Talk:Octahedron

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Thanks @David Eppstein for splitting this page out. It generally seems like a good idea. Do you think we could still keep some kind of summary section called regular octahedron near the top here? I expect many inbound links to octahedron will be looking for regular octahedron instead, as will many readers arriving from web search, etc. –jacobolus (t) 02:10, 10 June 2025 (UTC)Reply

As long as it's so short as to make it obvious that the other article is the main one and expansions should go there, I think this is a fine idea. Also I have not had time to make any efforts to fix the many inbound links that probably intended to go to regular octahedron and now go here, so that should also probably be done. —David Eppstein (talk) 07:43, 10 June 2025 (UTC)Reply

Sorry, but have one heard the name of dual of gyrobifastigium anyway in some sources? It seems I could not find it anyway, but the way of constructing it is available. Dedhert.Jr (talk) 16:32, 12 June 2025 (UTC)Reply

I didn't find a name, but I did at least find a source mentioning it. —David Eppstein (talk) 01:39, 20 June 2025 (UTC)Reply

Hexagonal frustum has obviously six quadrilaterals and two quadrilaterals. If this is not topological, so what would it be? Dedhert.Jr (talk) 01:27, 20 June 2025 (UTC)Reply

It has exactly the same pattern of faces, edges, and vertices as a hexagonal prism. We should not create separate entries for every minor distortion of a shape; I think one entry per topological equivalence class is enough. Prisms and frusta are in the same equivalence class, just as all of the different suggested geometries for Dürer's solid are all in the same equivalence class as each other. —David Eppstein (talk) 01:39, 20 June 2025 (UTC)Reply

Oh no wonder. Dedhert.Jr (talk) 02:28, 20 June 2025 (UTC)Reply

If we allow self-crossing polyhedra, there is one with six regular faces, obtained by removing two pairs of opposite faces from a regular octahedron and replacing them by squares. It is self-dual and topologically a sphere, but half turned inside-out through the crossing. It is related to the Tetrahemihexahedron, a 7-sided self-crossing polyhedron also obtained by replacing triangles with squares in the regular octahedron. Can it be sourced? —David Eppstein (talk) 18:03, 29 January 2026 (UTC)Reply

@David Eppstein. Not sure what you mean by the "self-dual self-crossing hexahedron". Do you have a source for this? I think your question might be suitable in Talk:Regular octahedron, since this article herein is about an eight-sided polyhedron in general. Dedhert.Jr (talk) 02:08, 30 January 2026 (UTC)Reply

I meant to post this in Talk:Hexahedron but we might as well continue here. No, I don't have a source. I was wondering whether anyone else did.

As for "not sure what you mean": I mean exactly what I said. Remove two pairs of opposite faces from a regular octahedron; there is only one way to do this, leaving two pairs of adjacent triangles dangling from two opposite vertices (say the top and bottom vertices). Reconnect these two pairs of triangles by two squares, the equators of the octahedron, crossing each other along a diagonal between the top and bottom vertices. The result is a self-crossing polyhedron, which can be seen to be orientable and (by calculating Euler's formula) topologically a sphere. If you embed the same system of vertices and faces without crossings on a topological sphere you can find its dual graph which turns out to be the same graph, rotated by 90°. —David Eppstein (talk) 02:31, 30 January 2026 (UTC)Reply

Not quite what you are describing, but a search turned a different interesting thing (doi:10.3390/sym4010001):

An appealing polyhedral model for ${\displaystyle \{3,8\}_{6}}$ with self-intersections but full tetrahedral symmetry can be obtained from a pair of homothetic octahedra by first omitting on each octahedron a set of alternate faces, different sets on the two octahedra, and then joining the resulting triangular circuits by tunnels, one tunnel for each pair of circuits in parallel planes (see [34]). This model has maximum possible symmetry among all polyhedral models of ${\displaystyle \{3,8\}_{6}}$ and is significantly more symmetric than the polyhedral embedding with maximal symmetry.

Reference 34 is: Schulte, E.; Wills, J.M. Geometric realizations for Dyck’s regular map on a surface of genus 3. Discret. Comput. Geom. 1986, 1, 141–153.

–jacobolus (t) 02:22, 30 January 2026 (UTC)Reply

This looks closer (doi:10.1017/S0025557200179653):

The octahedron has three diagonal planes passing through its centre. Two of these are seen as b in the facetting diagram shown as Figure 11 (the third is parallel to the plane of the diagram).

