211:
204:
29:
1698:
1143:
771:
1578:
1303:
933:
616:
1661:
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.
1413:
1174:
599:
526:
295:
262:
390:
328:
1345:
1138:{\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 }}~.}
766:{\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 )}.}
921:
465:
1405:
194:
1616:
1641:
1385:
1365:
1166:
898:
878:
1955:
1573:{\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 }~.}
1298:{\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)}}~.}
543:
333:
Certain proportions of the rhombs give rise to some well-known special cases. These typically occur in both prolate and oblate forms.
1948:
1941:
467:, with one vertex at the origin (0, 0, 0), and with one edge lying along the x-axis, the three generating vectors are
73:
2362:
482:
2357:
1871:
267:
234:
2154:
2095:
1347:, and as the height of a rhombohedron is given by its volume divided by the area of its base, the height
362:
300:
141:
1311:
2184:
2144:
2179:
2174:
1670:
Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an
1671:
2285:
2280:
2159:
2065:
1866:
1688:
The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a
431:
2149:
2090:
2080:
2025:
1689:
903:
447:
196:. There are two general forms of the rhombohedron, oblate (flattened) and prolate (stretched.
2169:
2085:
2040:
1988:
1390:
179:
2129:
2055:
2003:
1683:
1589:
425:
145:
8:
2295:
2164:
2139:
2124:
2060:
2008:
1928:
100:
2367:
2310:
2275:
2134:
2029:
1978:
1888:
1798:
1763:
1626:
1370:
1350:
1151:
883:
863:
776:
The other coordinates can be obtained from vector addition of the 3 direction vectors:
2290:
2100:
2075:
2019:
1911:
78:
28:
2229:
1880:
1844:
1790:
39:
1914:
1754:
Miller, William A. (January 1989). "Maths
Resource: Rhombic Dodecahedra Puzzles".
1710:
149:
108:
2050:
1973:
924:
133:
2351:
2255:
2111:
2045:
444:
For a unit (i.e.: with side length 1) rhombohedron, with rhombic acute angle
419:
409:
1831:
Solid geometry: with chapters on space-lattices, sphere-packs and crystals
1767:
2320:
2208:
1998:
1965:
1892:
1802:
210:
104:
96:
203:
2315:
2305:
2250:
2234:
2070:
1919:
1884:
1794:
2201:
1933:
1730:
117:
2325:
2300:
137:
50:
594:{\displaystyle {\biggl (}\cos \theta ,\sin \theta ,0{\biggr )},}
1697:
165:
1929:
Volume
Calculator https://rechneronline.de/pi/rhombohedron.php
1781:
Inchbald, Guy (July 1997). "The
Archimedean honeycomb duals".
1674:, and all orthocentric tetrahedra can be formed in this way.
1993:
342:
161:
1869:(October 1934), "Notes on the orthocentric tetrahedron",
148:
with rhombohedral cells. A rhombohedron has two opposite
1665:
1909:
1629:
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1393:
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1353:
1314:
1177:
1154:
936:
906:
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866:
619:
546:
485:
450:
365:
303:
270:
237:
182:
176:
The common angle at the two apices is here given as
164:is a special case of a rhombohedron with all sides
1635:
1610:
1572:
1399:
1379:
1359:
1339:
1297:
1160:
1137:
915:
892:
872:
765:
593:
520:
459:
384:
322:
289:
256:
188:
755:
622:
583:
549:
510:
488:
2349:
880:of a rhombohedron, in terms of its side length
1367:of a rhombohedron in terms of its side length
1308:As the area of the (rhombic) base is given by
1949:
21:
1956:
1942:
521:{\displaystyle {\biggl (}1,0,0{\biggr )},}
18:
923:, is a simplification of the volume of a
160:has an obtuse angle at these vertices. A
1780:
26:
1677:
2350:
1753:
290:{\displaystyle \theta <90^{\circ }}
257:{\displaystyle \theta >90^{\circ }}
152:at which all face angles are equal; a
1937:
1910:
1865:
1828:
136:in which all six faces are congruent
1963:
156:has this common angle acute, and an
1666:Relation to orthocentric tetrahedra
385:{\displaystyle \theta =90^{\circ }}
323:{\displaystyle \theta =90^{\circ }}
16:Polyhedron with six rhombi as faces
13:
1340:{\displaystyle a^{2}\sin \theta ~}
14:
2379:
1903:
1822:
439:
1696:
209:
202:
171:
27:
140:. It can be used to define the
92:
72:
64:
56:
45:
35:
1859:
1837:
1809:
1774:
1747:
1723:
1450:
1432:
1060:
1039:
1030:
1011:
971:
953:
1:
1872:American Mathematical Monthly
1740:
2336:Degenerate polyhedra are in
1733:is a two-dimensional figure.
