814:
801:
517:) 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?)
84:
74:
53:
827:
176:
158:
1101:
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
Knowledge until then).
1310:
1484:
1585:
22:
1210:
1576:
1333:
1105:
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.
1079:
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.
1068:. 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).
1683:
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,
1356:
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
478:
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
1682:
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
1104:
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
771:
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
1503:
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
1153:
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
1078:
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
851:
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
1100:
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,
1071:
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.
1435:
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
1450:
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
1465:
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
842:...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.
1601:(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,
340:
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.
1086:
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.
1609:) 11:29, 14 January 2012 (UTC) 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,
1567:
1285:
1830:
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
1838:
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
1765:
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,
1416:
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,
1366:, so get automatically transcluded to the equivalent Knowledge filenames. Hence the original gif on Knowledge was redundant. Where both versions exist, svg is the preferred format for Knowledge, hence the page update to use the svg.
1596:
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
1638:, 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.
1760:
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
1357:
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
Knowledge.
1652:
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
986:
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...
1667:
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.
435:. 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.
1868:
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.
1371:
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.
600:
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").
140:
1742:, 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
1394:
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.
845:
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)
753:
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?
1149:
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
1123:
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
1176:
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:
1927:
224:
1806:, 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?
1315:
The currently used image has two color image, shows distinct tetrahedra, and which edges and vertices exist in the tetrahedra, versus face intersections.
1917:
230:
130:
1912:
1251:
324:
1922:
1362:
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
772:
stellation, but I think "great" is just a convenience used for naming the regular star polyhedra/polychora, and otherwise ambiguous. So PERHAPS
106:
852:(along with "cube" instead of "hexahedron" etc.). However, any information to be added should be widely-accepted (and preferably sourced)...
328:
1266:
1134:
1010:
969:
889:
754:
668:
557:
521:
200:
1075:
Since the days of Kepler we have allowed the faces of polyhedra, and compounds, to self-intersect such that part remains hidden inside.
379:
1799:
1870:
1197:
If anyone finds a Stub-class polyhedron-related article, please add the following to the bottom of the page, just above "References":
1127:
an octahedron. That's why I think "great octahedron" is a better term for this, um, whatchamacall it. 19:51, 19 November 2011 (UTC)
97:
58:
870:
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
1724:
In Google scholar, there are significantly more hits for stella octangula. In the regular web Google, it's the other way around. —
1097:
The faces of Kelper's stella octangula are the eight large triangles bounded by sides which run from tip to tip of the points.
183:
163:
446:
faces or vertices. This, along with the fact that it is self-dual, sets the stellated octahedron apart from the other four.
1244:
321:
33:
604:
397:
I reverted this (below) from an anonymous editor, added without references, unclear to me, likely original research.
1743:
1035:
I didn't mean it in that sense, but rather "it's an uninteresting consequence of a property of something else"...
1530:
would be good for different purposes, and in many contexts would not be reasonable substitutes for each other...
1527:
1523:
1791:
1551:
Okay, for fun, I recolored SteelPillow's cube SVG w/o cube, and blue like png, not same orientation, but close!
1729:
1673:
1513:
1441:
1270:
1237:
1154:
Cayley long before Conway, is generally included as one of the many stellations of the icosahedron.) — Cheers,
1138:
1014:
973:
893:
813:
758:
672:
570:
No the concept of a regular polyhedron was defined by man. God creates shapes and man tries to classify them.--
561:
525:
314:
1798:
has four large and twelve small stellated octahedral sculptures. You can see this in pictures of the plaza in
1090:
Kepler stellated - or greatened as Conway would say - the regular octahedron to obtain a figure he called the
1082:
The process of stellation was originally described by Kepler, as extending the edges or faces of a polyhedron
800:
21:
750:
Then sue the guy that wrote it there! Because the faces of a great dodecahedron are inverted. Look carefully.
383:
1874:
1505:
318:
1751:
1714:
805:
257:
882:
What modern mysticism are you talking about? Great! We can put that here! 19:47, 19 November 2011 (UTC)
1771:
1688:
1614:
1606:
1422:
1381:
1159:
1113:
39:
83:
1658:
1598:
1471:
1130:
1009:
Then it should go under the "trivia" section. I need some ideas as to how to improve this article...
