909:
1283:
838:
831:
1666:
1415:
1324:
874:
321:
106:
1645:
934:
508:
1221:
1006:
1522:
37:
789:
1100:
676:
1527:
135:
1271:
394:
580:
1095:
671:
376:. Each edge of the tessellation is surrounded by either four or six disphenoids, according to whether it forms the base or one of the sides of its adjacent isosceles triangle faces respectively. When an edge forms the base of its adjacent isosceles triangles, and is surrounded by four disphenoids, they form an irregular
1638:
A cell can be seen as 1/48 of a translational cube with vertices positioned: one corner, one edge center, one face center, and the cube center. The edge colors and labels specify how many cells exist around the edge. It is one 1/6 of a smaller cube, with 6 phyllic disphenoidal cells sharing a common
1215:
A cell can be seen positioned within a translational cube, with 4 vertices mid-edge and 2 vertices in opposite faces. Edges are colored and labeled by the number of cells around the edge.
486:), then squashing it along the main diagonal until the distance between the points (0, 0, 0) and (1, 1, 1) becomes the same as the distance between the points (0, 0, 0) and (0, 0, 1).
315:
A cell can be seen as 1/12 of a translational cube, with its vertices centered on two faces and two edges. Four of its edges belong to 6 cells, and two edges belong to 4 cells.
1509:
1081:
657:
236:
1235:
cells. The original cubic honeycomb walls are removed, joining pairs of square pyramids into square bipyramids (octahedra). Its vertex and edge framework is identical to the
480:
454:
428:
783:
Cells can be seen in a translational cube, using 4 vertices on one face, and the cube center. Edges are colored by how many cells are around each of them.
380:. When an edge forms one of the two equal sides of its adjacent isosceles triangle faces, the six disphenoids surrounding the edge form a special type of
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1693:
1827:
1771:
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1562:
1554:
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1534:
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1145:
1107:
967:
721:
683:
541:
531:
166:
156:
83:
73:
1155:
1135:
1127:
1117:
987:
977:
957:
731:
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703:
693:
561:
551:
146:
93:
63:
1567:
1295:
845:
1577:
1549:
1539:
1395:
1385:
1375:
1150:
1140:
1122:
1112:
982:
972:
962:
726:
716:
698:
688:
556:
546:
536:
161:
151:
88:
78:
68:
1657:
1593:
889:
908:
1282:
328:
246:
949:
523:
1315:
1166:
893:
865:
742:
837:
47:
1818:; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "21. Naming Archimedean and Catalan Polyhedra and Tilings".
1236:
483:
1362:
900:
385:
55:
1846:
289:
110:
1478:
1050:
626:
205:
1786:
1624:
1455:
1197:
885:
769:
373:
353:
281:
1741:
Symmetry of Things, Table 21.1. Prime
Architectonic and Catopric tilings of space, p. 293, 298.
1729:
Symmetry of Things, Table 21.1. Prime
Architectonic and Catopric tilings of space, p. 293, 296.
1715:
Symmetry of Things, Table 21.1. Prime
Architectonic and Catopric tilings of space, p. 293, 295.
830:
1851:
139:
1665:
1815:
1803:
1628:
1446:
1419:
1201:
1027:
773:
603:
301:
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120:
1414:
17:
1823:
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1767:
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1243:
811:
285:
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1323:
873:
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105:
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1228:
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569:
401:
293:
260:
256:
888:
on their bases, another honeycomb is created with identical vertices and edges, called a
459:
433:
407:
400:
An orientation of the tetragonal disphenoid honeycomb can be obtained by starting with a
1644:
933:
507:
1764:
The
Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching
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128:
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393:
1620:
1526:
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765:
277:
134:
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1442:
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599:
372:: 24 disphenoids meet at each vertex. The union of these 24 disphenoids forms a
174:
377:
579:
1270:
1435:
1750:
Gibb, William (1990), "Paper patterns: solid shapes from metric paper",
1807:
1431:
1250:. These cut face-diagonally through the original cubes. There are also
1094:
670:
1799:
1334:
919:
493:
26:
1426:
1013:
327:
The tetrahedral disphenoid honeycomb is the dual of the uniform
1348:
931:
505:
1481:
1258:
passing through the centers of the octahedral cells.
