Talk:Stellated octahedron

Is this a polyhedron?

Is this a polyhedron? Double sharp (talk) 10:05, 14 August 2009 (UTC)

It's a compound_polyhedron, just like a hexagram is a compound polygon (2 triangles). Tom Ruen (talk) 10:50, 14 August 2009 (UTC)

Oh, if you only consider the visible surface, it can be a concave polyhedron, with 24 triangles. Tom Ruen (talk) 11:49, 14 August 2009 (UTC)

It looks to me like a small triakis octahedron built with equilateral triangles. Professor M. Fiendish, Esq. 02:17, 4 September 2009 (UTC)

Yep - visually by surfaces its all of them, depending on intepretation! The visible differences is in the vertices. The compound has 8 vertices of the cube (as shown in image). The small triakis octahedron would have 6 more vertices at the edge-intersections. Tom Ruen (talk) 02:32, 4 September 2009 (UTC)

By the way, Kepler seems to have thought that it had quasi-mystical signficance. AnonMoos (talk) 12:01, 4 September 2009 (UTC)

This Form Also called Merkabah , and there are stuff of this form in new age stores. 192.116.88.44 (talk) 15:43, 24 November 2010 (UTC)

As a regular polyhedron?!

I reverted this (below) from an anonymous editor, added without references, unclear to me, likely original research. Tom Ruen (talk) 03:11, 9 November 2011 (UTC)

As a regular concave polyhedron, it can be seen as an octahedron with the faces enlarged and inverted, in much the same way that the faces of the great dodecahedron are enlarged and inverted faces of the dodecahedron, and the faces of the great icosahedron are enlarged and inverted faces of the icosahedron. Therefore, because the stellated octahedron consists of eight congruent (triangular) faces and eight congruent (triangular) vertices, and because it is concave, it can be considered a Kepler-Poinsot solid. However, the other four Kepler-Poinsot solids have pentagrammic faces or vertices. This, along with the fact that it is self-dual, sets the stellated octahedron apart from the other four.

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Odd that the writer of this doesn't know "faces enlarged and inverted" means "stellated"... AnonMoos (talk) 05:12, 9 November 2011 (UTC)

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The "Ghost Editor" States His Case... in the Court of Geometry!

Umm... yeah. I am the anonymous geek editor who had an unusual insight into the geometry of this solid. Actually, I meant "greatened", not "stellated." And I'm laughing right now because one of you guys undid my edit, and I thought it was funny that you did that. I won't declare edit war on the culprit, but I'll bring it to court. And I mean the Supreme Court of Geometry.

"Stellated" means that edges are enlarged. For example, a pentagon is stellated into a pentagram. On the other hand, "greatened" means that faces are enlarged. For example, a dodecahedron is greatened into a great dodecahedron.

In this case, an octahedron is greatened into a stellated octahedron. See for yourself. Get a magnetic construction kit (like Geomag) and build a stellated octahedron by attaching tetrahedra to the exterior of an octahedron. Remove one of the tetrahedra. You should see a larger triangle, and there should be a smaller triangle inside it. Just like the other Kepler spiky-ball-things, the faces of the stella octangula intersect each other. That's why I'm going to call it a Kepler solid, even though it's self-dual and doesn't have any pentagrams in it. (Is it a requirement that a solid contain pentagrams in order for it to be a Kepler solid?)

I rest my case. If you still disagree with me, that's fine.68.173.113.106 (talk) 01:39, 18 November 2011 (UTC)

It can't really be a Kepler solid because it's composite... AnonMoos (talk) 00:53, 14 November 2011 (UTC)

Says who? Only GOD can define what a regular polyhedron is!

Stop telling God what to do!!!68.173.113.106 (talk) 01:39, 18 November 2011 (UTC)

No the concept of a regular polyhedron was defined by man. God creates shapes and man tries to classify them.--Salix (talk): 20:32, 19 November 2011 (UTC)

At least make a note of it.

To phrase it in a different way, I think we should at least mention that if the restriction that a regular polytope cannot be a compound of two polytopes were removed, the stella octangula would count as one of the Kepler-Poinsot solids (or "3-spikeballs").

The page on Schläfli-Hess polychora (which are "4-spikeballs") gives this definition of the following terms:

stellation – replaces edges by longer edges in same lines. (Example: a pentagon stellates into a pentagram) greatening – replaces the faces by large ones in same planes. (Example: an icosahedron greatens into a great icosahedron) aggrandizement – replaces the cells by large ones in same 3-spaces.

