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simplices, corresponding to the symmetry of certain mathematical groups. (IIRC there's one of interest in 36 dimensions, but I forget what it corresponds with.) Besides these, there is also the relatively unexplored area of higher-dimensional
Catalan polytopes and other cell-transitive polytopes that are not necessarily uniform.—
422:. 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1; respectively these five expressions correspond to the nonprismatic polychora (4), the polyhedron prisms (3+1), the duoprisms (2+2), prisms built on polygonal prisms (2+1+1) (equivalent to duoprisms in which one of the factors is a square), and the hypercube (1+1+1+1). —
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And I disagree that beyond polychora not much novelty is to be found: the Gosset uniform polytopes, for example, are of quite some interest, up to 8 dimensions or so. Beyond that, there are certain dimensions with sporadically uniform polytopes besides the usual ones derived from the cube/cross and
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of the remaining vertices; put another way, this means inscribing pentagons in the decagons and digons (degenerate polygons with only two vertices) in the squares, and putting triangles in place of the missing vertices. The result isn't uniform, but it can be distorted into uniformity (by shifting
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As I understand, q-gonal-p-gonal prism is the same as p-gonal-q-gonal prism. I mean, p and q are interchangeable, they define the same figure. Is it right? If I am right, may be it should be mentioned? For example, in the picture of
Duoprisms (p,q = 3 - 8), that pictures with yellow "centers" (p
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It is a bit confusing. A prism is a product of a line-segment {} and a polygon {p}, so {}x{p} is a p-gon in a plane extruded into the 3rd dimension into a solid prism. A duoprism is similar except in 4D, you can extrude in 2 dimensions from a plane, so you can extrude a polygon product {p}x{q}! I
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which is in his yet unpublished book on the subject (I asked him if he had a higher term!). Jonathan Bowers goes higher, BUT they also work a bit differently on the mixed forms. Anyway, nothing published by Bowers at all. Klitzing's website is the only "public" source at all, referencing the
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Same thing goes. Among the partitions of 5 are 4 + 1 (polychoral prism); 3 + 1 + 1 (polyhedral prism prism); 3 + 2 (polyhedron-polygon duoprism); 2 + 2 + 1 (duoprism prism); 2 + 1 + 1 + 1 (cube-polygon prism). Once you get to 5 dimensions and beyond, the lack of standard terminology makes it
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Hm. If that's uniform, then what you get if you remove the "frontmost" vertices of the v.f. and take its convex hull also ought to be uniform (and convex), each vertex having an octahedron surrounded alternately by T and 5-antiprisms. Maybe that can't exist for a topological reason.
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But if we start from a duoprism – let's say a hexagon × octagon – and do the same alternation, then in general you can't get uniformity, because three degrees of freedom are not enough to assure a solution to five constraints (equating six numbers).
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is a 4-4 duoprism, represented as {4}x{4}. It can also be a cubic prism {4,3}x{}. If you're talking uniform polytopes, {4,3,3}={4,3}x{}={4}x{4}={}x{}x{}x{}. But these lower forms have degrees of freedom, like {}x{} can be a
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Got it! So you can still contruct it by flattening the
Johnson solid gyrobifastigium until all points lie on a sphere, and that'll leave different edge lengths for the triangles. (And same calculation can work for any p,q)
1230:"Multiplying" an n-polytope by an "edge" (1-polytope) gives an (n+1)-prism. Multiplying a prism by a point (0-polytope) leaves it unchanged (or isomorphic, anyway). In both these cases the "m+n rule" works.
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Yes, {}x{}={4}, edge x edge = square, but as a special case, with {}={4,3} in general, so any prismatic polytope can be reduced to at most one {} product, and thus can be considered either a prism or not a
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https://web.archive.org/web/20030121092141/http://etext.lib.virginia.edu:80/etcbin/toccer-new2?id=ManFour.sgm&images=images/modeng&data=/texts/english/modeng/parsed&tag=public&part=all
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The only convex uniform solution is p=q=4; allowing rationals gives one more, p=10 and q=10/3. The general case cannot be made uniform; these two work only because there is some higher symmetry (the
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difficult to express exactly what you're talking about; writing it out as a
Cartesian product helps to nail it down precisely (e.g., square * triangle * tetrahedron = 7-dimensional multiprism).—
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If both p and q are even, the resulting duoprism has only even-sided faces, and thus alternatable. Are all the possible alternations, other than the alternated 4-4 duoprism (which is the
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Because no one has added them yet? If you wanted to, you could add them. You'd have to be able to supply enough material to justify devoting an entire article to each subject, though.
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Oh, already says that at the section start "The p-q duoprisms are identical to the q-p duoprisms, but look different because they are projected in the center of different cells."
