222:
215:
40:
1709:
1154:
782:
1589:
1314:
944:
627:
1672:
The body diagonal between the acute-angled vertices is the longest. By rotational symmetry about that diagonal, the other three body diagonals, between the three pairs of opposite obtuse-angled vertices, are all the same length.
1424:
1185:
610:
537:
306:
273:
401:
339:
1356:
1149:{\displaystyle V=a^{3}(1-\cos \theta ){\sqrt {1+2\cos \theta }}=a^{3}{\sqrt {(1-\cos \theta )^{2}(1+2\cos \theta )}}=a^{3}{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }}~.}
777:{\displaystyle {\biggl (}\cos \theta ,{\cos \theta -\cos ^{2}\theta \over \sin \theta },{{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }} \over \sin \theta }{\biggr )}.}
932:
476:
1416:
205:
1627:
1652:
1396:
1376:
1177:
909:
889:
1966:
1584:{\displaystyle h=a~{(1-\cos \theta ){\sqrt {1+2\cos \theta }} \over \sin \theta }=a~{{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }} \over \sin \theta }~.}
1309:{\displaystyle V=2{\sqrt {3}}~a^{3}\sin ^{2}\left({\frac {\theta }{2}}\right){\sqrt {1-{\frac {4}{3}}\sin ^{2}\left({\frac {\theta }{2}}\right)}}~.}
554:
344:
Certain proportions of the rhombs give rise to some well-known special cases. These typically occur in both prolate and oblate forms.
1959:
1952:
478:, with one vertex at the origin (0, 0, 0), and with one edge lying along the x-axis, the three generating vectors are
84:
2373:
493:
2368:
1882:
278:
245:
17:
2165:
2106:
1358:, and as the height of a rhombohedron is given by its volume divided by the area of its base, the height
373:
311:
152:
1322:
2195:
2155:
2190:
2185:
1681:
Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an
1682:
2296:
2291:
2170:
2076:
1877:
1699:
The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a
442:
2160:
2101:
2091:
2036:
1700:
914:
458:
207:. There are two general forms of the rhombohedron, oblate (flattened) and prolate (stretched.
2180:
2096:
2051:
1999:
1401:
190:
2140:
2066:
2014:
1694:
1600:
436:
156:
8:
2306:
2175:
2150:
2135:
2071:
2019:
1939:
111:
2378:
2321:
2286:
2145:
2040:
1989:
1899:
1809:
1774:
1637:
1381:
1361:
1162:
894:
874:
787:
The other coordinates can be obtained from vector addition of the 3 direction vectors:
2301:
2111:
2086:
2030:
1922:
89:
39:
2240:
1891:
1855:
1801:
50:
1925:
1765:
Miller, William A. (January 1989). "Maths
Resource: Rhombic Dodecahedra Puzzles".
1721:
160:
119:
2061:
1984:
935:
144:
2362:
2266:
2122:
2056:
455:
For a unit (i.e.: with side length 1) rhombohedron, with rhombic acute angle
430:
420:
1842:
Solid geometry: with chapters on space-lattices, sphere-packs and crystals
1778:
2331:
2219:
2009:
1976:
1903:
1813:
221:
115:
107:
214:
2326:
2316:
2261:
2245:
2081:
1930:
1895:
1805:
2212:
1944:
1741:
128:
2336:
2311:
148:
61:
605:{\displaystyle {\biggl (}\cos \theta ,\sin \theta ,0{\biggr )},}
1708:
176:
1940:
Volume
Calculator https://rechneronline.de/pi/rhombohedron.php
1792:
Inchbald, Guy (July 1997). "The
Archimedean honeycomb duals".
1685:, and all orthocentric tetrahedra can be formed in this way.