This makes "great octahedron" and "stellated octahedron" acceptable terms for this solid.

Note the terminology I am using. "n-spikeball" means a non-convex (particularly starlike) polytope in n-D. (It's for the purpose of making fun of the solids.) 68.173.113.106 (talk) 02:19, 18 November 2011 (UTC)

I don't think so. I think the definition of great is flawed, should say larger ones with the same orientation in the same plane.. The stellated octahedron has triangles which are in their dual position (180 degrees rotated). Tom Ruen (talk) 02:54, 18 November 2011 (UTC)

p.s. star polygon, star polyhedron, star polytope are the terms Coxeter would use for your spike-balls. Tom Ruen (talk) 02:56, 18 November 2011 (UTC)

Then sue the guy that wrote it there! Because the faces of a great dodecahedron are inverted. Look carefully.

I know the technical terms. It's for satirical purposes. Why don't people these days take a joke, especially from a guy who expects a near-perfect score on the shsat? 68.173.113.106 (talk) 03:48, 18 November 2011 (UTC)

Hmmmm... I apologize. You are right - the great dodecahedron has inverted pentagons, as seen in the stellation diagrams. In 2D, a stellated pentagon is a pentagram, while a stellated triangle actually doesn't exist. I'm not sure how this relates to 3D. Stellated octahedron is accurate as the first stellation, but I think "great" is just a convenience used for naming the regular star polyhedra/polychora, and otherwise ambiguous. So PERHAPS great octahedron is valid name but it would still need to be sourced to be added to the article. Tom Ruen (talk) 05:36, 18 November 2011 (UTC)

Stellation diagrams

stellated dodecahedron First stellation

great dodecahedron Second stellation

Stellated octahedron

I've got a reason...

...for all this. This page is a STUB. It really needs our help. Maybe if we start a search for authoritative sources on the stella octangula that would verify this information, Jimmy Wales &c. (or whoever manages the math department) would have to re-evaluate it.

Because some people (like me) are interested in this stuff and would probably like to have more info about some of the not-so-famous solids. 21:18, 18 November 2011 (UTC)

This is really not one of the obscurer polyhedra, since it was considered to be important by Kepler, and also by certain types of modern mysticism -- and it's in very elite company in being one of the few polyhedra to have a common name ("stella octangula") which is not derived by the usual rules (along with "cube" instead of "hexahedron" etc.). However, any information to be added should be widely-accepted (and preferably sourced)... AnonMoos (talk) 23:16, 18 November 2011 (UTC)

That's a good point. But the farthest I've seen in middle school textbooks (or high school textbooks, for that matter—I'm not even in high school yet) is discussion of Platonic solids. No mention of the Kepler-Poinsot solids or the Archimedean solids. At my school they don't even mention the Platonic solids once! What modern mysticism are you talking about? Great! We can put that here! 19:47, 19 November 2011 (UTC) — Preceding unsigned comment added by 68.173.113.106 (talk)

Wipkipedia used to have something tucked away at the end of article Merkabah (see here) and probably the more strictly mathematically-minded would prefer that it not be added here. File:Merkavah1d.gif still exists, but does not seem to be used on English Wikipedia... AnonMoos (talk) 01:26, 20 November 2011 (UTC)

P.S. See the anonymous IP comment of "15:43, 24 November 2010" above... AnonMoos (talk) 12:54, 20 November 2011 (UTC)

On Wolfram MathWorld's article about the stella octangula, it says that the solid "can be constructed using eight of the 20 vertices of the dodecahedron." That's worth including. 68.173.113.106 (talk) 22:03, 19 November 2011 (UTC)

It's fairly trivial, though -- a cube fits within a dodecahedron's vertices (as seen from the compound of five cubes within a dodecahedron), and the vertices of a stella octangula are the vertices of a cube... AnonMoos (talk) 12:54, 20 November 2011 (UTC)

Then it should go under the "trivia" section. I need some ideas as to how to improve this article... 68.173.113.106 (talk) 20:42, 20 November 2011 (UTC)

I didn't mean it in that sense, but rather "it's an uninteresting consequence of a property of something else"... AnonMoos (talk) 01:01, 21 November 2011 (UTC)

Reality check

Hi all, thanks to Tom Ruen for alerting me to this discussion.

