856:
1112:
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159:
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175:
2702:
1414:
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632:
2647:
29:
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1371:
1326:
868:
538:
2528:
1073:
without rotating and tilling space so that it fills the entire face. There are five three-dimensional primary parallelohedrons, one of which is the truncated octahedron. More generally, every permutohedron and parallelohedron is
429:
784:
626:
1911:. But as he writes (p. 429), Voronoi's conjecture for dimensions at most four was already proven by Delaunay. For the classification of three-dimensional parallelohedra into these five types, see
840:
684:
132:
404:
1166:
The truncated octahedron (in fact, the generalized truncated octahedron) appears in the error analysis of quantization index modulation (QIM) in conjunction with repetition coding.
2149:
Perez-Gonzalez, F.; Balado, F.; Martin, J.R.H. (2003). "Performance analysis of existing and new methods for data hiding with known-host information in additive channels".
721:
974:
1018:
930:
1048:
335:
558:
424:
355:
2764:
540:
The surface area of a truncated octahedron can be obtained by summing all polygonals' area, six squares and eight hexagons. Considering the edge length
644:. In other words, it has a highly symmetric and semi-regular polyhedron with two or more different regular polygonal faces that meet in a vertex. The
563:
1926:
2555:
2757:
729:
1313:
2226:
2133:
2102:
2008:
1820:
1723:
1674:
2750:
1509:
789:
2441:
2418:
2317:
1355:
533:{\displaystyle V={\frac {\sqrt {2}}{3}}(3a)^{3}-6\cdot {\frac {\sqrt {2}}{6}}a^{3}=8a^{3}{\sqrt {2}}\approx 11.3137.}
227:(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as a
199:
by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular
96:
1340:
3176:
2548:
1963:
1658:
1748:
658:
106:
855:
1111:
368:
888:, meaning it can be represented with even more symmetric coordinates in four dimensions: all permutations of
2372:
1696:
Mathematical
Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27–31 October 2010
1267:
310:
by cutting off all vertices. This resulting polyhedron has six squares and eight hexagons, leaving out six
249:. If the original truncated octahedron has unit edge length, its dual tetrakis hexahedron has edge lengths
2026:"Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses"
1993:
Image
Analysis: 21st Scandinavian Conference, SCIA 2019, Norrköping, Sweden, June 11–13, 2019, Proceedings
406:. Because six equilateral square pyramids are removed by truncation, the volume of a truncated octahedron
3171:
2963:
2904:
2695:
2541:
1255:
3181:
3166:
2993:
2953:
2305:
2125:
1709:
1283:
358:
2988:
2983:
693:
3094:
3089:
2968:
2874:
2673:
2408:
2254:
2208:
1971:
1917:
1907:
are combinatorially equivalent to
Voronoi tilings, and Erdahl proves this in the special case of
1292:
935:
2958:
2899:
2889:
2834:
2684:
2616:
1385:
1070:
979:
891:
652:. They both have the same three-dimensional symmetry group as the regular octahedron does, the
2484:
2285:
2204:
2119:
2092:
1988:
1967:
1903:. Voronoi conjectured that all tilings of higher dimensional spaces by translates of a single
2978:
2894:
2849:
2797:
2627:
2583:
1802:
1713:
1691:
1652:
1534:
1427:
1140:
In chemistry, the truncated octahedron is the sodalite cage structure in the framework of a
2938:
2812:
2236:
2158:
2037:
1949:
1898:
1771:
1637:
1502:
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8:
3104:
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1152:
1095:
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246:
220:
143:
101:
2162:
2041:
1595:
317:
3186:
3119:
3084:
2943:
2838:
2787:
2186:"Adventures Among the Toroids – Chapter 5 – Simplest (R)(A)(Q)(T) Toroids of genus p=1"
2068:
2025:
1857:
1853:
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1447:
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235:
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1913:
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2393:
2313:
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2222:
2129:
2098:
2073:
2055:
2004:
1989:"Zonohedral Approximation of Spherical Structuring Element for Volumetric Morphology"
1987:
Jensen, Patrick M.; Trinderup, Camilia H.; Dahl, Anders B.; Dahl, Vedrana A. (2019).
1816:
1779:
1739:
1719:
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1263:
1207:
1179:
641:
204:
192:
86:
39:
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1991:. In Felsberg, Michael; Forssén, Per-Erik; Sintorn, Ida-Maria; Unger, Jonas (eds.).
1940:
1921:
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1996:
1935:
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166:
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1945:
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1021:
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242:
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138:
76:
66:
44:
2706:
2859:
2782:
2701:
2594:
1545:
1190:
1183:
1156:
311:
208:
2622:
2611:
2578:
2218:
2000:
1812:
1666:
1413:
932:
form the vertices of a truncated octahedron in the three-dimensional subspace
426:
is obtained by subtracting the volume of a regular octahedron from those six:
3160:
3064:
2920:
2854:
2638:
2600:
2589:
2059:
1498:
1083:
1053:
The truncated octahedron can be used as a tilling space. It is classified as
881:
687:
362:
228:
174:
150:
49:
2170:
2050:
1248:
by a hyperplane so that its sliced cross-section is a truncated octahedron.
