## Quickfur's renders

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

### Re: Quickfur's renders

quickfur wrote:
mr_e_man wrote:On your Uniform Polyhedra page, the "portrait" description is missing the 20-gon prism in the 2nd row.

Haha, I'm not particularly inclined to add it, since there are an infinite number of prisms, and the ones shown are just a few selected ones.

I mean the description doesn't match the picture. The 20-gon prism is in the picture, but it's not in the description.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
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### Re: Quickfur's renders

viewtopic.php?p=19492#p19492
quickfur wrote:In the past few days I kinda got back into dabbling with 4D stuff -- considering how to represent polytope cell complexes as a way of interactively finding 4D CRFs by hand ("4D lego", if you will). Found some interesting theorems; but I'll have to post about that another time.

Have you posted that anywhere yet? (Or, do you even remember what you were talking about then?)

viewtopic.php?p=26001#p26001
quickfur wrote:Also, when coloring a very large polytope, often the surtope subset that needs coloring contains elements that are touching each other, so individual elements have to be assigned different colors. The program currently has no automation for this, so I end up having to solve the graph coloring problem by hand over and over, once per subset of cells to color.

You could add a slight random variation to the colours, so that different cells look different, but still similar enough if they're in the same subset.

viewtopic.php?p=26132#p26132
Mercurial, the Spectre wrote:
quickfur wrote:
quickfur wrote:Nothing new here, but this is the last of the known scaliform polychora on my website, so it seems like a natural thing to follow after spidrox.

Actually, there's one more convex scaliform polytope you haven't posted yet - tutcup (truncated tetrahedral cupolipirsm), which has 2 truncated tetrahedra, 6 tetrahedra, and 8 triangular cupolae. It is a segmentotope whose bases are two oppositely oriented truncated tetrahedra.

Ooh! Thanks for the tip! I'll be sure to post that next month!

Are all of the convex scaliforms already known? Or are these only the currently-known ones?

Definitely yes in 4D. They are related to a specific convex uniform polychoron (ex. tutcup-sircope, bidex-ex, and prissi-prico).

quickfur wrote:Not sure I understand. Are you saying we know all of them, or these are the only ones we know of? If the former, do we have proof that there are no others?

I also want an answer to this. I didn't find anything on Klitzing's site saying that all 4D convex scaliforms have been found. (Surely not all 4D scaliforms have been found, as the search for 4D non-convex uniforms is on-going.)
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
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### Re: Quickfur's renders

mr_e_man wrote:viewtopic.php?p=26132#p26132
Mercurial, the Spectre wrote:
quickfur wrote:
quickfur wrote:Nothing new here, but this is the last of the known scaliform polychora on my website, so it seems like a natural thing to follow after spidrox.

Actually, there's one more convex scaliform polytope you haven't posted yet - tutcup (truncated tetrahedral cupolipirsm), which has 2 truncated tetrahedra, 6 tetrahedra, and 8 triangular cupolae. It is a segmentotope whose bases are two oppositely oriented truncated tetrahedra.

Ooh! Thanks for the tip! I'll be sure to post that next month!

Are all of the convex scaliforms already known? Or are these only the currently-known ones?

Definitely yes in 4D. They are related to a specific convex uniform polychoron (ex. tutcup-sircope, bidex-ex, and prissi-prico).

quickfur wrote:Not sure I understand. Are you saying we know all of them, or these are the only ones we know of? If the former, do we have proof that there are no others?

I also want an answer to this. I didn't find anything on Klitzing's site saying that all 4D convex scaliforms have been found. (Surely not all 4D scaliforms have been found, as the search for 4D non-convex uniforms is on-going.)

No proof is known to me either. Wrt. the convex polychora it is. For the non-convex ones the search obviously is on-going.
--- rk
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### Re: Quickfur's renders

I've been meaning to do this, but haven't gotten to around to it until now: my website has moved. The website itself hasn't changed, but I've switched to a different domain name: https://www.qfbox.info/4d/. The old domain name will remain active until next year, then it will be permanently deleted. Please update your bookmarks! All URL paths remain the same, except for the hostname. So it shouldn't be too hard to update any existing bookmarks/links you have. (In theory, anyway. )
quickfur
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### Re: Quickfur's renders

quickfur wrote:I've been meaning to do this, but haven't gotten to around to it until now: my website has moved. The website itself hasn't changed, but I've switched to a different domain name: https://www.qfbox.info/4d/. The old domain name will remain active until next year, then it will be permanently deleted. Please update your bookmarks! All URL paths remain the same, except for the hostname. So it shouldn't be too hard to update any existing bookmarks/links you have. (In theory, anyway. )

Someone on the Polytope Wiki already seems to have changed it a while ago. I let Klitzing know elsewhere and all his links should be fixed whenever he next updates. I let Bowers know too so he hopefully will when he updates in the near future (we've been doing a lot of non-convex searching lately, he's got a lot of things to update). I can't think of any other links that need updated...
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### Re: Quickfur's renders

It's been a while, but I finally got around to updating my website again. I'm starting the Catalan project -- to cover all the Catalans: 3D first, then the dual prisms and antiprisms, then the 4D catalans. Starting out small, with the relatively small triakis tetrahedron:

I actually already have the next 3 Catalans lined up, but will post only one per month to give myself enough buffer to keep things going. Hopefully I'll be able to keep going until I finish the 4D catalans!
quickfur
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### Re: Quickfur's renders

Don't know if anybody still follows this thread, but I've posted the next Catalan solid on my website:

This is, of course, the rhombic dodecahedron, which everybody here should already know about. But it's nice to finally have an actual page to link to in the various mentions of the rhombic dodecahedron, like in the projection of the tesseract and the 24-cell.