2004:
353:
172:
1880:(October 1934), "Notes on the orthocentric tetrahedron",
159:
with rhombohedral cells. A rhombohedron has two opposite
1676:
1920:
1640:
1603:
1427:
1404:
1384:
1364:
1325:
1188:
1165:
947:
917:
897:
877:
630:
557:
496:
461:
376:
314:
281:
248:
193:
187:
The common angle at the two apices is here given as
175:is a special case of a rhombohedron with all sides
1646:
1621:
1583:
1410:
1390:
1370:
1350:
1308:
1171:
1148:
926:
903:
883:
776:
604:
531:
470:
395:
333:
300:
267:
199:
766:
633:
594:
560:
521:
499:
2360:
891:of a rhombohedron, in terms of its side length
1378:of a rhombohedron in terms of its side length
1319:As the area of the (rhombic) base is given by
1960:
32:
1967:
1953:
532:{\displaystyle {\biggl (}1,0,0{\biggr )},}
29:
934:, is a simplification of the volume of a
171:has an obtuse angle at these vertices. A
1791:
37:
1688:
14:
2361:
1764:
301:{\displaystyle \theta <90^{\circ }}
268:{\displaystyle \theta >90^{\circ }}
163:at which all face angles are equal; a
1948:
1921:
1876:
1839:
147:in which all six faces are congruent
1974:
167:has this common angle acute, and an
1677:Relation to orthocentric tetrahedra
396:{\displaystyle \theta =90^{\circ }}
334:{\displaystyle \theta =90^{\circ }}
27:Polyhedron with six rhombi as faces
24:
1351:{\displaystyle a^{2}\sin \theta ~}
25:
2390:
1914:
1833:
450:
1707:
220:
213:
182:
38:
151:. It can be used to define the
103:
83:
75:
67:
56:
46:
1870:
1848:
1820:
1785:
1758:
1734:
1461:
1443:
1071:
1050:
1041:
1022:
982:
964:
13:
1:
1883:American Mathematical Monthly
1751:
2347:Degenerate polyhedra are in
1744:is a two-dimensional figure.
1398:and its rhombic acute angle
911:and its rhombic acute angle
425:
409:
364:
233:
228:
7:
2166:pentagonal icositetrahedron
2107:truncated icosidodecahedron
1830:Third Edition. Dover. p.26.
1715:
1659:is the third coordinate of
153:rhombohedral lattice system
10:
2395:
2196:pentagonal hexecontahedron
2156:deltoidal icositetrahedron
1692:
1159:We can express the volume
2345:
2279:
2254:
2236:
2229:
2204:
2191:disdyakis triacontahedron
2186:deltoidal hexecontahedron
2120:
2028:
1983:
143:) is a special case of a
1794:The Mathematical Gazette
1727:
1683:orthocentric tetrahedron
927:{\displaystyle \theta ~}
471:{\displaystyle \theta ~}
275:and in the prolate case
2374:Space-filling polyhedra
2297:gyroelongated bipyramid
2171:rhombic triacontahedron
2077:truncated cuboctahedron
1411:{\displaystyle \theta }
443:rhombic triacontahedron
200:{\displaystyle \theta }
2292:truncated trapezohedra
2161:disdyakis dodecahedron
2127:(duals of Archimedean)
2102:rhombicosidodecahedron
2092:truncated dodecahedron
1858:. Wolfram. 17 May 2016
1701:trigonal trapezohedron
1648:
1623:
1585:
1412:
1392:
1372:
1352:
1310:
1173:
1150:
928:
905:
885:
778:
606:
533:
472:
397:
341:the figure is a cube.
335:
302:
269:
201:
2181:pentakis dodecahedron
2097:truncated icosahedron
2052:truncated tetrahedron
1844:. Dover Publications.
1767:Mathematics in School
1649:
1624:
1622:{\displaystyle h=a~z}
1586:
1413:
1393:
1373:
1353:
1311:
1174:
1151:
929:
906:
886:
779:
607:
534:
473:
398:
336:
303:
270:
202:
2369:Prismatoid polyhedra
2141:rhombic dodecahedron
2067:truncated octahedron
1695:Rhombohedral lattice
1689:Rhombohedral lattice
1638:
1601:
1425:
1402:
1382:
1362:
1323:
1186:
1163:
945:
915:
895:
875:
628:
555:
494:
459:
437:rhombic dodecahedron
374:
361:Golden Rhombohedron
312:
279:
246:
235:Prolate rhombohedron
191:
165:prolate rhombohedron
139:or, inaccurately, a
2176:triakis icosahedron
2151:tetrakis hexahedron
2136:triakis tetrahedron
2072:rhombicuboctahedron
1179:another way :
411:Ratio of diagonals
242:In the oblate case
230:Oblate rhombohedron
169:oblate rhombohedron
2146:triakis octahedron
2031:Archimedean solids
1923:Weisstein, Eric W.