1123:
1020:
and each edge represents a single pairwise swap of two elements. It has the
2350:
Gaiha, P. & Guha, S.K. (1977). "Adjacent vertices on a permutohedron".
2077:
1889:
1872:
1762:
1743:
1574:
1563:
1549:
1522:
1477:
1058:
1054:
54:
2148:
1189:
Removing the central octahedron and 2 or 4 triangular cupolae creates two
726:
The dihedral angle of a truncated octahedron between square-to-hexagon is
2529:
Editable printable net of a truncated octahedron with interactive 3D view
1559:
1538:
1518:
1490:
2310:
The
Geometrical Foundation of Natural Structure: A Source Book of Design
1836:
Crisman, Karl-Dieter (2011). "The
Symmetry Group of the Permutahedron".
1715:
The
Geometrical Foundation of Natural Structure: A Source Book of Design
3129:
3017:
2807:
2774:
2646:
1692:"Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes"
1530:
1301:
1175:
1075:
1062:
631:
216:
196:
59:
2333:
867:
686:. A square and two hexagons surround each of its vertex, denoting its
28:
3124:
3114:
3059:
3043:
2879:
2711:
2633:
2533:
2510:
2489:
2471:
2290:
1245:
1141:
2363:
1573:
Three different
Hamiltonian cycles described by the three different
1266:. The truncated octahedron is one of five three-dimensional primary
3010:
2742:
2479:
2280:
1908:
184:
2330:"Figure 5.5: Uniform space-filling using only truncated octahedra"
3134:
3109:
1391:
1145:
365:). From the equilateral square pyramid's property, its volume is
200:
1873:"Zonotopes, dicings, and Voronoi's conjecture on parallelohedra"
1376:
2457:
1133:, showing symmetry labels for high symmetry lines and points.
314:. Considering that each length of the regular octahedron is
2802:
1332:
779:{\textstyle \arccos(-1/{\sqrt {3}})\approx 125.26^{\circ }}
621:{\displaystyle (6+12{\sqrt {3}})a^{2}\approx 26.7846a^{2}.}
1098:, a polyhedron with either hexagonal or pentagonal faces.
207:), 36 edges, and 24 vertices. Since each of its faces has
1616:
Berman, Martin (1971). "Regular-faced convex polyhedra".
1174:
The truncated octahedron can be dissected into a central
976:. Therefore, each vertex corresponds to a permutation of
1986:
2500:
792:
732:
373:
371:
2327:
1029:
982:
938:
894:
696:
661:
566:
546:
432:
412:
343:
320:
109:
2349:
1800:
234:The truncated octahedron was called the "mecon" by
1042:
1012:
968:
924:
861:Truncated octahedron as a permutahedron of order 4
834:
778:
715:
678:
620:
552:
532:
418:
398:
349:
329:
126:
2213:, Universitext, New York: Springer, p. 109,
835:{\textstyle \arccos(-1/3)\approx 109.47^{\circ }}
3158:
2406:
2378:Virtual Polyhedra: The Encyclopedia of Polyhedra
2094:Chemical Processes for Environmental Engineering
1912:
845:
640:The truncated octahedron is one of the thirteen
2030:Proceedings of the National Academy of Sciences
880:The truncated octahedron can be described as a
786:, and that between adjacent hexagonal faces is
2202:
2758:
2549:
2394:"The Uniform Polyhedra: Truncated Octahedron"
2203:Borovik, Alexandre V.; Borovik, Anna (2010),
2121:Introduction to the Electron Theory of Metals
1927:Bulletin of the American Mathematical Society
1399:
337:, and the edge length of a square pyramid is
306:A truncated octahedron is constructed from a
2432:. United Kingdom: Cambridge. pp. 79–86
2427:
2271:Master Thesis, University of TĂĽbingen, 2018
2023:
1801:Johnson, Tom; Jedrzejewski, Franck (2014).
1089:
679:{\displaystyle \mathrm {O} _{\mathrm {h} }}
127:{\displaystyle \mathrm {O} _{\mathrm {h} }}
2765:
2751:
2556:
2542:
2370:
2335:Nanomedicine, Volume I: Basic Capabilities
2248:
1962:
1956:
1193:, with dihedral and tetrahedral symmetry:
301:
173:
157:
27:
2522:"3D convex uniform polyhedra x3x4o - toe"
2067:
2049:
1939:
1888:
1761:
399:{\textstyle {\tfrac {\sqrt {2}}{6}}a^{3}}
2391:
2304:
2117:
2111:
1708:
1689:
1611:
1609:
1568:
1117:The structure of the faujasite framework
629:
2097:. Imperial College Press. p. 338.