That, and also I'm using these relatively simple 3D catalans as buffer to work on the pages for the 4D catalans. Well, relatively simple except for the pentagonal icositetrahedron (dual of the snub cube), which needed a bit of work to wrangle with due to the cubic equations involved, and the pentagonal hexecontahedron (dual of the snub dodecahedron), which gave me a big headache for maybe a week or two because it involves 12th degree polynomials (square roots of cubics involving the golden ratio). I'm still not 100% sure I've completely picked it apart yet, but at least I've figured out enough to have algebraic expressions for its coordinates and edge length ratios. The pages for all the catalans are in fact already ready, I'm just publishing them one per month.

I'm also doing the duals of uniform prisms and antiprisms, and already one of them will definitely feature as cells in one of the 4D catalans, with at least another one that I suspect will also be cells to another 4D catalan.
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### Re: Quickfur's renders

Will you ever do nonconvex stuff?
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### Re: Quickfur's renders

Mecejide wrote:Will you ever do nonconvex stuff?

I would, if I could find a simple way to construct them. Currently all of the polytopes on my website are made from specifying vertices or hyperplanes and passing them through a convex hull algorithm followed by a face lattice enumeration algorithm that creates the entire structure automatically. But for non-convex stuff a convex hull algorithm obviously wouldn't work. But without that it would be too tedious to construct the face lattice by hand (esp. for the 600-cell family polytopes, which have a large number of elements). For symmetrical polytopes I suppose I could just do faceting... but still, currently I don't really have a good solution for constructing these things in a way that doesn't involve tons of error-prone manual work.
quickfur
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### Re: Quickfur's renders

There is a free polytope software miratope (https://github.com/galoomba1/miratope-rs), which allows to generate the full element structure of a given polytope and even allows to facet a given polytope. In fact it was set up (and there even can be joined to further code it) by members of the polytope discord.

--- rk
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### Re: Quickfur's renders

I generally dislike non-convex polytopes, because we can't even agree on what they are! There are many different definitions, and many different ways to render them, even in 3D.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
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### Re: Quickfur's renders

mr_e_man wrote:I generally dislike non-convex polytopes, because we can't even agree on what they are! There are many different definitions, and many different ways to render them, even in 3D.

Reminds me of Grünbaum's Are Your Polyhedra the Same as My Polyhedra?. I think eventually this boils down to the divide between abstract polyhedra (polytopes) vs actual realizations of them as concrete polyhedra that exist in some kind of ambient space (be it euclidean, hyperbolic, spherical, or I dunno, parabolic or projective). A concrete polyhedron (polytope) always has an associated abstract polyhedron (polytope), but the reverse may not always be true (some abstract polytopes may be impossible to realize faithfully in any kind of space, e.g., Coxeter's 11-cell). Sorta reminds me of the split between ordinals and cardinals in set theory. As long as you stay within the realm of the finite (resp. convex) the two remain equivalent. But once you step out into the realm of the infinite (resp. nonconvex) the two become inherently, irreconcilibly different, and their relationship with each other becomes complicated. And "monsters" like the Cantor set appear (resp. Bowers' "wild" polychora, with fissary / degenerate / etc. cases -- or for that matter, the 11-cell and other such things that can't be faithfully realized in any space).
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### Re: Quickfur's renders

This month's polytope is the tetrakis hexahedron, dual of the truncated octahedron:

It's also the projection envelope of the vertex-first perspective projection of the 24-cell into 3D.

I actually already have one of the 4D catalans rendered and ready, but decided to give myself a bit more time to polish it before posting it publicly. Probably will do it next month.
quickfur
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### Re: Quickfur's renders

Your Catalan pages don't mention some nice properties that follow from duality:
• All vertex figures are regular polygons.
• All 2-dyad (dihedral) angles are equal.
• All 1-dyad angles around a given vertex are equal.
The tetrakis cube could be constructed directly (not dually), by erecting pyramids on the cube's faces, and varying the height of the pyramids until all dihedral angles are equal.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
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### Re: Quickfur's renders

mr_e_man wrote:Your Catalan pages don't mention some nice properties that follow from duality:
• All vertex figures are regular polygons.
• All 2-dyad (dihedral) angles are equal.
• All 1-dyad angles around a given vertex are equal.

The tetrakis cube could be constructed directly (not dually), by erecting pyramids on the cube's faces, and varying the height of the pyramids until all dihedral angles are equal.

Nice!
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