giving an error if anything fractional appears (which would indicate a wrong number entered), and rotates the matrix to calculate the dual. As for referencing lower dimensional polytopes, it doesn't even bother checking the references make sense - I just tell it the name of the polytope and as long as a polytope with that name exists, it links it for me. So I have to do that bit manually. Mainly, the purpose of the script is to act as a database frontend, and lay the matrices out nicely with the element labels (Va, Ea, ka, Cka Hn.ka) and links to polytopes it uses and that use it.

[quickfur]

No polytope this month?
[...]

Oh, there is! You didn't see it? I just haven't posted it here yet.

Well anyway, here it is:



It's the cantitruncated tesseract: the last of the tesseract family of uniforms that I posted on my website. Now remains only the last few 120-cell family members, and the (convex) uniform polychora will have been covered!

[Marek14]

I was wondering: would it be possible to display some polytera as a 2D array of polyhedral 3D cuts obtained by cutting the polyteron with a plane? Similar to lace city description, but with smaller steps so the transitions would be easier to see?
The two axes don't need to be at right angles, either...

[Klitzing]

That should have been Jonathan's post
- have a look at http://www.polytope.net/hedrondude/polytera.htm.

--- rk