885:
575:
439:
1725:
1669:
1509:
1496:
1437:
913:
826:
268:
861:
513:
is greatened into a stellated octahedron. See for yourself. Get a magnetic construction kit (like
199:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
1893:
1535:
1296:
1187:
1040:
992:
943:
921:
857:
818:
540:
503:
459:
420:
361:
89:
912:) and probably the more strictly mathematically-minded would prefer that it not be added here.
73:
52:
1853:
1812:
1747:
1710:
1643:
1556:
1456:
1399:
1343:
1320:
781:
737:
699:
639:
428:
402:
346:
300:
280:
253:
1767:
1739:
1684:
1624:
1610:
1602:
1418:
1377:
1309:
1155:
1109:
454:
Odd that the writer of this doesn't know "faces enlarged and inverted" means "stellated"...
874:
high school yet) is discussion of
Platonic solids. No mention of the Kepler-Poinsot solids
175:
157:
1654:
1467:
717:
656:
This makes "great octahedron" and "stellated octahedron" acceptable terms for this solid.
571:
1823:
1627:, a bit more work to compute face intersections which I've not done! And I can also add
1632:
1221:
1180:
1636:
1483:
1906:
1889:
1531:
1292:
1036:
988:
939:
917:
853:
721:
536:
455:
357:
960:
1849:
1808:
1639:
1628:
1552:
1452:
1395:
1339:
1316:
965:
964:, it says that the solid "can be constructed using eight of the 20 vertices of the
777:
733:
713:
695:
499:
424:
398:
342:
296:
276:
878:
the Archimedean solids. At my school they don't even mention the Platonic solids
419:
with the faces enlarged and inverted, in much the same way that the faces of the
356:
By the way, Kepler seems to have thought that it had quasi-mystical signficance.
1762:
1500:
1338:
This SVG is similar, but uncolored. I wonder if that's why the PNG was removed?
776:
is valid name but it would still need to be sourced to be added to the article.
635:
432:
102:
1897:
1878:
1858:
1817:
1775:
1755:
1733:
1718:
1692:
1677:
1662:
1647:
1618:
1584:
1566:
1539:
1517:
1475:
1460:
1445:
1426:
1403:
1385:
1347:
1324:
1300:
1284:
1274:
1163:
1117:
1044:
1018:
996:
977:
947:
925:
785:
762:
741:
703:
676:
579:
565:
544:
529:
463:
406:
387:
365:
350:
331:
304:
284:
261:
1885:
1217:
616:
510:
416:
79:
625:
491:
443:
196:
192:
1746:
may be the most neutral and appropriate name for this polyhedron compound.
1526:
is not trying to be quasi-photographic, and a quasi photographic image and
1795:
1623:
I wrote my own 3D viewer, with an SVG export option for convex blocks by
905:
621:
607:(which are "4-spikeballs") gives this definition of the following terms:
487:
371:
272:
1803:
1575:
1332:
938:
P.S. See the anonymous IP comment of "15:43, 24 November 2010" above...
682:
I don't think so. I think the definition of great is flawed, should say
1209:
725:
375:
188:
1504:
which a flat-faced polyhedral model would be a better choice) but for
694:
has triangles which are in their dual position (180 degrees rotated).
1738:
Thanks. This leads me to think that perhaps the article should be at
514:
904:
Wipkipedia used to have something tucked away at the end of article
916:
still exists, but does not seem to be used on English Knowledge...
663:-spikeball" means a non-convex (particularly starlike) polytope in
1495:
If we are going to use bitmaps then something with higher quality
1283:
634:– replaces the faces by large ones in same planes. (Example: an
1061:
Hi all, thanks to Tom Ruen for alerting me to this discussion.
470:
The "Ghost Editor" States His Case... in the Court of Geometry!
15:
1482:
1451:
intersections, and true vertices versus edge intersections.
620:– replaces edges by longer edges in same lines. (Example: a
596:
To phrase it in a different way, I think we should at least
553:
Says who? Only GOD can define what a regular polyhedron is!