Several subsymmetric facettings may be obtained, of which two are described here. The tetrahemihexahedron has alternate faces of the octahedron removed to expose the square diagonal planes, giving it tetrahedral symmetry. Its vertex figure is shown in Figure 12. Another facetting has two opposing faces of the octahedron removed, giving it the symmetry of the triangular antiprism. Facets B are sub-facetted similarly to Figure 8 and the vertex figure is shown in Figure 13. Note how, again, the vertex figures are contained in the facetting diagram.

–jacobolus (t) 02:32, 30 January 2026 (UTC)Reply

Ping @Steelpillow:? —David Eppstein (talk) 02:37, 30 January 2026 (UTC)Reply

The figure is familiar enough to me, it was one of the other subsymmetric facettings I had in mind when I wrote the above quote. But I cannot recall any specific source or any name for it. Not much use, but if I can trace an RS I'll let you know. — Cheers, Steelpillow (Talk) 13:46, 30 January 2026 (UTC)Reply

Update: a fair old search has found only Hessel's pair of joined tetrahedra. Probably could do with naming and publishing. A four-chamfered crossed cube, perhaps? — Cheers, Steelpillow (Talk) 19:49, 1 February 2026 (UTC)Reply

I would call it something like a "lemniscate hexahedron" maybe because of the crossed-square cross-section through its equator. I suppose I could make a blog post about it, but that wouldn't count as a reliable source for Wikipedia purposes. —David Eppstein (talk) 20:24, 1 February 2026 (UTC)Reply

That name seems kinda confusing to me. I'd call the cross-section a 'bowtie' shape. –jacobolus (t) 20:37, 1 February 2026 (UTC)Reply

Ok, but if we're brainstorming names for the whole polyhedron, do you have any suggestions? Bowtie hexahedron? —David Eppstein (talk) 21:20, 1 February 2026 (UTC)Reply

Maybe. Or something about chopping out two quadrants? I don't know. –jacobolus (t) 21:52, 1 February 2026 (UTC)Reply

Aha! "Bowtie" led me to the Bowtie tegum article on the Polytope Wiki, which also calls it the bowtie bipyramid or bobipyr. "Tegum" is Kreiger's term, "bobipyr" is Bowers' shorthand term. Besides Bowers, the article also references this page on Klitzing's site, which adds "crossed-square bipyramid". His observation that it is abstractly self-dual is intriguing. Hardly WP:RS, but maybe a starting point. Note that the "bowtie prism", a crossed cube, is another bowtie hexahedron. Personally, I find "bowtie" a bit kitsch. How about "crossed tetradihedron?" — Cheers, Steelpillow (Talk) 16:39, 2 February 2026 (UTC)Reply

That's the one! I'm not a fan of these made-up words like bobipyr and tegum. "Crossed tetradihedron" is much better. But now we at least have a web page for it. We would need a published source to include it in the non-convex section of hexahedron (where it belongs, in the absence of the even better sourcing needed for a separate article). —David Eppstein (talk) 18:49, 2 February 2026 (UTC)Reply

I thought the self-crossing hexahedron you meant was removing the alternating triangles, which apparently looks like the way of constructing the tetrahemihexahedron but has two intersectional squares instead, but oh well :(.

Anyway, I found a source for this one [1], the website, which is not reliable. Dedhert.Jr (talk) 02:51, 3 February 2026 (UTC)Reply

"Hedron Dude" is a pseudonym used by Bowers; that is his website. Steelpillow (Talk) 19:22, 3 February 2026 (UTC)Reply

These names are awful, lol. "Tricu", "Bocuco", "Thah", "Squippy", .... –jacobolus (t) 19:36, 3 February 2026 (UTC)Reply

Forget the name. Description is important. Dedhert.Jr (talk) 01:11, 4 February 2026 (UTC)Reply

@Jacobolus. And you might want to look other Bowers style acronyms [2], like "medial omnicircumfacetopental trishecatonicosachoron" (a.k.a. "mom fapathi")? Dedhert.Jr (talk) 13:13, 4 February 2026 (UTC)Reply

Why? One does reach a point where a rigorous mathematical notation would be more useful than alphabet soup. — Cheers, Steelpillow (Talk) 19:23, 4 February 2026 (UTC)Reply

Or even just a numbering scheme. —David Eppstein (talk) 19:37, 4 February 2026 (UTC)Reply

