Talk:Cupola (geometry)

4D cupola

Could anyone draw the 4D cupolas? They are composed of a cantellated polyhedron for the base (outside) and this polyhedron for the top (inside), linked by tetrahedrons and prisms.

For exemple we can see one in the runcinated 5-cell (a runcinated 5-cell is composed of two cuboctahedral cupolas), or in the runcinated tesseract, which is composed of two rhombicuboctahedral cupolas separed by a rhombicuboctahedral prism.

Thanks a lot. Padex 12:31, 5 Nov 2008

Hello. I would like to say that a year or two back I started making "Johnson analogues" using this exact same construction for the cupolae, and as of June '08 I created a render of the tetrahedral cupola. Firstly, if this information is listed elsewhere, PLEASE inform me, as I am on a constant quest to see if the analogues have already been made. Secondly, is there any way my page could be used to supplement the article? I don't know if I qualify as a "reliable source", but at least some independent verification is better than nothing, right? Here's the page. LokiClock (talk) 23:15, 15 July 2009 (UTC)

Hi LokiClock. About Johnson analogues, on Wikipedia, I believe there is nothing but the hypercupolae. So, every contribution is useful, isn't it? I think you should ask Tom Ruen, who has made a lot for polytopes on Wikipedia, for more informations.

However, I discovered on internet (Convex Segmentochora) a list of several Johnson analogues. I hope that will help you! Padex (talk) 17:19, 7 August 2009 (UTC)

Hmm.. They list a connection of two octahedra via triangular prisms for 4.11.1, but no mention is made of the gyroelongation method that I use. I don't have time to read all of that right now, but thanks for posting that! LokiClock (talk) 21:44, 7 August 2009 (UTC)

Two or more types of 4D cupolas?!

It seems like there are two or more ways to extend cupola to 4D. The section now seems to be a regular polyhedron and its Expansion (geometry) {p,q} and t0,2{p,q}, while another extension could be a polyhedron and its truncation (geometry), {p,q} and t01{p,q}. There's a third option might be a polyhedron and its Rectification (geometry) {p,q} and p1{p,q}, and a final forth option as a polyhedron and its omnitruncation {p,q} and t012{p,q}! Or wait, one more, a polyhedron and is snub (geometry): {p,q} and s{p,q}! No, wait, there's more a polyhedron and its dual polyhedron {p,q} and {q,p}, and a truncation and dual truncation t01{p,q} and t01{q,p}. So not sure how to account for all the side-cells in these cases or what they might be called, but its not clear which deserves the name "cupola" or something else. Tom Ruen (talk) 03:42, 29 August 2013 (UTC)

I find transitioning to the full duals and snubs disagreeable, because they look round and I'm not sure if the full snubs fit, but that could just be a sign of the ability of 4D things to be round in separate ways, and the greater number of layers in 4D uniform solids. However, it looks like the omnitruncated solids would be attached to the platonic solids by cupolae of the platonic's face! However, alternation is not itself defined in a way that allows transitional application. The connection from the regular polyhedron to its expansion is the same as to its rectification when viewing the stages of truncation as the slices of the polychoron as one moves from archimedean base to platonic cap. If I recall, they are particularly slices of the runcinated 5-cell, runcinated tesseract, runcinated 24-cell (though half a rectified 16-cell is just as good!), runcinated 120-cell (the rectified 120-cell is also appealing), and rectified 600-cell. Because the bases of these match the solids halved to make the cupolae, these are strong analogues. ᛭ LokiClock (talk) 12:41, 29 August 2013 (UTC)

I don't think there's any limitation on connectivity and they're all convex. The regular-dual combination could be considered a hyper-antiprism, but also could a truncation and dual truncation I think. Incidentally my current interest is that I was looking at the vertex figures for the uniform 4-honeycombs, and found one as {3,3},t1{3,3}, at first thought it was a hypercupola, but then a different one regular-rectified. Tom Ruen (talk) 21:22, 29 August 2013 (UTC)

Oh, are you talking about the rectified 5-cell in the 5-cell honeycomb? The connection by cells is typical of a gyroelongation as I had carried it out, and you'd get a gyroelongated tetrahedron by merging the bottom triangle in this picture to a point. The other modifications of the 5-cell are very cupola-like indeed. But suppose we limit the solids to ones with at most two hyperplanes that don't cut any face short, or rather, take the slices between those separating hyperplanes - that is, take convex segmentochoron slices. Further, we could limit the slices to ones where the interior angles to one bounding hyperplane are all acute. By the by, I just found our expanded cupolae as co || tet - which they describe as actually being a rotunda, sirco || oct, sirco || oct, srid || doe - giving a picture and model, as well as confirming it's a slice of the runcinated 120-cell, a problem I built a net folding program to solve; which is a much harder problem than building the runcinated 120-cell and cutting it - and rather than srid || ike, we have id || ike, which is the actual segment of the rectified 600-cell. Their general description of cupolae is as the Stott expansion of the pyramid, a process they define, and they arrive at doe || id for example, instead of the one given here. They also have a process for antiprisms, which differs from mine, using duals instead of rotations - which might match the cupolae you mentioned. The results are oct||cube and doe||ike; the rectified 5-cell would of course be produced analogously. The reasoning is described here. They also produce lunes there. ᛭ LokiClock (talk) 03:12, 31 August 2013 (UTC)

Cool, I see I have to do some studying here [1] Tom Ruen (talk) 04:01, 31 August 2013 (UTC)

Negatively curved cupolae with regular polygons

Since a cupola, being a polyhedron, can be thought of as a "flattening" of a spherical tiling, then I suppose you could extend the sequence further through the "cupolae" that appear as flat in hyperbolic space, in the cantellated tilings rr{p,3}. Furthermore, I suppose you could also generalise them to alternate squares and other polygons than triangles, so that they would appear in every cantellated polyhedron or tiling rr{p,q}. But I am not sure if any RS has remarked on this, since it seems fairly obvious and trivial. Double sharp (talk) 16:09, 30 September 2018 (UTC)