1828:Regular Polytopes.
1644:
1619:
1581:
1408:
1388:
1368:
1348:
1306:
1169:
1146:
938:, and is given by
924:
901:
881:
774:
602:
529:
468:
441:Dissection of the
435:Dissection of the
393:
331:
298:
265:
197:
137:rhombic hexahedron
2356:
2355:
2275:
2274:
2112:snub dodecahedron
2087:icosidodecahedron
1856:"Vector Addition"
1840:Lines, L (1965).
1740:More accurately,
1647:{\displaystyle z}
1615:
1577:
1573:
1560:
1508:
1498:
1484:
1439:
1391:{\displaystyle a}
1371:{\displaystyle h}
1347:
1302:
1298:
1292:
1265:
1243:
1207:
1203:
1172:{\displaystyle V}
1142:
1138:
1074:
1005:
923:
904:{\displaystyle a}
884:{\displaystyle V}
762:
749:
693:
467:
448:
447:
240:
239:
125:
124:
99:, , (×), order 2
16:(Redirected from
2386:
2234:
2233:
2230:Dihedral uniform
2205:Dihedral regular
2128:
2044:
1993:
1969:
1962:
1955:
1946:
1945:
1936:
1935:
1908:
1906:
1874:
1868:
1867:
1865:
1863:
1852:
1846:
1845:
1837:
1831:
1824:
1818:
1817:
1800:(491): 213–219.
1789:
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1153:
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742:
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637:
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611:
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564:
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538:
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524:
503:
502:
477:
475:
474:
469:
465:
402:
400:
399:
394:
392:
391:
358:√2 Rhombohedron
347:
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340:
338:
337:
332:
330:
329:
307:
305:
304:
299:
297:
296:
274:
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198:
42:
30:
21:
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2387:
2385:
2384:
2383:
2359:
2358:
2357:
2352:
2341:
2280:Dihedral others
2271:
2250:
2225:
2200:
2129:
2126:
2125:
2116:
2045:
2034:
2033:
2024:
1987:
1985:Platonic solids
1979:
1973:
1917:
1912:
1911:
1896:10.2307/2300415
1875:
1871:
1861:
1859:
1854:
1853:
1849:
1838:
1834:
1825:
1821:
1806:10.2307/3619198
1790:
1786:
1763:
1759:
1754:
1749:
1748:
1739:
1735:
1730:
1722:Lists of shapes
1718:
1697:
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243:
192:
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188:
185:
135:(also called a
120:parallelohedron
97:
28:
23:
22:
15:
12:
11:
5:
2392:
2382:
2381:
2376:
2371:
2354:
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2199:
2198:
2193:
2188:
2183:
2178:
2173:
2168:
2163:
2158:
2153:
2148:
2143:
2138:
2132:
2130:
2123:Catalan solids
2121:
2118:
2117:
2115:
2114:
2109:
2104:
2099:
2094:
2089:
2084:
2079:
2074:
2069:
2064:
2062:truncated cube
2059:
2054:
2048:
2046:
2029:
2026:
2025:
2023:
2022:
2017:
2012:
2007:
2002:
1996:
1994:
1981:
1980:
1972:
1971:
1964:
1957:
1949:
1943:
1942:
1937:
1926:"Rhombohedron"
1916:
1915:External links
1913:
1910:
1909:
1890:(8): 499–502,
1869:
1847:
1832:
1826:Coxeter, HSM.