1835:
1738:
1718:. Dover Publications, Inc. p. 78.
1702:
1683:
1533:of the truncated octahedron. It has 24
1304:nets often include truncated octahedra.
3159:
2563:
2338:. Georgetown, Texas: Landes Bioscience
2151:IEEE Transactions on Signal Processing
1870:
1829:
1732:
1650:
1615:
2746:
2537:
2501:
2480:
2281:
1864:
1744:"Convex polyhedra with regular faces"
1644:
1606:
873:Truncated octahedron in tilling space
2772:
2519:
2373:"VRML model of truncated octahedron"
2269:Engineering Linear Layouts with SAT.
1980:
2352:SIAM Journal on Applied Mathematics
2249:Read, R. C.; Wilson, R. J. (1998),
2090:
2084:
1995:. Springer. p. 131–132.
1163:lattice is a truncated octahedron.
1057:, meaning it can be defined as the
13:
2183:
2017:
1577:for the truncated octahedral graph
670:
664:
636:3D model of a truncated octahedron
296:
118:
112:
14:
3198:
2451:
2413:. Berlin: Springer. p. 539.
2024:Schein, S.; Gayed, J. M. (2014).
1877:European Journal of Combinatorics
1794:
1618:Journal of the Franklin Institute
648:of a truncated octahedron is the
2723:
2718:
2705:
2700:
2689:
2678:
2667:
2656:
2645:
2632:
2621:
2610:
2599:
2588:
2577:
1412:
1384:
1369:
1354:
1339:
1324:
1312:
1291:
1282:
1234:
1227:
1122:
1110:
1065:. The plesiohedron includes the
866:
854:
2274:
2261:
2242:
2196:
2177:
2142:
1941:10.1090/S0273-0979-1980-14827-2
1850:10.4169/college.math.j.42.2.135
1838:The College Mathematics Journal
1749:Canadian Journal of Mathematics
1698:. World Scientific. p. 48.
1654:Multi-shell Polyhedral Clusters
1101:
630:
2328:Freitas, Robert A. Jr (1999).
1922:"Tilings with congruent tiles"
1690:Koca, M.; Koca, N. O. (2013).
1588:
1510:Table of graphs and parameters
1094:The truncated octahedron is a
1007:
983:
919:
895:
816:
799:
760:
739:
586:
567:
461:
451:
211:the truncated octahedron is a
1:
1976:. Springer. pp. 349–359.
1581:
1169:
1082:that can be defined by using
846:As a tilling space polyhedron
3145:Degenerate polyhedra are in
2485:"Truncated octahedral graph"
2286:"Truncated octahedral graph"
1630:10.1016/0016-0032(71)90071-8
1566:in multiple ways: , , and .
716:{\displaystyle 4\cdot 6^{2}}
7:
2964:pentagonal icositetrahedron
2905:truncated icosidodecahedron
2696:Truncated icosidodecahedron
2312:. Dover Publications, Inc.
1562:, it can be represented by
1531:graph of vertices and edges
1264:body-centered cubic lattice
1256:bitruncated cubic honeycomb
195:that arises from a regular
10:
3203:
2994:pentagonal hexecontahedron
2954:deltoidal icositetrahedron
2126:Cambridge University Press
2118:Mizutani, Uichiro (2001).
1527:truncated octahedral graph
1407:Truncated octahedral graph
1400:Truncated octahedral graph
1273:
1244:It is possible to slice a
1205:
969:{\displaystyle x+y+z+w=10}
357:(the square pyramid is an
3143:
3077:
3052:
3034:
3027:
3002:
2989:disdyakis triacontahedron
2984:deltoidal hexecontahedron
2918:
2826:
2781:
2571:
2407:Alexandrov, A.D. (1958).
2219:10.1007/978-0-387-79066-4
2001:10.1007/978-3-030-20205-7
1813:10.1007/978-3-0348-0554-4
1667:10.1007/978-3-319-64123-2
1508:
1486:
1476:
1466:
1456:
1446:
1436:
1426:
1411:
1406:
1361:model made with Polydron
1013:{\displaystyle (1,2,3,4)}
925:{\displaystyle (1,2,3,4)}
172:
165:
156:
149:
137:
95:
85:
75:
65:
35:
26:
21:
1807:. Springer. p. 15.