I have no comment about this anyway. My bad. Dedhert.Jr (talk) 01:57, 5 February 2026 (UTC)Reply

A further search for 'tetrahemihexahedron' turns up more results. –jacobolus (t) 02:36, 30 January 2026 (UTC)Reply

@David Eppstein. I have tried to draw the regular octahedron with two pairs of adjacent triangles and two squares. But I don't think it is very well, particularly in two-dimensional illustration with translucent edges. The reason is that, when I tried to use coral for the squares, the jasmine triangles would mix into orange. It will make more sense to use GIF, rotated around the axis of Antipodal points (top and bottom). Do you have any other idea besides mine? Dedhert.Jr (talk) 03:21, 5 February 2026 (UTC)Reply

Rotating gif makes sense as a way to understand it. I'm not well set up for making images like that but the green povrays might be the way to go. —David Eppstein (talk) 06:09, 5 February 2026 (UTC)Reply

I won't mind it, as long as the illustration has a good perspective, just like the Herschel enneahedron. Dedhert.Jr (talk) 11:36, 5 February 2026 (UTC)Reply

Is this any better? I prefer monochrome, unless the colour is providing significant information. tetradi.png? (I usually find it easiest to trace an image in Inkscape, style it manually for best presentation, and if need be export to png or whatever.) — Cheers, Steelpillow (Talk) 11:49, 5 February 2026 (UTC)Reply

@Steelpillow. I never expected that can be drawn in Inkscape with that. But, should there be another non-convex polygon on the other side? You drew one vertical line in front, but not behind the solid. Or in other words, you did not remove another vertical edge behind. Because if you did, it is not a hexahedron anymore, but rather an octahedron. Maybe I still miscomprehend again. Dedhert.Jr (talk) 13:00, 5 February 2026 (UTC)Reply

The vertical line is not an edge but where the two square faces cross. It lies along an axis of symmetry, there is no second one "behind" it. I have edited the image to try and show this, any better now? tetradi.png — Cheers, Steelpillow (Talk) 14:27, 5 February 2026 (UTC)Reply

@Steelpillow. Okay, so that it is counted as a 7-sided self-crossing polyhedron? Nice! Dedhert.Jr (talk) 14:46, 5 February 2026 (UTC)Reply

Not 7 but 6. Note that 4 triangles + 2 squares = 6 faces. Which is why I called it a tetradihedron. Not to be confused with the 7-faced one which has a third, horizontal square as well, the triangles shuffled around, and a different topology altogether. But I'm glad you like it! — Cheers, Steelpillow (Talk) 17:20, 5 February 2026 (UTC)Reply

You forgot to take out the edge in the back. But I think a monochromatic picture for this shape is going to inevitably be pretty hard to interpret for someone who doesn't already know what they're supposed to be seeing. –jacobolus (t) 18:02, 5 February 2026 (UTC)Reply

Thank you, well spotted. Now fixed. I tried colouring it, but the hidden faces or parts of faces tint the ones in front, so it doesn't really help. — Cheers, Steelpillow (Talk) 11:49, 6 February 2026 (UTC)Reply

Another helpful possibility would be to change the camera angle to be more from the "top", so that neither of the cut-away sections is entirely obscured. –jacobolus (t) 19:49, 6 February 2026 (UTC)Reply

Try this - also, colour works better with it. cross-tetradis.png. — Cheers, Steelpillow (Talk) 06:30, 7 February 2026 (UTC)Reply

Nice! —David Eppstein (talk) 07:41, 7 February 2026 (UTC)Reply

I'll take this as a reference for my drawing, the lighting, although I am still suck at this, so I don't particularly care about it anyway. Dedhert.Jr (talk) 08:07, 7 February 2026 (UTC)Reply

That's better. One other thing to try: make the triangles gray and use two different colors for the squares; that might make it clearer that they cross each-other. –jacobolus (t) 18:16, 7 February 2026 (UTC)Reply

Tried it. Just tuns it into a colour salad. — Cheers, Steelpillow (Talk) 20:50, 7 February 2026 (UTC)Reply

Gyrobifastigium has eight faces and eight vertices, and its dual is eight faces as well. Since the dual corresponds any vertex to any face, perhaps the dual of elongated triangular bipyramid and biaugmented triangular prism can be listed here. Dedhert.Jr (talk) 01:21, 18 July 2026 (UTC)Reply