1819:
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1713:
1712:
1693:Main article:
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936:parallelepiped
920:
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854:
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827:
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451:Solid geometry
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145:parallelepiped
123:
122:
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101:
100:
93:
87:
85:Symmetry group
81:
80:
77:
73:
72:
69:
65:
64:
58:
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53:
48:
44:
43:
35:
34:
26:
9:
6:
4:
3:
2:
2391:
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2239:
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2235:
2232:
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2222:
2221:
2217:
2215:
2214:
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2209:
2207:
2203:
2197:
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2189:
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2179:
2177:
2174:
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2169:
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2159:
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2149:
2147:
2144:
2142:
2139:
2137:
2134:
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2131:
2124:
2119:
2113:
2110:
2108:
2105:
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2100:
2098:
2095:
2093:
2090:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2068:
2065:
2063:
2060:
2058:
2057:cuboctahedron
2055:
2053:
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2049:
2047:
2042:
2038:
2032:
2027:
2021:
2018:
2016:
2013:
2011:
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2001:
1998:
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1995:
1991:
1986:
1982:
1978:
1970:
1965:
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1932:
1927:
1924:
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1879:
1873:
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1836:
1829:
1823:
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1811:
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1803:
1799:
1795:
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1780:
1776:
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1757:
1743:
1737:
1733:
1723:
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1710:
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1705:
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1702:
1696:
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1513:
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1500:
1494:
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1478:
1475:
1472:
1469:
1466:
1458:
1455:
1452:
1449:
1446:
1434:
1431:
1428:
1421:
1420:
1419:
1405:
1385:
1365:
1342:
1339:
1336:
1331:
1327:
1303:
1294:
1289:
1286:
1281:
1277:
1272:
1268:
1262:
1259:
1254:
1251:
1245:
1240:
1237:
1232:
1228:
1223:
1219:
1213:
1209:
1200:
1195:
1192:
1189:
1182:
1181:
1180:
1166:
1143:
1135:
1132:
1127:
1123:
1119:
1116:
1113:
1110:
1105:
1101:
1097:
1094:
1091:
1084:
1080:
1076:
1068:
1065:
1062:
1059:
1056:
1053:
1045:
1037:
1034:
1031:
1028:
1025:
1015:
1011:
1007:
1002:
999:
996:
993:
990:
987:
979:
976:
973:
970:
967:
959:
955:
951:
948:
941:
940:
939:
937:
918:
898:
878:
869:
867:
866:
858:
857:
849:
848:
840:
839:
831:
830:
822:
821:
813:
812:
804:
803:
795:
794:
771:
758:
755:
752:
746:
743:
738:
734:
730:
727:
724:
721:
716:
712:
708:
705:
702:
695:
689:
686:
683:
678:
675:
670:
666:
662:
659:
656:
653:
647:
644:
641:
638:
623:
622:
615:
614:
599:
589:
586:
583:
580:
577:
574:
571:
568:
565:
550:
549:
542:
541:
526:
516:
513:
510:
507:
504:
489:
488:
481:
480:
479:
462:
444:
440:
438:
434:
432:
431:Regular solid
429:
426:
422:
419:
416:
413:
410:
406:
404:
388:
384:
380:
377:
370:
365:
360:
357:
355:
352:
349:
348:
345:
342:
326:
322:
318:
315:
293:
289:
285:
282:
260:
256:
252:
249:
236:
231:
227:
223:
219:
216:
212:
211:
208:
194:
183:Special cases
180:
178:
174:
170:
166:
162:
158:
154:
150:
146:
142:
138:
134:
130:
121:
117:
113:
109:
106:
102:
98:
96:
92:
88:
86:
82:
78:
74:
70:
66:
63:
59:
55:
52:
49:
45:
41:
36:
33:Rhombohedron
31:
19:
2348:
2267:trapezohedra
2218:
2211:
2015:dodecahedron
1929:
1887:
1881:
1878:Court, N. A.
1872:
1860:. Retrieved
1850:
1841:
1835:
1827:
1822:
1797:
1793:
1787:
1773:(1): 18–24.