1258:can also be seen as the
1129:First Brillouin zone of
1090:As a Goldberg polyhedron
3177:Space-filling polyhedra
3095:gyroelongated bipyramid
2969:rhombic triacontahedron
2875:truncated cuboctahedron
2674:Truncated cuboctahedron
2255:Oxford University Press
2210:Mirrors and Reflections
2171:10.1109/TSP.2003.809368
2051:10.1073/pnas.1310939111
1858:college.math.j.42.2.135
1537:and 36 edges, and is a
1078:, a polyhedron that is
302:As an Archimedean solid
3090:truncated trapezohedra
2959:disdyakis dodecahedron
2925:(duals of Archimedean)
2900:rhombicosidodecahedron
2890:truncated dodecahedron
2685:Rhombicosidodecahedron
2617:Truncated dodecahedron
1890:10.1006/eujc.1999.0294
1871:Erdahl, R. M. (1999).
1763:10.4153/cjm-1966-021-8
1651:Diudea, M. V. (2018).
1596:"Truncated Octahedron"
1578:
1069:, a polyhedron can be
1044:
1014:
970:
926:
836:
780:
717:
680:
637:
622:
554:
534:
420:
400:
351:
331:
128:
2979:pentakis dodecahedron
2895:truncated icosahedron
2850:truncated tetrahedron
2628:Truncated icosahedron
2584:Truncated tetrahedron
2428:Cromwell, P. (1997).
1572:
1222:, , (*332), order 24
1045:
1043:{\displaystyle S_{4}}
1015:
971:
927:
837:
781:
718:
681:
635:
623:
555:
535:
421:
401:
352:
332:
129:
2939:rhombic dodecahedron
2865:truncated octahedron
2606:Truncated octahedron
2463:Truncated octahedron
2091:Yen, Teh F. (2007).
1968:"8.1 Parallelohedra"
1260:Voronoi tessellation
1213:, , (2*3), order 12
1186:above the vertices.
1182:on each face, and 6
1027:
980:
936:
892:
790:
730:
694:
659:
564:
544:
430:
410:
369:
341:
318:
189:truncated octahedron
107:
22:Truncated octahedron
2974:triakis icosahedron
2949:tetrakis hexahedron
2934:triakis tetrahedron
2870:rhombicuboctahedron
2663:Rhombicuboctahedron
2520:Klitzing, Richard.
2163:2003ITSP...51..960P
2042:2014PNAS..111.2920S
1319:ancient Chinese die
1161:face-centered cubic
1153:solid-state physics
1096:Goldberg polyhedron
1080:centrally symmetric
654:octahedral symmetry
650:tetrakis hexahedron
247:tetrakis hexahedron
221:Goldberg polyhedron
144:tetrakis hexahedron
102:octahedral symmetry
3172:Archimedean solids
2944:triakis octahedron
2829:Archimedean solids
2565:Archimedean solids
2503:Weisstein, Eric W.
2482:Weisstein, Eric W.
2459:Weisstein, Eric W.
2434:Archimedean solids
2283:Weisstein, Eric W.
2251:An Atlas of Graphs
1804:Looking at Numbers
1740:Johnson, Norman W.
1579:
1180:triangular cupolae
1178:, surrounded by 8
1040:
1010:
966:
922:
832:
776:
713:
676:
642:Archimedean solids
638:
618:
550:
530:
416:
396:
384:
347:
330:{\displaystyle 3a}
327:
308:regular octahedron
236:Buckminster Fuller
124:
3182:Truncated tilings
3167:Uniform polyhedra
3154:
3153:
3073:
3072:
2910:snub dodecahedron
2885:icosidodecahedron
2740:
2739:
2735:
2734:
2730:Snub dodecahedron
2652:Icosidodecahedron
2467:Archimedean solid
2228:978-0-387-79065-7
2135:978-0-521-58709-9
2104:978-1-86094-759-9
2010:978-3-030-20205-7
1964:Alexandrov, A. D.
1822:978-3-0348-0554-4
1725:978-0-486-23729-9
1676:978-3-319-64123-2
1600:Wolfram Mathworld
1542:Archimedean graph
1515:
1514:
1418:3-fold symmetric
1242:
1241:
758:
584:
553:{\displaystyle a}
522:
489:
485:
449:
445:
419:{\displaystyle V}
383:
379:
350:{\displaystyle a}
219:. It is also the
193:Archimedean solid
181:
180:
40:Archimedean solid
16:Archimedean solid
3194:
3032:
3031:
3028:Dihedral uniform
3003:Dihedral regular
2926:
2842:
2791:
2767:
2760:
2753:
2744:
2743:
2727:
2722:
2709:
2704:
2693:
2682:
2671:
2660:
2649:
2636:
2625:
2614:
2603:
2592:
2581:
2574:
2573:
2558:
2551:
2544:
2535:
2534:
2525:
2516:
2515:
2495:
2494:
2476:
2447:
2424:
2410:Konvexe Polyeder
2403:
2401:
2400:
2388:
2386:
2385:
2371:Hart, George W.