1053:
629:
462:
436:
410:
208:
1794:(5), Mathematical Association of America: 227–243,
1242:There is one type of plane with faces: a flattened
1231:with each cube subdivided by a center point into 6
799:with each cube subdivided by a center point into 6
1503:
1075:
651:
474:
448:
422:
230:
806:There are two types of planes of faces: one as a
1766:, Cambridge University Press, pp. 363–366,
8:
30:Tetragonal disphenoid tetrahedral honeycomb
1337:
922:
868:with octahedral and truncated cubic cells:
496:
274:tetragonal disphenoid tetrahedral honeycomb
29:
1495:
1484:
1483:
1480:
1318:with octahedral and cuboctahedral cells:
1067:
1056:
1055:
1052:
643:
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631:
628:
461:
435:
409:
222:
211:
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207:
1784:(1981), "Which tetrahedra fill space?",
1737:
1735:
1725:
1723:
1721:
1711:
1709:
1679:Architectonic and catoptric tessellation
1260:
820:
34:
1705:
1694:Triakis truncated tetrahedral honeycomb
814:with half of the triangles removed as
1822:. A K Peters, Ltd. pp. 292–298.
899:It is analogous to the 2-dimensional
7:
352:lattice, which is also known as the
25:
482:(i.e. subdividing each cube into
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1504:{\displaystyle {\tilde {C}}_{3}}
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1076:{\displaystyle {\tilde {C}}_{3}}
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652:{\displaystyle {\tilde {C}}_{3}}
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231:{\displaystyle {\tilde {C}}_{3}}
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18:Disphenoid tetrahedral honeycomb
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404:, subdividing it at the planes
252:
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197:
187:
173:
127:
116:
100:
54:
43:
1762:Pritchard, Chris, ed. (2003),
1617:phyllic disphenoidal honeycomb
1489:
1339:Phyllic disphenoidal honeycomb
1331:Phyllic disphenoidal honeycomb
1246:with half of the triangles as
1061:
880:If the square pyramids of the
637:
216:
1:
1658:omnitruncated cubic honeycomb
1594:Omnitruncated cubic honeycomb
1293:
1262:
1254:plane that exist as nonface
1190:square bipyramidal honeycomb
923:Square bipyramidal honeycomb
916:Square bipyramidal honeycomb
890:square bipyramidal honeycomb
843:
822:
1619:is a uniform space-filling
1192:is a uniform space-filling
764:is a uniform space-filling
329:bitruncated cubic honeycomb
247:Bitruncated cubic honeycomb
1868:
1316:rectified cubic honeycomb
1167:Rectified cubic honeycomb
894:rectified cubic honeycomb
866:truncated cubic honeycomb
743:Truncated cubic honeycomb
1820:The Symmetries of Things
1627:) in Euclidean 3-space.
1200:) in Euclidean 3-space.
772:) in Euclidean 3-space.
48:convex uniform honeycomb
1363:Coxeter-Dynkin diagrams
1357:Dual uniform honeycomb
1237:hexakis cubic honeycomb
950:Coxeter–Dynkin diagrams
944:Dual uniform honeycomb
762:hexakis cubic honeycomb
524:Coxeter–Dynkin diagrams
518:Dual uniform honeycomb
497:Hexakis cubic honeycomb
490:Hexakis cubic honeycomb
334:Its vertices form the A
290:tetragonal disphenoidal
1505:
1077:
901:tetrakis square tiling
653:
476:
450:
424:
386:trigonal trapezohedron
232:
56:Coxeter-Dynkin diagram
1752:Mathematics in School
1506:
1078:
892:, or the dual of the
654:
477:
451:
425:
288:made up of identical
233:
111:Tetragonal disphenoid
1787:Mathematics Magazine
1479:
1227:It can be seen as a
1051:
795:It can be seen as a
627:
460:
434:
408:
374:rhombic dodecahedron
306:oblate tetrahedrille
206:
1206:oblate octahedrille
926:Oblate octahedrille
475:{\displaystyle y=z}
449:{\displaystyle x=z}
423:{\displaystyle x=y}
354:body-centered cubic
276:is a space-filling
140:tetrakis hexahedron
1782:Senechal, Marjorie
1656:It is dual to the
1652:Related honeycombs
1633:Eighth pyramidille
1629:John Horton Conway
1501:
1447:Fibrifold notation
1420:Phyllic disphenoid
1343:Eighth pyramidille
1314:It is dual to the
1310:Related honeycombs
1202:John Horton Conway
1073:
1028:Fibrifold notation
864:It is dual to the
860:Related honeycombs
774:John Horton Conway
649:
604:Fibrifold notation
472:
446:
420:
302:John Horton Conway
298:isosceles triangle
228:
121:isosceles triangle
1829:978-1-56881-220-5
1613:
1612:
1492:
1307:
1306:
1289:triangular tiling
1244:triangular tiling
1186:
1185:
1064:
857:
856:
812:triangular tiling
758:
757:
640:
364:This honeycomb's
296:with 4 identical
292:cells. Cells are
286:Euclidean 3-space
270:
269:
265:vertex-transitive
219:
16:(Redirected from
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1451:Coxeter notation
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1208:or shortened to
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1001:Square bipyramid
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308:or shortened to
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1816:Conway, John H.