Let's begin with, "Only GOD can define what a regular polyhedron is!". True - but not really relevant, because only MANKIND can define what the words "regular polyhedron" mean. Having estabished our chosen meaning, we turn to God and Mathematics to tell us where that takes us (I will take God and Mathematics as synonymous, without being concerned as to which, if either, takes primacy).

Definitions of a polyhedron (such as Coxeter's) commonly require that the boundary be a single contiguous surface. A polyhedral figure whose boundary divides into several closed parts is called a "polyhedral compound" and is no more a single polyhedron than a married couple are a single person.

Since the days of Kepler we have allowed the faces of polyhedra, and compounds, to self-intersect such that part remains hidden inside.

We now turn to regularity. We first define the term "flag" to indicate a set of adjacent elements, one from each dimensionality, that is, the 3D interior or body, a 2D face, a 1D edge of that face, a 0D vertex of that edge (and, for set-theoretic reasons, the -1D "null polytope"). We now define a regular polyhedron as one which is flag-transitive, so that by rotating and/or reflecting a copy and exactly superimposing it on the original, we can superimpose any flag of the copy over any flag of the original. Among the polyhedra whose faces do not self-intersect, God has provided us with five regular examples - the regular octahedron being one.

The process of stellation was originally described by Kepler, as extending the edges or faces of a polyhedron until they meet again to create a new polyhedron. Conway later described this face extension as "greatening", and in the higher-dimensional world of polychora defined aggrandizement in an analogous way (it involves two 3-spaces meeting in 4-space) that does not concern us here.

Kepler stellated - or greatened as Conway would say - the regular octahedron to obtain a figure he called the stella octangula.

The faces of Kelper's stella octangula are the eight large triangles bounded by sides which run from tip to tip of the points.

God now decrees a remarkable thing. The stella octangula is flag-transitive, while none of the other so-called regular compounds (of ten tetrahedra, five octahedra or five cubes) are. Here, Cromwell blunders. He assumes that if each polyhedron in a compound is flag-transitive (or as he calls it, totally transitive), then the compound will be too. But this is not generally the case, since the symmetries of the polyhedra and of the compound are not the same. It turns out to be true only for the stella octangula. So although Kepler's figure is not a "polyhedron" in its own right, it stands unique among such outcasts as being truly regular. (I have not seen this result in print, which is a shame - we cannot explain it on Wikipedia until then).

Meanwhile, prior to Kepler, Pacioli had augmented the octahedron by adding pyramids to create a similar-looking polyhedron with 24 smaller triangular faces. This polyhedron is not regular, and I am uncertain whether it should truly be called a stella octangula, since it was described before Kepler discovered and named his compound. If we do accept the name, then it follows that the stella octangula can, depending on which form we are talking about, be either a polyhedron or regular but not both at the same time - a view which I find rather confusing, so I'd prefer to refer to Pacioli's figure as an "augmented octahedron" and treat it as a distinct figure.

HTH — Cheers, Steelpillow (Talk) 13:07, 19 November 2011 (UTC)

Kepler didn't stellate the octahedron, if we are to use Conway's definition. You cannot stellate an octahedron as all non-adjacent edges are parallel. You can only greaten an octahedron. That's why I think "great octahedron" is a better term for this, um, whatchamacall it. 19:51, 19 November 2011 (UTC) — Preceding unsigned comment added by 68.173.113.106 (talk)

Strictly, since the figure is a compound and not an octahedron, it cannot be a "great" one or any other kind of one. It might more correctly be called a greatening of the octahedron, but even so, Conway's terminology is not fully consistent with tradition and seems to be mainly used for naming the newer uniform star polychora. "Stellation", in Kepler's original sense, remains the accepted general term for polyhedra. (For example the great icosahedron, named by Cayley long before Conway, is generally included as one of the many stellations of the icosahedron.) — Cheers, Steelpillow (Talk) 22:30, 19 November 2011 (UTC)

Great news!

We're officially out of Stub class! But that doesn't mean that this or any other article is perfect. Here's a list of some Stub-class polyhedron-related articles:

• Truncated cube(stub)
• Its dual, the triakis octahedron (stub)

etc.