535:
It can't really be a Kepler solid because it's composite...
1888:, which does not have the text "Stella Octangula" in it...
520:
I rest my case. If you still disagree with me, that's fine.
667:-D. (It's for the purpose of making fun of the solids.)
1225:
909:
479:
it to court. And I mean the Supreme Court of Geometry.
415:
As a regular concave polyhedron, it can be seen as an
1847:
This refers to the twelve small stellated octahedra.
295:, it can be a concave polyhedron, with 24 triangles.
1304:
648:– replaces the cells by large ones in same 3-spaces.
187:, a collaborative effort to improve the coverage of
101:, a collaborative effort to improve the coverage of
1066:"Only GOD can define what a regular polyhedron is!"
229:This article has not yet received a rating on the
1363:
1841:a lamppost, surmounted by two joined tetrahedra.
1833:una farola, coronada por dos tetraedros unidos.
1245:
8:
494:. On the other hand, "greatened" means that
1884:The relevant Knowledge article seems to be
19:
1800:commons:Category:Plaza de Europa, Zaragoza
1631:lighting. So someday maybe I'll try more.
1252:
1238:
152:
47:
1705:Stellated octahedron or stella octangula?
958:On Wolfram MathWorld's article about the
1561:
1436:represented better in vector graphics. —
1199:
792:
609:
410:
431:are enlarged and inverted faces of the
423:are enlarged and inverted faces of the
374:, and there are stuff of this form in
154:
49:
1928:Unknown-importance Polyhedra articles
291:Oh, if you only consider the visible
275:is a compound polygon (2 triangles).
7:
1206:
1204:
181:This article is within the scope of
95:This article is within the scope of
38:It is of interest to the following
659:Note the terminology I am using. "
317:built with equilateral triangles.
14:
1918:Low-priority mathematics articles
1786:Plaza de Europa (Zaragoza, Spain)
115:Knowledge:WikiProject Mathematics
1913:Start-Class mathematics articles
1583:
1574:
1565:
1331:
1308:
1208:
837:
825:
812:
799:
174:
156:
118:Template:WikiProject Mathematics
82:
72:
51:
20:
209:Knowledge:WikiProject Polyhedra
135:This article has been rated as
1923:Start-Class Polyhedra articles
556:Stop telling God what to do!!!
212:Template:WikiProject Polyhedra
1:
1275:21:33, 22 November 2011 (UTC)
1164:22:30, 19 November 2011 (UTC)
1118:13:07, 19 November 2011 (UTC)
1045:01:01, 21 November 2011 (UTC)
1019:20:42, 20 November 2011 (UTC)
997:12:54, 20 November 2011 (UTC)
978:22:03, 19 November 2011 (UTC)
948:12:54, 20 November 2011 (UTC)
926:01:26, 20 November 2011 (UTC)
862:23:16, 18 November 2011 (UTC)
786:05:36, 18 November 2011 (UTC)
763:03:48, 18 November 2011 (UTC)
742:02:56, 18 November 2011 (UTC)
704:02:54, 18 November 2011 (UTC)
677:02:19, 18 November 2011 (UTC)
580:20:32, 19 November 2011 (UTC)
566:01:39, 18 November 2011 (UTC)
545:00:53, 14 November 2011 (UTC)
530:01:39, 18 November 2011 (UTC)
498:are enlarged. For example, a
486:are enlarged. For example, a
388:15:43, 24 November 2010 (UTC)
366:12:01, 4 September 2009 (UTC)
351:02:32, 4 September 2009 (UTC)
332:02:17, 4 September 2009 (UTC)
203:and see a list of open tasks.
109:and see a list of open tasks.