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as polytope products as most studied, and that's what's in the table, but otherwise, I've leaned for wider inclusiveness. BUT I'm also supposed to be on wikibreak again!
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are about regular and uniform polytopes, so there could be 2 sets of articles like for lower dimensions, one focused on uniform figures. I started a division on
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948:. It's all rather unstable ground, with a firm theory, but the only reference to the figures comes from Dr. Richard Klitzing's webpages, currently hosted at
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http://etext.lib.virginia.edu/etcbin/toccer-new2?id=ManFour.sgm&images=images/modeng&data=/texts/english/modeng/parsed&tag=public&part=all
1223:. Any product P X Q is polytope when rank(P) < 2 or Rank(Q) < 2, or both. Of course, many of these are trivial - and uninteresting. Will return to
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Ah sorry, the article already states that they are generally nonuniform, but there should be some examples by
Schlegel diagram, aren't they? --
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the vertices), because there are two degrees of freedom with which to solve two equations (to make three numbers equal).
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Yes, they are interchangeable, but shown in complementary projection directions. It could be mentioned for clarity.
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And if we start with a pentagonal one instead? Now you can't label the vertices alternately, so what happens now?
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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1217:-ist, where the smallest polytope is the null polytope of rank -1, and all polytopes have a (-1)-face.
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Actually, the PRIMARY reason to end at 5-polytopes is Norman
Johnson's truncation terminology stops at
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family to the uniform polyteron, but never finished the tables for other familes, given on a test page
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So perhaps the definition about should say a product of two polytopes, each dimension two or higher?
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Since the tesseract is a cubic prism, and the cube itself is a square prism, is it also a duoprism?
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1879:. In particular {∞}×{∞} = {4,4} as an infinite stack of apeirogonal prisms, a construction using A
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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before doing mass systematic removals. This message is updated dynamically through the template
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p.s. s{5,2,5/3} isn't a valid symbol by
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I am not sure "prism" means anything other than a product of an edge and another polytope.
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you can refer to an n-polytope that is the product of a j-pol and a k-pol with j, k : -->
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p.s. The infinite (tessellation/honeycombs) forms are fun too, especially 3d space ones.
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apeirogonal prisms, each one connecting along a stack of squares that forms one of the
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I changed it a little bit. I think, it is not the best, but it is better, than it was.
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If you found an error with any archives or the URLs themselves, you can fix them with
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Front and back views of vertex figure uploaded, in case it helps answer the question.
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1353:< q) are redundant - the same as p-gonal-q-gonal prism, but different projection.
1552:{5,2,5/3}) is the only way I know. s{2,2,2} is also invalid, but s{2} or s{2} work.
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could be justified, but I don't think going higher than that has much more novelty.
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OK, I've updated the definition accordingly. Hopefully the wording is clear enough.—
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I tried that in Stella4D. The polychoron begins to generate, but doesn't close up.
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I see now. Thank you. I didn't notice it because in my comp the site looked messy
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admit it is confusing since there's no other clear lower dimensional analogy.
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and names by Jonathan Bowers. I'll be going offline for a week. Have fun!
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How would you call the cartesian product of a disk and a polygon?
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A polytope whose number of vertices is prime is a prime polytope
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Apathy? Beyond polychora there's not much novelty to be found. —
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sides of the prismatic tower, as described by Klitzing on his
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The vertices of the gyrobifastigium do not lie on a sphere. —
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When you have finished reviewing my changes, please set the
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for additional information. I made the following changes:
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Can anyone explain why it works as a uniform polychoron?
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to explain fully; it's easier to begin with a decagonal
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1867:-gonal prisms where the rest of 3D space is filled by
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can be considered tessellations of the surface of the
187:, a collaborative effort to improve the coverage of
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1724:using the archive tool instructions below. Editors
843:This isn't really relevant, but there are articles
1431:s{3,2,3} can't be uniform? Its vertex figure is a
951:. They are a bit cryptic, using an ASCII "inline"
229:This article has not yet received a rating on the
1848:If we were to actually take something like {∞}×{
1242:that the article be renamed "Product polytope".
1071:, and Richard Klitzing lists forms up to 8D at:
1401:http://comp.chem.umn.edu/~averkiev/Duoprism.png
1956:Participate in the deletion discussion at the
1852:}, the result must have an infinite number of
1710:This message was posted before February 2018.
1221:There is no reason to exclude lower dimensions
977:Convex uniform honeycombs in hyperbolic space
871:doesn't even exist. Can someone tell me why?