2367:
2346:
2344:
2343:
2323:
2306:Williams, Robert
2297:
2296:
2295:
2278:
2272:
2265:
2259:
2258:
2246:
2240:
2239:
2200:
2194:
2193:
2181:
2175:
2174:
2146:
2140:
2139:
2115:
2109:
2108:
2088:
2082:
2081:
2071:
2053:
2036:(8): 2920–2925.
2021:
2015:
2014:
1984:
1978:
1977:
1973:Convex Polyhedra
1960:
1954:
1953:
1943:
1914:GrĂĽnbaum, Branko
1902:
1892:
1868:
1862:
1861:
1833:
1827:
1826:
1798:
1792:
1791:
1765:
1736:
1730:
1729:
1710:Williams, Robert
1706:
1700:
1699:
1687:
1681:
1680:
1648:
1642:
1641:
1613:
1604:
1603:
1592:
1458:Chromatic number
1420:Schlegel diagram
1416:
1404:
1403:
1388:
1373:
1363:construction set
1358:
1343:
1328:
1316:
1295:
1286:
1238:
1231:
1196:
1195:
1126:
1114:
1049:
1047:
1046:
1041:
1039:
1038:
1019:
1017:
1016:
1011:
975:
973:
972:
967:
931:
929:
928:
923:
870:
858:
841:
839:
838:
833:
831:
830:
812:
785:
783:
782:
777:
775:
774:
759:
754:
752:
722:
720:
719:
714:
712:
711:
685:
683:
682:
677:
675:
674:
673:
667:
634:
627:
625:
624:
619:
614:
613:
598:
597:
585:
580:
559:
557:
556:
551:
539:
537:
536:
531:
523:
518:
516:
515:
500:
499:
490:
481:
480:
469:
468:
450:
441:
440:
425:
423:
422:
417:
405:
403:
402:
397:
395:
394:
385:
375:
374:
356:
354:
353:
348:
336:
334:
333:
328:
292:
291:
286:
284:
283:
280:
277:
270:
269:
264:
262:
261:
258:
255:
177:
161:
133:
131:
130:
125:
123:
122:
121:
115:
31:
19:
18:
3202:
3201:
3197:
3196:
3195:
3193:
3192:
3191:
3157:
3156:
3155:
3150:
3139:
3078:Dihedral others
3069:
3048:
3023:
2998:
2927:
2924:
2923:
2914:
2843:
2832:
2831:
2822:
2785:
2783:Platonic solids
2777:
2771:
2741:
2736:
2728:
2710:
2694:
2683:
2672:
2661:
2650:
2637:
2626:
2615:
2604:
2593:
2582:
2567:
2562:
2506:"Permutohedron"
2454:
2444:
2421:
2398:
2396:
2383:
2381:
2364:10.1137/0132025
2341:
2339:
2320:
2301:
2300:
2279:
2275:
2267:Wolz, Jessica;
2266:
2262:
2247:
2243:
2229:
2205:"Exercise 14.4"
2201:
2197:
2182:
2178:
2147:
2143:
2136:
2128:. p. 112.
2116:
2112:
2105:
2089:
2085:
2022:
2018:
2011:
1985:
1981:
1961:
1957:
1918:Shephard, G. C.
1905:convex polytope
1869:
1865:
1834:
1830:
1823:
1799:
1795:
1737:
1733:
1726:
1707:
1703:
1688:
1684:
1677:
1649:
1645:
1614:
1607:
1594:
1593:
1589:
1584:
1422:
1402:
1395:
1389:
1380:
1374:
1365:
1359:
1350:
1344:
1335:
1329:
1320:
1317:
1308:
1307:
1306:
1305:
1298:
1297:
1296:
1288:
1287:
1276:
1253:cell-transitive
1220:
1211:
1191:Stewart toroids
1184:square pyramids
1172:
1138:
1137:
1136:
1135:
1134:
1127:
1119:
1118:
1115:
1104:
1092:
1067:parallelohedron
1061:of a symmetric
1034:
1030:
1028:
1025:
1024:
1022:symmetric group
981:
978:
977:
937:
934:
933:
893:
890:
889:
886:4-permutohedron
878:
877:
876:
875:
874:
871:
863:
862:
859:
848:
826:
822:
808:
791:
788:
787:
770:
766:
753:
748:
731:
728:
727:
707:
703:
695:
692:
691:
669:
668:
663:
662:
660:
657:
656:
646:dual polyhedron
609:
605:
593:
589:
579:
565:
562:
561:
545:
542:
541:
517:
511:
507:
495:
491:
479:
464:
460:
439:
431:
428:
427:
411:
408:
407:
390:
386:
372:
370:
367:
366:
342:
339:
338:
319:
316:
315:
312:square pyramids
304:
299:
297:Classifications
289:
287:
281:
278:
275:
274:
272:
267:
265:
259:
256:
253:
252:
250:
243:dual polyhedron
226:
139:Dual polyhedron
117:
116:
111:
110:
108:
105:
104:
58:
53:
48:
45:Parallelohedron
43:
17:
12:
11:
5:
3200:
3190:
3189:
3184:
3179:
3174:
3169:
3152:
3151:
3144:
3141:
3140:
3138:
3137:
3132:
3127:
3122:
3117:
3112:
3107:
3102:
3097:
3092:
3087:
3081:
3079:
3075:
3074:
3071:
3070:
3068:
3067:
3062:
3056:
3054:
3050:
3049:
3047:
3046:
3041:
3035:
3029:
3025:
3024:
3022:
3021:
3014:
3006:
3004:
3000:
2999:
2997:
2996:
2991:
2986:
2981:
2976:
2971:
2966:
2961:
2956:
2951:
2946:
2941:
2936:
2930:
2928:
2921:Catalan solids
2919:
2916:
2915:
2913:
2912:
2907:
2902:
2897:
2892:
2887:
2882:
2877:
2872:
2867:
2862:
2860:truncated cube
2857:
2852:
2846:
2844:
2827:
2824:
2823:
2821:
2820:
2815:
2810:
2805:
2800:
2794:
2792:
2779:
2778:
2770:
2769:
2762:
2755:
2747:
2738:
2737:
2733:
2732:
2715:
2714:
2698:
2687:
2676:
2665:
2654:
2642:
2641:
2630:
2619:
2608:
2597:
2595:Truncated cube
2586:
2572:
2569:
2568:
2561:
2560:
2553:
2546:
2538:
2532:
2531:
2526:
2517:
2498:
2497:
2496:
2453:
2452:External links
2450:
2449:
2448:
2442:
2425:
2419:
2404:
2392:Mäder, Roman.
2389:
2368:
2358:(2): 323–327.
2347:
2325:
2318:
2299:
2298:
2273:
2260:
2241:
2227:
2195:
2190:www.doskey.com
2184:Doskey, Alex.
2176:
2157:(4): 960–980.
2141:
2134:
2110:
2103:
2083:
2016:
2009:
1979:
1955:
1934:(3): 951–973.
1930:. New Series.
1883:(6): 527–549.
1863:
1844:(2): 135–139.
1828:
1821:
1793:
1731:
1724:
1701:
1682:
1675:
1661:. p. 39.
1643:
1624:(5): 329–352.
1605:
1586:
1585:
1583:
1580:
1546:book thickness
1513:
1512:
1506:
1505:
1503:zero-symmetric
1488:
1484:
1483:
1480:
1474:
1473:
1470:
1468:Book thickness
1464:
1463:
1460:
1454:
1453:
1450:
1444:
1443:
1440:
1434:
1433:
1430:
1424:
1423:
1417:
1409:
1408:
1401:
1398:
1397:
1396:
1390:
1383:
1381:
1375:
1368:
1366:
1360:
1353:
1351:
1345:
1338:
1336:
1330:
1323:
1321:
1318:
1311:
1300:
1299:
1290:
1289:
1281:
1280:
1279:
1278:
1277:
1275:
1272:
1268:parallelohedra
1240:
1239:
1232:
1224:
1223:
1218:
1214:
1209:
1204:
1203:
1200:
1171:
1168:
1157:Brillouin zone
1128:
1121:
1120:
1116:
1109:
1108:
1107:
1106:
1105:
1103:
1100:
1091:
1088:
1037:
1033:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
965:
962:
959:
956:
953:
950:
947:
944:
941:
921:
918:
915:
912:
909:
906:
903:
900:
897:
884:of order 4 or
872:
865:
864:
860:
853:
852:
851:
850:
849:
847:
844:
829:
825:
821:
818:
815:
811:
807:
804:
801:
798:
795:
773:
769:
765:
762:
757:
751:
747:
744:
741:
738:
735:
710:
706:
702:
699:
672:
666:
617:
612:
608:
604:
601:
596:
592:
588:
583:
578:
575:
572:
569:
549:
529:
526:
521:
514:
510:
506:
503:
498:
494:
488:
484:
478:
475:
472:
467:
463:
459:
456:
453:
448:
444:
438:
435:
415:
393:
389:
382:
378:
346:
326:
323:
303:
300:
298:
295:
224:
209:point symmetry
179:
178:
170:
169:
163:
162:
154:
153:
147:
146:
141:
135:
134:
120:
114:
99:
97:Symmetry group
93:
92:
89:
83:
82:
79:
73:
72:
69:
63:
62:
37:
33:
32:
24:
23:
15:
9:
6:
4:
3:
2:
3199:
3188:
3185:
3183:
3180:
3178:
3175:
3173:
3170:
3168:
3165:
3164:
3162:
3148:
3142:
3136:
3133:
3131:
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3111:
3108:
3106:
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3082:
3080:
3076:
3066:
3063:
3061:
3058:
3057:
3055:
3051:
3045:
3042:
3040:
3037:
3036:
3033:
3030:
3026:
3020:
3019:
3015:
3013:
3012:
3008:
3007:
3005:
3001:
2995:
2992:
2990:
2987:
2985:
2982:
2980:
2977:
2975:
2972:
2970:
2967:
2965:
2962:
2960:
2957:
2955:
2952:
2950:
2947:
2945:
2942:
2940:
2937:
2935:
2932:
2931:
2929:
2922:
2917:
2911:
2908:
2906:
2903:
2901:
2898:
2896:
2893:
2891:
2888:
2886:
2883:
2881:
2878:
2876:
2873:
2871:
2868:
2866:
2863:
2861:
2858:
2856:
2855:cuboctahedron
2853:
2851:
2848:
2847:
2845:
2840:
2836:
2830:
2825:
2819:
2816:
2814:
2811:
2809:
2806:
2804:
2801:
2799:
2796:
2795:
2793:
2789:
2784:
2780:
2776:
2768:
2763:
2761:
2756:
2754:
2749:
2748:
2745:
2731:
2726:
2721:
2717:
2716:
2713:
2708:
2703:
2699:
2697:
2692:
2688:
2686:
2681:
2677:
2675:
2670:
2666:
2664:
2659:
2655:
2653:
2648:
2644:
2643:
2640:
2639:Cuboctahedron
2635:
2631:
2629:
2624:
2620:
2618:
2613:
2609:
2607:
2602:
2598:
2596:
2591:
2587:
2585:
2580:
2576:
2575:
2570:
2566:
2559:
2554:
2552:
2547:
2545:
2540:
2539:
2536:
2530:
2527:
2523:
2518:
2513:
2512:
2507:
2504:
2499:
2492:
2491:
2486:
2483:
2478:
2477:
2474:
2473:
2468:
2464:
2460:
2456:
2455:
2445:
2443:0-521-55432-2
2439:
2435:
2431:
2426:
2422:
2420:3-540-23158-7
2416:
2412:
2411:
2405:
2395:
2390:
2380:
2379:
2374:
2369:
2365:
2361:
2357:
2353:
2348:
2337:
2336:
2331:
2326:
2324:(Section 3–9)
2321:
2319:0-486-23729-X
2315:
2311:
2307:
2303:
2302:
2293:
2292:
2287:
2284:
2277:
2270:
2264:
2257:, p. 269
2256:
2252:
2245:
2238:
2234:
2230:
2224:
2220:
2216:
2212:
2211:
2206:
2199:
2191:
2187:
2180:
2172:
2168:
2164:
2160:
2156:
2152:
2145:
2137:
2131:
2127:
2123:
2122:
2114:
2106:
2100:
2096:
2095:
2087:
2079:
2075:
2070:
2065:
2061:
2057:
2052:
2047:
2043:
2039:
2035:
2031:
2027:
2020:
2012:
2006:
2002:
1998:
1994:
1990:
1983:
1975:
1974:
1969:
1965:
1959:
1951:
1947:
1942:
1937:
1933:
1929:
1928:
1923:
1919:
1915:
1910:
1906:
1900:
1896:
1891:
1886:
1882:
1878:
1874:
1867:
1859:
1855:
1851:
1847:
1843:
1839:
1832:
1824:
1818:
1814:
1810:
1806:
1805:
1797:
1789:
1785:
1781:
1777:
1773:
1769:
1764:
1759:
1755:
1751:
1750:
1745:
1741:
1735:
1727:
1721:
1717:
1716:
1711:
1705:
1697:
1693:
1686:
1678:
1672:
1668:
1664:
1660:
1656:
1655:
1647:
1639:
1635:
1631:
1627:
1623:
1619:
1612:
1610:
1601:
1597:
1591:
1587:
1576:
1575:LCF notations
1571:
1567:
1565:
1561:
1558:
1553:
1551:
1547:
1543:
1540:
1536:
1532:
1528:
1524:
1520:
1511:
1507:
1504:
1500:
1496:
1492:
1489:
1485:
1481:
1479:
1475:
1471:
1469:
1465:
1461:
1459:
1455:
1451:
1449:
1448:Automorphisms
1445:
1441:
1439:
1435:
1431:
1429:
1425:
1421:
1415:
1410:
1405:
1393:
1387:
1382:
1378:
1372:
1367:
1364:
1357:
1352:
1348:
1342:
1337:
1334:
1331:sculpture in
1327:
1322:
1315:
1310:
1309:
1303:
1294:
1285:
1271:
1269:
1265:
1261:
1257:
1254:
1249:
1247:
1237:
1233:
1230:
1226:
1225:
1221:
1215:
1212:
1206:
1201:
1198:
1197:
1194:
1192:
1187:
1185:
1181:
1177:
1167:
1164:
1162:
1158:
1154:
1149:
1147:
1143:
1132:
1125:
1113:
1099:
1097:
1087:
1085:
1084:Minkowski sum
1081:
1077:
1072:
1068:
1064:
1060:
1056:
1051:
1035:
1031:
1023:
1004:
1001:
998:
995:
992:
989:
986:
963:
960:
957:
954:
951:
948:
945:
942:
939:
916:
913:
910:
907:
904:
901:
898:
887:
883:
882:permutohedron
869:
857:
843:
827:
823:
819:
813:
809:
805:
802:
796:
793:
771:
767:
763:
755:
749:
745:
742:
736:
733:
724:
708:
704:
700:
697:
689:
688:vertex figure
655:
651:
647:
643:
633:
628:
615:
610:
606:
602:
599:
594:
590:
581:
576:
573:
570:
547:
527:
524:
519:
512:
508:
504:
501:
496:
492:
486:
482:
476:
473:
470:
465:
457:
454:
446:
442:
436:
433:
413:
391:
387:
380:
376:
364:
363:Johnson solid
360:
344:
324:
321:
313:
309:
294:
248:
244:
239:
237:
232:
230:
229:permutohedron
222:
218:
214:
210:
206:
202:
198:
194:
190:
186:
176:
171:
168:
164:
160:
155:
152:
151:Vertex figure
148:
145:
142:
140:
136:
103:
100:
98:
94:
90:
88:
84:
80:
78:
74:
70:
68:
64:
61:
56:
51:
50:Permutohedron
46:
41:
38:
34:
30:
25:
20:
3146:
3065:trapezohedra
3016:
3009:
2864:
2813:dodecahedron
2605:
2509:
2488:
2470:
2433:
2429:
2409:
2397:. Retrieved
2382:. Retrieved
2377:
2355:
2351:
2340:. Retrieved
2334:
2309:
2289:
2276:
2268:
2263:
2250:
2244:
2209:
2198:
2189:
2179:
2154:
2150:
2144:
2120:
2113:
2093:
2086:
2033:
2029:
2019:
1992:
1982:
1972:
1958:
1931:
1925:
1880:
1876:
1866:
1841:
1837:
1831:
1803:
1796:
1753:
1747:
1734:
1714:
1704:
1695:
1685:
1653:
1646:
1621:
1617:
1599:
1590:
1564:LCF notation
1554:
1550:queue number
1526:
1523:graph theory
1519:mathematical
1516:
1478:Queue number
1347:Rubik's Cube
1250:
1243:
1188:
1173:
1165:
1155:, the first
1150:
1139:
1102:Applications
1093:
1059:Voronoi cell
1055:plesiohedron
1052:
885:
879:
725:
639:
361:, the first
305:
240:
233:
212:
188:
182:
55:Plesiohedron
2835:semiregular
2818:icosahedron
2798:tetrahedron
1756:: 169–200.
1560:cubic graph
1557:Hamiltonian
1495:Hamiltonian
1131:FCC lattice
560:, this is:
359:equilateral
3161:Categories
3130:prismatoid
3060:bipyramids
3044:antiprisms
3018:hosohedron
2808:octahedron
2399:2006-09-08
2384:2006-09-08
2342:2006-09-08
1788:0132.14603
1582:References
1487:Properties
1302:Jungle gym
1176:octahedron
1170:Dissection
1148:crystals.
1076:zonohedron
1071:translated
1063:Delone set
217:zonohedron
197:octahedron
60:Zonohedron
3187:Zonohedra
3125:birotunda
3115:bifrustum
2880:snub cube
2775:polyhedra
2712:Snub cube
2511:MathWorld
2490:MathWorld
2472:MathWorld
2430:Polyhedra
2291:MathWorld
2060:0027-8424
1909:zonotopes
1780:122006114
1544:. It has
1521:field of
1246:tesseract
1144:-type of
1142:faujasite
828:∘
820:≈
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764:≈
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600:≈
525:≈
477:⋅
471:−
3105:bicupola
3085:pyramids
3011:dihedron
2308:(1979).
2078:24516137
1966:(2005).
1920:(1980).
1742:(1966).
1712:(1979).
1659:Springer
1535:vertices
1428:Vertices
1202:Genus 3
1199:Genus 2
528:11.3137.
201:hexagons
185:geometry
87:Vertices
3147:italics
3135:scutoid
3120:rotunda
3110:frustum
2839:uniform
2788:regular
2773:Convex
2237:2561378
2159:Bibcode
2069:3939887
2038:Bibcode
1950:0585178
1899:1703597
1772:0185507
1638:0290245
1529:is the
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1499:regular
1394:crystal
1392:Boleite
1379:crystal
1349:variant
1274:Objects
1262:of the
1159:of the
1146:zeolite
603:26.7846
288:√
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205:squares
191:is the
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3053:duals:
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