If anyone finds a Stub-class polyhedron-related article, please add the following to the bottom of the page, just above "References":

Thanks! —The Doctahedron, 68.173.113.106 (talk) 21:33, 22 November 2011 (UTC)

Other image

Could be used... AnonMoos (talk) 04:02, 12 January 2012 (UTC)

	The currently used image has two color image, shows distinct tetrahedra, and which edges and vertices exist in the tetrahedra, versus face intersections. Tom Ruen (talk) 05:19, 12 January 2012 (UTC)
	This SVG is similar, but uncolored. I wonder if that's why the PNG was removed? Tom Ruen (talk) 05:49, 12 January 2012 (UTC)

The red-and-yellow images uses rod-and-ball styling which I find distracting. It can help to emphasise which edges/vertices are true and which are false, but there are more graphically simple (and hence easier for other editors to mimic) conventions that do the job just as well. The particular images here are generated by a proprietary program so any editor wishing to duplicate the style must buy that program (unless anybody knows different?). I find this notably distasteful and against the spirit of Wikipedia.

The monochrome image is designed specifically to show the relationship as a facetting of the cube. Both svg and gif versions are available on the Commons, so get automatically transcluded to the equivalent Wikipedia filenames. Hence the original gif on Wikipedia was redundant. [updated] Where both versions exist, svg is the preferred format for Wikipedia, hence the page update to use the svg.

I like the new blue image. It is a good depiction of the augmented octahedron (24 smaller faces), less satisfactory for the regular compound since it does not distinguish true and false edges/vertices.

— Cheers, Steelpillow (Talk) 13:15, 12 January 2012 (UTC)

The Stella images don't require purchase of the program for reuse or modification, only attribution as its source. The (very old and unused) blue image is nice, but pretty low resolution. Tom Ruen (talk) 20:48, 12 January 2012 (UTC)

Main issue with the rod-and-ball images is if an editor wishes to create a new image of a new polyhedron. Stella can perfectly well produce more conventional styling - a hi-res version of the blue Stella Octangula, for example. — Cheers, Steelpillow (Talk) 19:23, 13 January 2012 (UTC)

I prefer the contrasting coloring of the Stella image showing the two tetrahedra, but I don't generally prefer the rods and balls (since I think a polyhedron is a geometric object that doesn't actually have rods and balls on it) and I also don't like using bitmap formats for images that can be represented better in vector graphics. —David Eppstein (talk) 19:50, 13 January 2012 (UTC)

I agree on vector graphics, SVG, especially if they're computed rather than hand-traced from PNG, mostly since there's too many polyhedra/compounds. I support the rod/ball models (even if should be less intrusive) on the star/compounds so you can see the difference between true edges and face intersections, and true vertices versus edge intersections. Tom Ruen (talk) 20:34, 13 January 2012 (UTC)

In terms of visual quality I would say the red/yellow one are the better ones, there are no anti-aliasing effects and the shading is better. I agree about the rods and balls. Vector graphics would reduce quality somewhat as the subtile colour gradients are likely to be non-linear.--Salix (talk): 21:18, 13 January 2012 (UTC)

If we are going to use bitmaps then something with higher quality global illumination or at least ray tracing would be preferable to the images above, which all appear to use purely synthetic calculations of the colors of each face. It's even possible to use actual photography for this sort of subject; for instance, here's one I made recently that is actually a stella octangula, not for this article (for which a flat-faced polyhedral model would be a better choice) but for stella octangula number. —David Eppstein (talk) 21:46, 13 January 2012 (UTC)

Disagree -- File:Stella_octangula.png is not trying to be quasi-photographic, and a quasi photographic image and File:Stella_octangula.png would be good for different purposes, and in many contexts would not be reasonable substitutes for each other... AnonMoos (talk) 03:16, 14 January 2012 (UTC)

Okay, for fun, I recolored SteelPillow's cube SVG w/o cube, and blue like png, not same orientation, but close! Tom Ruen (talk)

Old PNG

Old SVG

New SVG

For "diagram" style svg vector is better, for true "photo" lighting bitmap is necessary. I prefer diagram style, since it is easier to work across toolsets and manually tweak specific features, such as the transparencies used in File:CubeAndStel.svg (the "Old SVG" above). But I traced that whole image by hand from a png. Is there any 3D software that calculates lighting shades on different faces and outputs an svg graphic? Failing that, I think we are stuck with bitmap for most of our 3D images. — Cheers, Steelpillow (Talk) 11:29, 14 January 2012 (UTC) [Update] Having said that, I don't think it pushes us towards photographic rendering as suggested above - just, allows it if need be. In general there is much to be said for a "flat colour" diagrammatic representation. — Cheers, Steelpillow (Talk) 11:32, 14 January 2012 (UTC)