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2026:Unknown-importance Polyhedra articles
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1863:. This must therefore be a stack of
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1530:Probably a similar reason why the
1302:. The article was started for the
1209:"Duoprism" term and its definition
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2016:Low-priority mathematics articles
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1981:p-gonal-cylindrical prism
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1142:
1138:
1137:
1131:
1128:
1125:
1121:
1120:
1114:
1111:
1108:
1104:
1103:
1100:
1097:
1094:
1093:Ring-Interval
1088:
1087:
1075:
1061:
1060:
1059:
1058:
1057:
1056:
1028:
1027:
1026:
1025:
1004:
1003:
968:
967:
940:. I added the
921:
920:
904:
903:
899:
898:
840:
837:
836:
835:
834:
833:
832:
831:
830:
829:
828:
827:
826:
825:
824:
823:
822:
821:
820:
819:
818:
817:
803:star antiprism
780:
779:
778:
777:
776:
775:
774:
773:
772:
771:
770:
769:
768:
767:
766:
765:
764:
763:
727:
726:
725:
724:
723:
722:
721:
720:
719:
718:
717:
716:
715:
714:
713:
712:
697:
666:
665:
664:
663:
662:
661:
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659:
658:
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656:
655:
654:
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606:
605:
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603:
602:
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600:
599:
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597:
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595:
561:
560:
559:
558:
557:
556:
555:
554:
553:
552:
529:
528:
527:
526:
525:
524:
523:
522:
490:
489:
488:
487:
486:
485:
484:
483:
461:
460:
459:
458:
457:
456:
437:
436:
435:
434:
410:Generally, in
405:
404:
388:
387:
386:
385:
384:
383:
364:
363:
362:
361:
344:
343:
342:
341:
329:works for me —
324:
323:
309:
300:rather than a
292:
273:
272:
249:
246:
243:
242:
239:
238:
235:
234:
227:
221:
220:
218:
201:the discussion
179:
167:
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149:
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133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
2038:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2009:
2008:
2006:
1999:
1998:
1994:
1990:
1986:
1982:
1974:
1972:
1971:
1967:
1963:
1959:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1932:
1929:
1927:
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1901:
1899:
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1890:
1878:
1874:
1870:
1866:
1862:
1859:
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1835:
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1827:
1823:
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1818:
1817:
1816:
1812:
1808:
1803:
1802:
1798:
1794:
1790:
1782:
1780:
1779:
1774:
1769:
1768:
1757:
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1750:
1746:
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1744:
1737:
1731:
1727:
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1713:
1708:
1704:
1698:
1694:
1690:
1683:
1679:
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1667:
1663:
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1569:
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1533:
1529:
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1518:
1509:
1501:
1494:
1488:
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1473:
1469:
1465:
1461:
1460:
1459:
1458:
1454:
1450:
1446:
1442:
1438:
1437:Johnson solid
1434:
1430:
1422:
1416:
1412:
1408:
1404:
1402:
1398:
1397:
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1395:
1392:
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1234:
1231:
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1197:
1188:
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1177:
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1123:
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1080:
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1038:
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1024:
1020:
1016:
1012:
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1007:
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1002:
998:
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990:
986:
982:
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974:
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966:
962:
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950:
947:
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939:
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931:
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745:
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743:
742:
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740:
739:
738:
737:
736:
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734:
733:
732:
731:
730:
729:
728:
711:
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703:
698:
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690:
686:
682:
681:
680:
679:
678:
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675:
674:
673:
672:
671:
670:
669:
668:
667:
652:
648:
644:
639:
633:
628:
625:instead of a
624:
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619:
618:
617:
616:
615:
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613:
612:
611:
610:
609:
608:
607:
594:
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586:
582:
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571:
570:
569:
568:
567:
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563:
562:
551:
547:
543:
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536:
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531:
530:
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510:
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498:
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491:
482:
478:
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455:
451:
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442:
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433:
429:
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417:
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328:
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278:
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232:
226:
223:
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219:
202:
198:
194:
190:
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185:
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177:
173:
172:
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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:
1984:
1980:
1978:
1955:
1905:
1889:Double sharp
1872:
1868:
1864:
1857:
1853:
1849:
1847:
1830:Double sharp
1804:
1786:
1764:
1761:
1736:source check
1715:
1709:
1696:
1692:
1688:
1686:
1659:
1656:
1605:Double sharp
1568:Double sharp
1517:Double sharp
1514:
1429:duoantiprism
1426:
1355:
1351:
1348:
1340:
1323:
1299:
1272:
1270:
1264:
1259:
1253:
1248:
1247:
1244:
1239:
1238:
1235:
1232:
1229:
1224:
1220:
1219:
1212:
1194:
1178:
1127:Cantellated
1010:
873:Double sharp
842:
688:
643:Double sharp
542:Double sharp
446:Double sharp
419:
411:
373:Double sharp
305:
301:
295:
274:
259:
253:
195:, and other
182:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
1807:Dannyu NDos
1793:Dannyu NDos
1703:Sourcecheck
1445:tetrahedron
1254:non-trivial
1179:Pentellated
1161:Stericated
1147:Prismated?