1814:
1800:10.2307/2689983
1780:
1774:
1761:
1760:, reprinted in
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1746:
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1719:
1714:
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1684:Cubic honeycomb
1675:
1654:
1639:diagonal axis.
1608:face-transitive
1604:Cell-transitive
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1229:cubic honeycomb
1181:Face-transitive
1177:Cell-transitive
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797:cubic honeycomb
753:Cell-transitive
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402:cubic honeycomb
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310:obtetrahedrille
294:face-transitive
261:face-transitive
257:cell-transitive
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1303:pmm, (*2222)
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1233:square pyramid
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1210:oboctahedrille
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1300:p4m, (*442)
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1276:Square tiling
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1044:Coxeter group
1041:
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850:p4m, (*442)
849:
847:
844:
839:
835:
832:
828:
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808:square tiling
804:
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634:
623:
621:
620:Coxeter group
617:
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370:tetrakis cube
367:
366:vertex figure
359:
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199:Coxeter group
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129:Vertex figure
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1847:3-honeycombs
1819:
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1763:
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1751:
1655:
1637:
1632:
1621:tessellation
1616:
1614:
1342:
1338:
1313:
1255:
1247:
1241:
1226:
1214:
1209:
1205:
1204:calls it an
1194:tessellation
1189:
1187:
925:
898:
881:
879:
863:
815:
805:
794:
782:
777:
766:tessellation
761:
759:
499:
399:
363:
333:
326:
314:
309:
305:
304:calls it an
278:tessellation
273:
271:
1689:Space frame
1443:Space group
1349:(No image)
1024:Space group
882:pyramidille
778:pyramidille
776:calls it a
600:Space group
500:Pyramidille
175:Space group
1852:Tetrahedra
1841:Categories
1700:References
1600:Properties
1287:flattened
1173:Properties
1018:Triangles
749:Properties
574:Isosceles
378:octahedron
253:Properties
117:Face types
1625:honeycomb
1490:~
1198:honeycomb
1062:~
770:honeycomb
638:~
384:called a
356:lattice.
282:honeycomb
217:~
101:Cell type
1758:(3): 2–4
1673:See also
1436:Triangle
1296:Symmetry
1278:"holes"
846:Symmetry
592:Triangle
360:Geometry
189:Symmetry
183:m (229)
1808:2689983
1460:m (229)
1432:Rhombus
1036:m (221)
803:cells.
612:m (221)
594:square
300:faces.
1826:
1806:
1770:
1266:plane
1264:Tiling
886:joined
826:plane
824:Tiling
456:, and
1804:JSTOR
1427:Faces
1256:holes
1248:holes
1014:Faces
816:holes
588:Faces
368:is a
284:) in
50:dual
1824:ISBN
1768:ISBN
1623:(or
1615:The
1590:Dual
1409:Cell
1354:Type
1196:(or
1188:The
1163:Dual
1038:4:2
996:Cell
941:Type
884:are
768:(or
760:The
739:Dual
614:4:2
570:Cell
515:Type
280:(or
272:The
243:Dual
123:{3}
44:Type
1796:doi
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1083:,
659:,
343:/ D
238:,
1843::
1802:,
1792:54
1790:,
1756:19
1754:,
1734:^
1720:^
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1660::
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1239:.
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470:z
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418:y
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