1770:
1766:
1760:
1736:
1698:
1680:
1671:
1661:
1660:
1655:
1630:
1593:
1418:is given by
1318:
1158:
870:
861:
860:
852:
851:
843:
842:
834:
833:
825:
824:
816:
815:
807:
806:
798:
797:
789:
788:
786:
617:
616:
544:
543:
483:
482:
454:
421:Golden ratio
368:constraints
343:
241:
234:
229:
186:
168:
164:
140:
136:
133:rhombohedron
132:
126:
94:
90:
18:Rhombohedral
2037:semiregular
2020:icosahedron
2000:tetrahedron
871:The volume
427:Occurrence
112:equilateral
2363:Categories
2332:prismatoid
2262:bipyramids
2246:antiprisms
2220:hosohedron
2010:octahedron
1752:References
116:zonohedron
104:Properties
2379:Zonohedra
2327:birotunda
2317:bifrustum
2082:snub cube
1977:polyhedra
1931:MathWorld
1570:θ
1567:
1558:θ
1555:
1536:θ
1533:
1517:−
1495:θ
1492:
1482:θ
1479:
1459:θ
1456:
1450:−
1406:θ
1343:θ
1340:
1287:θ
1278:
1255:−
1238:θ
1229:
1136:θ
1133:
1114:θ
1111:
1095:−
1069:θ
1066:
1038:θ
1035:
1029:−
1003:θ
1000:
980:θ
977:
971:−
919:θ
759:θ
756:
747:θ
744:
725:θ
722:
706:−
690:θ
687:
679:θ
676:
663:−
660:θ
657:
645:θ
642:
584:θ
581:
572:θ
569:
463:θ
389:∘
378:θ
327:∘
316:θ
294:∘
283:θ
261:∘
250:θ
195:θ
157:honeycomb
2307:bicupola
2287:pyramids
2213:dihedron
1779:30214564
1742:rhomboid
1716:See also
1634:, where
624: :
551: :
490: :
141:rhomboid
129:geometry
76:Vertices
2349:italics
2337:scutoid
2322:rotunda
2312:frustum
2041:uniform
1990:regular
1975:Convex
1904:2300415
1814:3619198
2302:cupola
2255:duals:
2241:prisms
1902:
1862:17 May
1812:
1777:
1614:
1594:Note:
1576:
1507:
1438:
1346:
1301:
1206:
1141:
922:
841:, and
466:
308:. For
177:square
161:apices
149:rhombi
108:convex
62:rhombi
1900:JSTOR
1810:JSTOR
1775:JSTOR
1728:Notes
366:Angle
350:Form
68:Edges
57:Faces
51:prism
2005:cube
1864:2016
354:Cube
286:<
253:>
173:cube
155:, a
131:, a
47:Type
2039:or
1892:doi
1802:doi
1564:sin
1546:cos
1524:cos
1489:sin
1476:cos
1453:cos
1337:sin
1269:sin
1220:sin
1124:cos
1102:cos
1063:cos
1032:cos
997:cos
974:cos
753:sin
735:cos
713:cos
684:sin
667:cos
654:cos
639:cos
578:sin
566:cos
417:√2
127:In
71:12
2365::
1928:.
1898:,
1888:41
1886:,
1808:.
1798:81
1796:.
1771:18
1769:.
1703::
868:.
859:+
850:+
832:+
823:,
814:+
805:,
796:+
414:1
385:90
323:90
290:90
257:90
179:.
118:,
114:,
110:,
79:8
60:6
2351:.
2043:)
2035:(
1992:)
1988:(
1968:e
1961:t
1954:v
1934:.
1907:.
1894::
1866:.
1816:.
1804::
1781:.
1668:.
1664:3
1662:e
1656:3
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1608:=
1605:h
1579:.
1550:3
1542:2
1539:+
1528:2
1520:3
1514:1
1504:a
1501:=
1473:2
1470:+
1467:1
1462:)
1447:1
1444:(
1435:a
1432:=
1429:h
1386:a
1366:h
1332:2
1328:a
1304:.
1295:)
1290:2
1282:(
1273:2
1263:3
1260:4
1252:1
1246:)
1241:2
1233:(
1224:2
1214:3
1210:a
1201:3
1196:2
1193:=
1190:V
1167:V
1144:.
1128:3
1120:2
1117:+
1106:2
1098:3
1092:1
1085:3
1081:a
1077:=
1072:)
1060:2
1057:+
1054:1
1051:(
1046:2
1042:)
1026:1
1023:(
1016:3
1012:a
1008:=
994:2
991:+
988:1
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968:1
965:(
960:3
956:a
952:=
949:V
899:a
879:V
864:3
862:e
855:2
853:e
846:1
844:e
837:3
835:e
828:2
826:e
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817:e
810:1
808:e
801:2
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792:1
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772:.
767:)
739:3
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728:+
717:2
709:3
703:1
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671:2
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634:(
620:3
618:e
600:,
595:)
590:0
587:,
575:,
561:(
547:2
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527:,
522:)
517:0
514:,
511:0
508:,
505:1
500:(
486:1
484:e
381:=
319:=
95:i
91:C
20:)
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