1144:Runcinated
1130:Rhombated?
1079:Sterication
1069:E8 polytope
930:10-polytope
693:convex hull
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
2005:Categories
1887:symmetry.
1773:Report bug
1441:octahedron
1277:SteveWoolf
1164:Cellated?
1113:Truncated
1110:Truncated
1102:Polytopes
1011:runcinated
938:5-polytope
926:6-polytope
416:partitions
248:definition
1756:this tool
1749:this tool
1041:Professor
942:5-simplex
910:Tetracube
748:Professor
581:rectangle
576:tesseract
473:Tetracube
351:Tetracube
206:Polyhedra
197:polytopes
193:polyhedra
164:Polyhedra
1762:Cheers.—
1662:Duoprism
1625:Tom Ruen
1554:Tom Ruen
1536:Tom Ruen
1479:Tom Ruen
1449:Tom Ruen
1383:Tom Ruen
1372:Tom Ruen
1308:Tom Ruen
1300:proprism
1196:Tom Ruen
1183:Terated
1096:Johnson
1047:Fiendish
993:Tom Ruen
985:3-sphere
957:Tom Ruen
754:Fiendish
585:Tom Ruen
512:Tom Ruen
394:Tom Ruen
313:Tom Ruen
306:triprism
302:duoprism
264:polytope
260:duoprism
256:geometry
189:polygons
1789:16-cell
1689:checked
1666:my edit
1591:Tamfang
1550:0,1,2,3
1464:Tamfang
1447:cells.
1443:and 18
1328:Tamfang
1252:0 as a
1227:point.
1099:Bowers
1015:Tamfang
888:Tamfang
807:Tamfang
702:Tamfang
574:Yes, a
424:Tamfang
331:Tamfang
277:Tamfang
139:on the
1697:failed
1296:Conway
1035:Maybe
979:. The
975:, and
623:cuboid
308:, etc.
291:prism.
36:scale.
1407:Bor75
1357:Bor75
1294:, or
1273:false
262:is a
1993:talk
1983:(or
1966:talk
1893:talk
1877:site
1834:talk
1811:talk
1797:talk
1693:true
1629:talk
1609:talk
1595:talk
1572:talk
1558:talk
1540:talk
1521:talk
1483:talk
1468:talk
1453:talk
1435:, a
1411:talk
1387:talk
1376:talk
1361:talk
1332:talk
1312:talk
1281:talk
1257:1.
1249:Then
1225:that
1200:talk
1189:...
1170:...
1153:...
1136:...
1119:...
1051:Esq.
1019:talk
997:talk
961:talk
914:talk
892:talk
877:talk
863:and
851:and
811:talk
758:Esq.
706:talk
689:anti
647:talk
627:cube
589:talk
546:talk
516:talk
477:talk
450:talk
428:talk
398:talk
377:talk
355:talk
335:talk
317:talk
281:talk
258:, a
1987:)?
1960:. —
1883:× A
1828:).
1730:RfC
1707:).
1695:or
1680:to
1298:'s
1013:. —
936:vs
928:to
418:of
254:In
225:???
131:Low
2007::
1995:)
1979:A
1968:)
1895:)
1836:)
1813:)
1799:)
1743:.
1738:}}
1734:{{
1705:}}
1701:{{
1631:)
1623:.
1611:)
1597:)
1574:)
1560:)
1542:)
1523:)
1485:)
1470:)
1455:)
1413:)
1389:)
1363:)
1334:)
1314:)
1283:)
1202:)
1175:5
1158:4
1141:3
1124:2
1107:1
1049:,
1044:M.
1021:)
999:)
991:.
963:)
916:)
894:)
879:)
859:,
855:.
847:,
813:)
756:,
751:M.
708:)
649:)
591:)
548:)
518:)
479:)
452:)
430:)
400:)
379:)
357:)
337:)
319:)
283:)
191:,
1991:(
1964:(
1891:(
1885:1
1881:1
1873:n
1869:n
1865:n
1858:n
1854:n
1850:n
1832:(
1809:(
1795:(
1775:)
1771:(
1758:.
1751:.
1627:(
1607:(
1593:(
1589:—
1570:(
1556:(
1538:(
1519:(
1481:(
1466:(
1451:(
1409:(
1385:(
1378:)
1374:(
1359:(
1330:(
1310:(
1279:(
1267:.
1198:(
1017:(
995:(
959:(
912:(
890:(
875:(
809:(
704:(
700:—
645:(
587:(
544:(
514:(
475:(
448:(
426:(
420:n
412:n
396:(
375:(
353:(
333:(
315:(
279:(
233:.
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