1898:02:48, 21 January 2024 (UTC)
1879:23:11, 20 January 2024 (UTC)
1859:10:17, 19 October 2021 (UTC)
1818:10:10, 19 October 2021 (UTC)
1693:12:35, 15 January 2012 (UTC)
1678:01:40, 15 January 2012 (UTC)
1663:01:19, 15 January 2012 (UTC)
1648:21:47, 14 January 2012 (UTC)
1619:11:32, 14 January 2012 (UTC)
1540:03:16, 14 January 2012 (UTC)
1518:21:46, 13 January 2012 (UTC)
1476:21:18, 13 January 2012 (UTC)
1461:20:34, 13 January 2012 (UTC)
1446:19:50, 13 January 2012 (UTC)
1427:19:23, 13 January 2012 (UTC)
1404:20:48, 12 January 2012 (UTC)
1386:13:15, 12 January 2012 (UTC)
1348:05:49, 12 January 2012 (UTC)
1325:05:19, 12 January 2012 (UTC)
1301:04:02, 12 January 2012 (UTC)
1224:. You can help Knowledge by
464:05:12, 9 November 2011 (UTC)
407:03:11, 9 November 2011 (UTC)
1822:Edit: There's a source! On
1635:has lots of 3D VRML models
968:." That's worth including.
592:At least make a note of it.
305:11:49, 14 August 2009 (UTC)
285:10:50, 14 August 2009 (UTC)
262:10:05, 14 August 2009 (UTC)
1944:
1804:Google Maps satellite view
1776:16:04, 24 April 2014 (UTC)
1756:15:24, 24 April 2014 (UTC)
1744:compound of two tetrahedra
1734:16:03, 23 April 2014 (UTC)
1719:15:20, 23 April 2014 (UTC)
1265:Thanks! —The Doctahedron,
1203:
797:
231:project's importance scale
1528:File:Stella_octangula.png
1524:File:Stella_octangula.png
393:As a regular polyhedron?!
228:
169:
134:
67:
46:
438:However, the other four
315:small triakis octahedron
141:project's priority scale
1506:stella octangula number
605:Schläfli-Hess polychora
482:"Stellated" means that
427:, and the faces of the
98:WikiProject Mathematics
1709:Which is more common?
1487:
1288:
1220:-related article is a
806:stellated dodecahedron
370:This Form Also called
313:It looks to me like a
252:Is this a polyhedron?
28:This article is rated
1486:
1287:
1084:until they meet again
831:Stellated octahedron
684:larger ones with the
440:Kepler-Poinsot solids
248:Is this a polyhedron?
184:WikiProject Polyhedra
1599:File:CubeAndStel.svg
838:I've got a reason...
794:Stellation diagrams
692:stellated octahedron
502:is greatened into a
490:is stellated into a
121:mathematics articles
1497:global illumination
914:File:Merkavah1d.gif
795:
728:would use for your
269:compound_polyhedron
1488:
1289:
1188:triakis octahedron
1064:Let's begin with,
822:Second stellation
819:great dodecahedron
793:
688:in the same plane.
504:great dodecahedron
421:great dodecahedron
215:Polyhedra articles
90:Mathematics portal
34:content assessment
1593:
1592:
1353:
1352:
1291:Could be used...
1263:
1262:
1233:
1232:
1133:comment added by
888:comment added by
835:
834:
809:First stellation
654:
653:
640:great icosahedron
624:stellates into a
509:In this case, an
451:
450:
429:great icosahedron
245:
244:
241:
240:
237:
236:
151:
150:
147:
146:
1935:
1857:
1816:
1740:stella octangula
1625:backface culling
1587:
1578:
1569:
1562:
1335:
1312:
1305:
1254:
1247:
1240:
1212:
1205:
1200:
1142:
1092:stella octangula
961:stella octangula
897:
829:
816:
803:
796:
774:great octahedron
686:same orientation
638:greatens into a
610:
411:
217:
216:
213:
210:
207:
178:
171:
170:
160:
153:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
1943:
1942:
1938:
1937:
1936:
1934:
1933:
1932:
1903:
1902:
1866:
1864:Miracle Musical
1848:
1807:
1792:Plaza de Europa
1788:
1707:
1282:
1259:
1258:
1174:
1128:
1059:
883:
840:
718:star polyhedron
594:
474:Umm... yeah. I
472:
395:
250:
214:
211:
208:
205:
204:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
1941:
1939:
1931:
1930:
1925:
1920:
1915:
1905:
1904:
1901:
1900:
1865:
1862:
1845:
1844:
1835:
1787:
1784:
1783:
1782:
1781:
1780:
1779:
1778:
1726:David Eppstein
1706:
1703:
1702:
1701:
1700:
1699:
1698:
1697:
1696:
1695:
1680:
1670:David Eppstein
1633:George_W._Hart
1591:
1590:
1588:
1581:
1579:
1572:
1570:
1549:
1548:
1547:
1546:
1545:
1544:
1543:
1542:
1510:David Eppstein
1481:
1480:
1479:
1478:
1448:
1438:David Eppstein
1433:
1432:
1431:
1430:
1429:
1409:
1408:
1407:
1406:
1389:
1388:
1373:
1372:
1368:
1367:
1359:
1358:
1351:
1350:
1336:
1328:
1327:
1313:
1281:
1278:
1267:68.173.113.106
1261:
1260:
1257:
1256:
1249:
1242:
1234:
1231:
1230:
1213:
1192:
1191:
1186:Its dual, the
1184:
1181:Truncated cube
1173:
1170:
1169:
1168:
1167:
1166:
1144:
1143:
1135:68.173.113.106
1108:HTH — Cheers,
1058:
1055:
1054:
1053:
1052:
1051:
1050:
1049:
1048:
1047:
1026:
1025:
1024:
1023:
1022:
1021:
1011:68.173.113.106
1002:
1001:
1000:
999:
981:
980:
970:68.173.113.106
955:
954:
953:
952:
951:
950:
931:
930:
929:
928:
899:
898:
890:68.173.113.106
867:
866:
865:
864:
839:
836:
833:
832:
830:
823:
821:
817:
810:
808:
804:
791:
790:
789:
788:
766:
765:
755:68.173.113.106
751:
747:
746:
745:
744:
724:are the terms
707:
706:
669:68.173.113.106
652:
651:
650:
649:
646:aggrandizement
643:
629:
593:
590:
589:
588:
587:
586:
585:
584:
583:
582:
558:68.173.113.106
548:
547:
522:68.173.113.106
471:
468:
467:
452:
449:
448:
394:
391:
354:
353:
337:
336:
335:
334:
308:
307:
288:
287:
271:, just like a
249:
246:
243:
242:
239:
238:
235:
234:
227:
221:
220:
218:
201:the discussion
179:
167:
166:
161:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
1940:
1929:
1926:
1924:
1921:
1919:
1916:
1914:
1911:
1910:
1908:
1899:
1895:
1891:
1887:
1883:
1882:
1881:
1880:
1876:
1872:
1863:
1861:
1860:
1855:
1851:
1843:
1842:
1836:
1834:
1829:
1828:
1827:
1825:
1820:
1819:
1814:
1810:
1805:
1801:
1797:
1793:
1785:
1777:
1773:
1769:
1764:
1759:
1758:
1757:
1753:
1749:
1745:
1741:
1737:
1736:
1735:
1731:
1727:
1723:
1722:
1721:
1720:
1716:
1712:
1704:
1694:
1690:
1686:
1681:
1679:
1675:
1671:
1666:
1665:
1664:
1660:
1656:
1651:
1650:
1649:
1645:
1641:
1637:
1634:
1630:
1626:
1622:
1621:
1620:
1616:
1612:
1608:
1604:
1600:
1595:
1594:
1586:
1582:
1577:
1573:
1568:
1564:
1563:
1560:
1558:
1554:
1541:
1537:
1533:
1529:
1525:
1521:
1520:
1519:
1515:
1511:
1507:
1502:
1498:
1494:
1493:
1492:
1491:
1490:
1489:
1485:
1477:
1473:
1469:
1466:non-linear.--
1464:
1463:
1462:
1458:
1454:
1449:
1447:
1443:
1439:
1434:
1428:
1424:
1420:
1415:
1414:
1413:
1412:
1411:
1410:
1405:
1401:
1397:
1393:
1392:
1391:
1390:
1387:
1383:
1379:
1375:
1374:
1370:
1369:
1365:
1361:
1360:
1355:
1354:
1349:
1345:
1341:
1337:
1334:
1330:
1329:
1326:
1322:
1318:
1314:
1311:
1307:
1306:
1303:
1302:
1298:
1294:
1286:
1279:
1277:
1276:
1272:
1268:
1255:
1250:
1248:
1243:
1241:
1236:
1235:
1229:
1227:
1223:
1219:
1214:
1211:
1207:
1202:
1201:
1198:
1195:
1189:
1185:
1182:
1179:
1178:
1177:
1171:
1165:
1161:
1157:
1152:
1148:
1147:
1146:
1145:
1140:
1136:
1132:
1126:
1122:
1121:
1120:
1119:
1115:
1111:
1106:
1102:
1098:
1095:
1093:
1088:
1085:
1080:
1076:
1073:
1069:
1067:
1062:
1057:Reality check
1056:
1046:
1042:
1038:
1034:
1033:
1032:
1031:
1030:
1029:
1028:
1027:
1020:
1016:
1012:
1008:
1007:
1006:
1005:
1004:
1003:
998:
994:
990:
985:
984:
983:
982:
979:
975:
971:
967:
963:
962:
957:
956:
949:
945:
941:
937:
936:
935:
934:
933:
932:
927:
923:
919:
915:
911:
907:
903:
902:
901:
900:
895:
891:
887:
881:
877:
873:
869:
868:
863:
859:
855:
850:
849:
848:
847:
846:
843:
828:
824:
820:
815:
811:
807:
802:
798:
787:
783:
779:
775:
770:
769:
768:
767:
764:
760:
756:
752:
749:
748:
743:
739:
735:
731:
727:
723:
722:star polytope
719:
715:
711:
710:
709:
708:
705:
701:
697:
693:
689:
687:
681:
680:
679:
678:
674:
670:
666:
662:
657:
647:
644:
641:
637:
633:
630:
627:
623:
619:
618:
614:
613:
612:
611:
608:
606:
601:
599:
591:
581:
577:
573:
569:
568:
567:
563:
559:
555:
554:
552:
551:
550:
549:
546:
542:
538:
534:
533:
532:
531:
527:
523:
518:
516:
512:
507:
505:
501:
497:
493:
489:
485:
480:
477:
469:
466:
465:
461:
457:
447:
445:
441:
436:
434:
430:
426:
422:
418:
413:
412:
409:
408:
404:
400:
392:
390:
389:
385:
381:
380:192.116.88.44
377:
373:
368:
367:
363:
359:
352:
348:
344:
339:
338:
333:
330:
326:
323:
320:
316:
312:
311:
310:
309:
306:
302:
298:
294:
290:
289:
286:
282:
278:
274:
270:
266:
265:
264:
263:
259:
255:
247:
232:
226:
223:
222:
219:
202:
198:
194:
190:
186:
185:
180:
177:
173:
172:
168:
165:
162:
159:
155:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1871:83.27.232.85
1867:
1846:
1840:
1837:
1832:
1821:
1789:
1748:Double sharp
1711:Double sharp
1708:
1629:flat shading
1550:
1522:Disagree --
1499:or at least
1290:
1264:
1226:expanding it
1215:
1196:
1193:
1175:
1150:
1129:— Preceding
1124:
1107:
1103:
1099:
1096:
1091:
1089:
1083:
1081:
1077:
1074:
1070:
1065:
1063:
1060:
966:dodecahedron
959:
884:— Preceding
879:
875:
871:
844:
841:
773:
729:
714:star polygon
691:
685:
683:
664:
660:
658:
655:
645:
631:
615:
603:The page on
602:
597:
595:
519:
508:
500:dodecahedron
495:
483:
481:
475:
473:
453:
444:pentagrammic
437:
425:dodecahedron
414:
396:
369:
355:
292:
254:Double sharp
251:
195:, and other
182:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
1852:(she/her •
1811:(she/her •
1768:Steelpillow
1763:deltahedron
1685:Steelpillow
1611:Steelpillow
1603:Steelpillow
1501:ray tracing
1419:Steelpillow
1378:Steelpillow
1280:Other image
1172:Great news!
