Quickfur's renders

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

Re: Quickfur's renders

Postby quickfur » Fri Sep 01, 2023 4:44 pm

Here's the next 4D Catalan, as promised:

Image

This is what the Polytope Wiki calls the Triangular Antitegmatic Hexacontatetrachoron, the dual of the runcinated tesseract. It's obviously the direct analogue of the deltoidal icositetrahedron, the dual of the rhombicuboctahedron.

Interestingly enough, the projection envelope of this 4D baby is exactly the deltoidal icositetrahedron, having exactly the same proportions. This leads to a conjecture that all duals of x4o...o3x have this projective relationship with each other. I haven't worked out the proof yet, nor have I checked the 5D analogue yet, but I'm fairly confident this is the case. I find this remarkable, because most of the 4D Catalans (that I've checked so far) don't have such direct proportions with their 3D analogues. It seems that x4o...o3x is one of those cases where there's a direct analogy across dimensions -- if this conjecture is true. It's also remarkable that this polychoron has 3 different edge lengths, whereas the deltoidal icositetrahedron has only 2 different edge lengths, yet in the projection the edge length proportions line up nicely. So this projective relationship isn't as trivial as it may seem. In any case, it's an interesting observation.
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Re: Quickfur's renders

Postby wendy » Sat Sep 02, 2023 3:12 am

that's right. with the runcinate xP..Qx, the dual is an antigum along the line between the vertices of xP...Qo and its dual. so the antitegum is based on what is the ...
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Re: Quickfur's renders

Postby mr_e_man » Fri Nov 03, 2023 8:22 am

Next Catalan?
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
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Re: Quickfur's renders

Postby quickfur » Fri Nov 03, 2023 6:32 pm

Oops, I actually had it ready 2 weeks ago, but got busy and totally forgot to post it on the website. :oops: I'll get around to it sometime today, hopefully.
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Re: Quickfur's renders

Postby quickfur » Fri Nov 03, 2023 6:43 pm

OK, got it set up faster than expected. Here it is:

Image

This is the square antitegmatic hecatontetracontatetrachoron (what a mouthful!), the dual of the runcinated 24-cell. This time round I actually have two series of structural renders, one the traditional layer-by-layer cell breakdown, and a bonus one showing the Hopf fibration structure of a subset of the cells. Very interesting, check it out!
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Re: Quickfur's renders

Postby quickfur » Wed Dec 06, 2023 5:48 pm

This month's POM (polytope of the month) is the dual of the uniform pentagonal prism, the (non-Johnson) pentagonal bipyramid:

Image

Nothing special in itself, but it does feature as a cell in a 4D Catalan which will be posted next month. So stay tuned! :)
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Re: Quickfur's renders

Postby Klitzing » Fri Dec 08, 2023 8:03 pm

That one then ought be the dual of the rectified 600-cell, cf. https://bendwavy.org/klitzing/incmats/o3m3o5o.htm, which also is known as "pentagonal-bipyramidal heptacosiicosachoron".
In fact yourself provided a first render in here already, cf. http://hi.gher.space/forum/viewtopic.php?p=16141#p16141, however the pic itself seems lost since (but then was saved and still is shown on my page, hehe).

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Re: Quickfur's renders

Postby quickfur » Fri Dec 08, 2023 8:19 pm

Very nice! So you already know what's coming. :lol:
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Re: Quickfur's renders

Postby quickfur » Tue Jan 02, 2024 5:36 pm

And here it is:

Image

This is the joined 120-cell, sporting 720 pentagonal bipyramids. It is the dual of the rectified 600-cell, one of the rare semiregular polychora.
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Re: Quickfur's renders

Postby quickfur » Wed Feb 14, 2024 5:02 pm

Hmm, apparently I neglected to post this month's POM:

Image

This is, of course, the famous rhombic triacontahedron, which is one of the projection images of the castellated rhombicuboctahedral prism, the first CRF we discovered that sported bilunabirotunda cells in a non-trivial manner. The order-5 vertices correspond to the apices of the castellated prism's pentagonal pyramid cells, and the order-3 vertices correspond with the tetrahedral cells.
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Re: Quickfur's renders

Postby Keiji » Tue Feb 20, 2024 7:30 pm

Thank you for going to the effort to continue to post these after all these years! :)

I'm hoping to spend a little more time reading up on these topics again. What would you say were the most interesting discoveries of the last 5 years or so?
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