I wrote my own 3D viewer, with an SVG export option for convex blocks by backface culling, a bit more work to compute face intersections which I've not done! And I can also add flat shading lighting. So someday maybe I'll try more. George_W._Hart has lots of 3D VRML models [1], although not all I'd want to replace. ALSO, SVG supports transparency, while Stella images don't, so another thing to look at for less symmetric polyhedra, basically just have to sort faces from far to near in drawing order. Tom Ruen (talk) 21:47, 14 January 2012 (UTC)

I do have some old code which can do the intersection steps, I might be able to dig it out. The perspective on the New SGV image doesn't quite look right, it seems a little squat, a slightly further fov might be better.--Salix (talk): 01:19, 15 January 2012 (UTC)

I think most of these are orthogonal projections — even for diagram-style graphics, I prefer true perspective, like the Stella one, and if you're rolling your own code anyway it's not very hard to calculate. —David Eppstein (talk) 01:40, 15 January 2012 (UTC)

Sometimes non-convex faces can overlap, with a different face being in front of the other in different places. This occurs for example in some stellations. So a simple "this face is behind the other" algorithm will not suffice. If anybody wants a given feature in Stella badly enough they can always ask the maintainer. But he won't release the code or even offer its use for free, so that's a bad road anyway. Tom says he based the new SVG image on my monochrome hand sketch - hence incorrect proportions. I do like the idea of a 3D viewer with svg export. Any chance of open-sourcing the code? — Cheers, Steelpillow (Talk) 12:35, 15 January 2012 (UTC)

Stellated octahedron or stella octangula?

Which is more common? Double sharp (talk) 15:20, 23 April 2014 (UTC)

In Google scholar, there are significantly more hits for stella octangula. In the regular web Google, it's the other way around. —David Eppstein (talk) 16:03, 23 April 2014 (UTC)

Thanks. This leads me to think that perhaps the article should be at stella octangula, as it originally was – especially since the regular web Google results may be partially driven by our using "stellated octahedron" as the title. Nevertheless, given the conflict, perhaps compound of two tetrahedra may be the most neutral and appropriate name for this polyhedron compound. Double sharp (talk) 15:24, 24 April 2014 (UTC)

I would caution against the compound of two octahedra. The same outer hull may also be obtained by erecting eight smaller tetrahedral pyramids on a single octahedron, a figure which is not a compound but a 24-sided deltahedron having sets of coplanar faces. IMHO both constructions should be treated in a single article: both are commonly understood as forms of the stellated octahedron. I'm not sure where the Stella Octangula falls in this distinction. — Cheers, Steelpillow (Talk) 16:04, 24 April 2014 (UTC)

Plaza de Europa (Zaragoza, Spain)

The Plaza de Europa in Zaragoza has four large and twelve small stellated octahedral sculptures. You can see this in pictures of the plaza in commons:Category:Plaza de Europa, Zaragoza, and in Google Maps satellite view, but I don't know if this is documented in any textual sources about the plaza. Is this something we need an additional source for or are the pictures fine? Qzekrom (she/her • talk) 10:10, 19 October 2021 (UTC)

Edit: There's a source! On this page:

El obelisco está situado en la vía pública, en el centro de un gran espacio circular de 30 metros de diámetro, una parte peatonal y ajardinada donde se ha dibujado en el pavimento una estrella de doce puntas, en cada una de las cuales se ha fijado una farola, coronada por dos tetraedros unidos.

Translated: The obelisk is located on the public road, in the center of a large circular space 30 meters in diameter, a pedestrian and landscaped part where a twelve-pointed star has been drawn on the pavement, each of which has been fixed a lamppost, surmounted by two joined tetrahedra.

This refers to the twelve small stellated octahedra. Qzekrom (she/her • talk) 10:17, 19 October 2021 (UTC)

Miracle Musical

The musical project mentions the "Stella Octangula" in it's only album, "Hawaii Part ii", on the 3rd track, titled "Black Rainbows." The symbol is also quite often used to represent the album and project. 83.27.232.85 (talk) 23:11, 20 January 2024 (UTC)

The relevant Wikipedia article seems to be Tally Hall, which does not have the text "Stella Octangula" in it... AnonMoos (talk) 02:48, 21 January 2024 (UTC)