1156:Steelpillow
1110:Steelpillow
730:spike-balls
636:icosahedron
433:icosahedron
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1907:Categories
1886:Tally Hall
1376:— Cheers,
1218:polyhedron
632:greatening
617:stellation
511:octahedron
417:octahedron
1824:this page
1802:, and in
1653:better.--
626:pentagram
492:pentagram
319:Professor
206:Polyhedra
197:polytopes
193:polyhedra
164:Polyhedra
1890:AnonMoos
1796:Zaragoza
1640:Tom Ruen
1589:New SVG
1580:Old SVG
1571:Old PNG
1553:Tom Ruen
1532:AnonMoos
1453:Tom Ruen
1396:Tom Ruen
1340:Tom Ruen
1317:Tom Ruen
1293:AnonMoos
1131:unsigned
1037:AnonMoos
989:AnonMoos
940:AnonMoos
918:AnonMoos
906:Merkabah
886:unsigned
854:AnonMoos
778:Tom Ruen
734:Tom Ruen
696:Tom Ruen
622:pentagon
537:AnonMoos
488:pentagon
456:AnonMoos
399:Tom Ruen
378:stores.
372:Merkabah
358:AnonMoos
343:Tom Ruen
325:Fiendish
297:Tom Ruen
277:Tom Ruen
273:hexagram
189:polygons
1850:Qzekrom
1809:Qzekrom
1364:Commons
1125:greaten
726:Coxeter
598:mention
376:new age
293:surface
267:It's a
139:on the
1190:(stub)
1183:(stub)
690:. The
515:Geomag
36:scale.
1655:Salix
1468:Salix
1216:This
1194:etc.
908:(see
880:once!
712:p.s.
572:Salix
496:faces
484:edges
442:have
1894:talk
1875:talk
1854:talk
1813:talk
1790:The
1772:Talk
1752:talk
1730:talk
1715:talk
1689:Talk
1674:talk
1659:talk
1644:talk
1615:Talk
1607:Talk
1557:talk
1536:talk
1514:talk
1472:talk
1457:talk
1442:talk
1423:Talk
1400:talk
1382:Talk
1344:talk
1321:talk
1297:talk
1271:talk
1222:stub
1160:Talk
1139:talk
1114:Talk
1041:talk
1015:talk
993:talk
974:talk
944:talk
922:talk
910:here
894:talk
858:talk
782:talk
759:talk
738:talk
700:talk
673:talk
576:talk
562:talk
541:talk
526:talk
460:talk
403:talk
384:talk
362:talk
347:talk
329:Esq.
301:talk
281:talk
258:talk
1794:in
1661:):
1508:. —
1474:):
578:):
225:???
131:Low
1909::
1896:)
1877:)
1826::
1774:)
1754:)
1732:)
1717:)
1691:)
1676:)
1646:)
1617:)
1559:)
1538:)
1516:)
1459:)
1444:)
1425:)
1402:)
1384:)
1346:)
1323:)
1299:)
1273:)
1162:)
1151:of
1141:)
1116:)
1094:.
1043:)
1017:)
995:)
976:)
946:)
924:)
896:)
876:or
872:in
860:)
784:)
761:)
740:)
732:.
720:,
716:,
702:)
675:)
564:)
543:)
528:)
506:.
476:am
462:)
405:)
386:)
364:)
349:)
327:,
322:M.
303:)
283:)
260:)
191:,
1892:(
1873:(
1856:)
1815:)
1770:(
1750:(
1728:(
1713:(
1687:(
1672:(
1668:—
1657:(
1642:(
1613:(
1605:(
1555:(
1534:(
1512:(
1470:(
1455:(
1440:(
1421:(
1398:(
1380:(
1342:(
1319:(
1295:(
1269:(
1253:e
1246:t
1239:v
1228:.
1158:(
1137:(
1112:(
1039:(
1013:(
991:(
972:(
942:(
920:(
892:(
856:(
780:(
757:(
736:(
698:(
671:(
665:n
661:n
642:)
628:)
574:(
560:(
539:(
524:(
458:(
401:(
382:(
360:(
345:(
299:(
279:(
256:(
233:.
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
