Talk:Tetrahemihexahedron

layout bug

This page doesn't display properly under IE 6, the graphics box completely obscures the main text. —Preceding unsigned comment added by 208.68.244.251 (talk) 23:53, 1 April 2008 (UTC)

Viewing the HTML page source in my browser options, the graphic box is standards-compliant HTML table markup, set to right-align. I know that IE6 is obsolescent and not standards-compliant, but I wouldn't expect it to be that bad. Check around. Maybe IE6 really is that bad by modern standards, or you need to reinstall. (I use Firefox and it's just fine). Sorry I can't help more. -- Steelpillow (talk) 17:50, 2 April 2008 (UTC)

Categories

I just removed this from the stellation and quasiregular categories.

Firstly, it is not a stellation of any convex core.

Then, yes it has regular faces of two types alternating around each vertex. But there are more subtle issues which mean that it is not usually regarded as quasiregular. For example its vertex figure is sometimes written as {3.4.3/2.4}. Personally I think it should be (for an even more subtle reason), but that is not what the rest of the world thinks.

steelpillow 15:05, 8 January 2007 (UTC)

I think I'll try to explain some subtle issues here. 3.4.3.4 is the cuboctahedron , but 3.4.3/2.4, the tetrahemihexahedron , means that the second triangle is instead wound backwards - try testing out what the Schläfli symbol {3/2} really means. Because it is wound backward it is not exactly quasiregular. But it still is a triangle, so Steelpillow, I agree with you that it should be quasiregular. Unfortunately, that is not what many people think, so we cannot put that into the article. Professor M. Fiendish, Esq. 06:52, 25 August 2009 (UTC)

A quasiregular polyhedron is one derived from a regular polyhedron by rectification. This thing, the hemicuboctahedron, is derived by rectification, not from a regular polyhedron, but from the hemicube (which is not a regular polyhedron, but it is a regular map in the projective plane). Therefore I believe that while the hemicuboctahedron is not a quasiregular polyhedron, it should still be regarded as quasiregular.

I don't consider the hemicuboctahedron as "wound backwards" in any way. I regard it as existing quite normally in the projective plane. The "winding backwards" occurs when we immerse (wrong word – see below) the hemicuboctahedron in R³. Arguably, "hemicuboctahedron" denotes the quasiregular map, and "tetrahemihexahedron" denotes its immersion in R³. Maproom (talk) 09:41, 10 April 2012 (UTC)

Exactly so (though I am always unclear whether an "immersion" is allowed to self-intersect). All we need is a verifiable reference. Wikipedia can't follow the logic through unless somebody else has already done so - and published. — Cheers, Steelpillow (Talk) 19:59, 10 April 2012 (UTC)

An immersion is like an embedding, except that it is allowed to self-intersect. I can't help with what matters, though, the citable reference. Maproom (talk) 21:30, 10 April 2012 (UTC)

Somebody else has already (although not exactly in the way Maproom did) shown the relationship between thah and co; see this page. Double sharp (talk) 13:35, 11 April 2012 (UTC)

Thanks for the clarification. I bet I forget it again though, sigh. These models of the projective plane, and many others, are well documented. What I cannot find is any study of their quasiregularity. — Cheers, Steelpillow (Talk) 20:28, 11 April 2012 (UTC)

George Hart considers the hemipolyhedra to be quasiregular. He also has a list of all the polyhedra that he considers to be quasiregular. Double sharp (talk) 11:40, 12 April 2012 (UTC)

OK, looking at WP:SELFPUBLISH, I think we can take his website as a reliable reference. Anybody got any problems with that? — Cheers, Steelpillow (Talk) 10:53, 15 April 2012 (UTC)

Have a look at the "elco" entry at Klitzing's page for Grünbaum–Coxeter polytopes: [1]. Double sharp (talk) 13:51, 29 May 2012 (UTC)

Immersions

Immersions can self-intersect but that is not the issue, the neighbourhoods of the six vertices are not homeomorphic to R^2. The article draws a good analogy to the roman surface which is also has six singular points. --174.118.1.24 (talk) 01:32, 18 April 2012 (UTC)

They are indeed disclike, or homeomorphic to R^2 as you describe them, as are the six points on the Roman surface as seen from within the surface. The immersion cannot have the same characteristics intrinsic to the surface - after all, it is not even an embedding. However, none of this is relevant to quasiregularity. HTH. — Cheers, Steelpillow (Talk) 21:18, 18 April 2012 (UTC)

Perhaps in the end not relevant, but for the record, that is not correct. The meaning of immersion isn't disputed, and it does imply a local diffeomorphism, in this case to R^2. Boy's surface is an immersion but not an embedding of the projective plane, for example, while the Roman surface is not even an immersion, and this is not either. --192.75.48.150 (talk) 22:02, 18 April 2012 (UTC)

Immersion vs. embedding always confuses me, but you are the first person ever to explain to me that the tetrahemihexahedron in R^3 is neither. What then is it? An injection, a projection, or what? I also fail to understand why the vertices are not locally homeomorphic to R^2 - they are after all embedded in a projective plane (or should that be diffeomorphic not homeomorphic? why mention both in the above, and what's the difference anyway? I do struggle with these long words which always seem to get explained in terms of each other.) — Cheers, Steelpillow (Talk) 21:07, 19 April 2012 (UTC)

Sorry, never mind the long words, I'd say the key word is locally. You might look at this immersion of the Klein bottle. The image self-intersects, so it is not an embedding. But no matter where you are on the original Klein bottle, if you consider a small area around you, the image of that area is non-self-intersecting. So it is a local embedding, AKA an immersion.

Away from the vertices, the image in R^3 of the tetrahemihexahedron is also an immersion. Three distant points of the original polyhedron (in the projective plane) are mapped to the centre. But as you move along an axis from the centre towards a vertex, the self-intersection is between closer and closer points of the original. At the vertex itself, no matter how small a scale you consider, the self-intersection is always in your neighbourhood. So it fails to be an embedding, even locally.

A fun fact is that you can pair up the vertices along three edges and perform an unwinding operation to get a genuine polyhedral immersion of the projective plane, that is very similar to Boy's surface. But this destroys "quasiregularity". --192.75.48.150 (talk) 21:22, 20 April 2012 (UTC)

Thank you for the clearest explanation yet. One last piece of the puzzle, if I want to talk about the tetrahemihexahedron as "a xxxxxxx of the projective plane in R^3", what word should I use for "xxxxxxx"? "Injection" or "mapping" are very broad, is there a more specific term? — Cheers, Steelpillow (Talk) 09:51, 21 April 2012 (UTC)

I'm afraid I don't know. Jon McCammond calls an image of a CW complex that would be an immersion but for the vertices (0-skeleton) a "near-immersion". Not a standard term though. Wikipedia calls Steiner's Roman surface "... a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry ..." A bit of a mouthful that. --192.75.48.150 (talk) 16:57, 24 April 2012 (UTC)

Thanks anyway. I am coming to suspect that there is no accepted term. Guess that leaves me free to invent one. — Cheers, Steelpillow (Talk) 21:32, 24 April 2012 (UTC)

Edge arrangement

I don't see anything wrong with my edit. My open-link edits like this were done over about 30 articles, getting started linking some common terminology, still useful even without linked-article yet. I purposely removed the vertex/edge counts since it includes ALL of them, left the face info since only half of triangle are shared. Tom Ruen 21:07, 11 May 2007 (UTC) See: Special:Whatlinkshere/Edge_arrangement and Special:Whatlinkshere/Vertex_arrangement.

I added back element counts, moved to first paragraph, and empty link to edge arrangement. Tom Ruen 21:13, 11 May 2007 (UTC)

OK with the broken links for now. It also has the same vertex arrangement as the octahedron, so I have added that in to replace the bit you deleted. -- Steelpillow 09:05, 12 May 2007 (UTC)

Wow!

It looks nice, but I don't quite see where the square faces are. --Johanneskepler (talk) 03:00, 10 August 2009 (UTC)

The squares intersect right through the center of the model. Tom Ruen (talk) 06:08, 10 August 2009 (UTC)

In the rod-and-ball image, the squares are yellow. You can only see half of a given square at a time. -- Cheers, Steelpillow (Talk) 20:13, 10 August 2009 (UTC)

Two quarters, that is. —Tamfang (talk) 06:36, 11 August 2009 (UTC)

“Demicross” polytope

This article claimed that the tetrahemihexahedron was the three-dimensional “demicross” polytope. I’ve commented out that claim, since I can’t find any other source replicating it, which violates WP:OR. However, it’s interesting to try to interpret that claim. Here’s my guess, which will hopefully inspire someone else to look into the matter further (and possibly even create a citation for that claim?)

By the same reason an n-hypercube’s vertices can be 2-colored, any n-hyperoctahedron’s facets can too. Furthermore, if we remove any two opposite vertices from an n-hyperoctahedron, we get an (n − 1)-hyperoctahedron’s vertices. We can then create an n-polytope with these two sets of (n − 1)-polytopes. I believe that this is what a demicross polytope is supposed to be.

Under this assumption, the demicross 2-polytope would be a crossed quadrilateral with the vertices of a square, and the demicross 3-polytope would effectively be the tetrahemihexahedron. The demicross 4-polytope would consist of 8 tetrahedra and 4 octahedra (and would in fact correspond to what Bowers has dubbed the tesseracthemioctachoron. The demicross 5-polytope would consist of 16 5-cells and 5 16-cells (and correspond to Bowers’ hexadecahemidecateron. And so on.

Seems like an interesting family, it’s a shame that uniform polytopes are still such an obscure field of study. OfficialURL (talk) 06:21, 30 March 2020 (UTC)

@OfficialURL: This is one of George Olshevsky's terms. Doesn't seem to have gotten any uptake, so I agree that it should be left out. But yes, this is an infinite family that is uniform if the number of dimensions is at least 3 (and is in fact semiregular). The vertex figure is the demicross polytope of one less dimension, which is uniform if we're already in 4D or above, so we can truncate and rectify such polytopes in 4D and up and still get uniform polytopes. Double sharp (talk) 03:13, 31 March 2020 (UTC)

@Double sharp: What a shame. There've been so many kinds of polytopes and properties thereof discovered during the last decade, and it's just unfortunate that none have yet been published. – OfficialURL (talk) 03:49, 31 March 2020 (UTC)