1387:and its rhombic acute angle
900:and its rhombic acute angle
414:
398:
353:
222:
217:
7:
2155:pentagonal icositetrahedron
2096:truncated icosidodecahedron
1819:Third Edition. Dover. p.26.
1704:
1648:is the third coordinate of
142:rhombohedral lattice system
10:
2384:
2185:pentagonal hexecontahedron
2145:deltoidal icositetrahedron
1681:
1148:We can express the volume
2334:
2268:
2243:
2225:
2218:
2193:
2180:disdyakis triacontahedron
2175:deltoidal hexecontahedron
2109:
2017:
1972:
132:) is a special case of a
1783:The Mathematical Gazette
1716:
1672:orthocentric tetrahedron
916:{\displaystyle \theta ~}
460:{\displaystyle \theta ~}
264:and in the prolate case
2363:Space-filling polyhedra
2286:gyroelongated bipyramid
2160:rhombic triacontahedron
2066:truncated cuboctahedron
1400:{\displaystyle \theta }
432:rhombic triacontahedron
189:{\displaystyle \theta }
2281:truncated trapezohedra
2150:disdyakis dodecahedron
2116:(duals of Archimedean)
2091:rhombicosidodecahedron
2081:truncated dodecahedron
1847:. Wolfram. 17 May 2016
1690:trigonal trapezohedron
1637:
1612:
1574:
1401:
1381:
1361:
1341:
1299:
1162:
1139:
917:
894:
874:
767:
595:
522:
461:
386:
330:the figure is a cube.
324:
291:
258:
190:
2170:pentakis dodecahedron
2086:truncated icosahedron
2041:truncated tetrahedron
1833:. Dover Publications.
1756:Mathematics in School
1638:
1613:
1611:{\displaystyle h=a~z}
1575:
1402:
1382:
1362:
1342:
1300:
1163:
1140:
918:
895:
875:
768:
596:
523:
462:
387:
325:
292:
259:
191:
2358:Prismatoid polyhedra
2130:rhombic dodecahedron
2056:truncated octahedron
1684:Rhombohedral lattice
1678:Rhombohedral lattice
1627:
1590:
1414:
1391:
1371:
1351:
1312:
1175:
1152:
934:
904:
884:
864:
617:
544:
483:
448:
426:rhombic dodecahedron
363:
350:Golden Rhombohedron
301:
268:
235:
224:Prolate rhombohedron
180:
154:prolate rhombohedron
128:or, inaccurately, a
2165:triakis icosahedron
2140:tetrakis hexahedron
2125:triakis tetrahedron
2061:rhombicuboctahedron
1168:another way :
400:Ratio of diagonals
231:In the oblate case
219:Oblate rhombohedron
158:oblate rhombohedron
2135:triakis octahedron
2020:Archimedean solids
1912:Weisstein, Eric W.
1817:Regular Polytopes.
1633:
1608:
1570:
1397:
1377:
1357:
1337:
1295:
1158:
1135:
927:, and is given by
913:
890:
870:
763:
591:
518:
457:
430:Dissection of the
424:Dissection of the
382:
320:
287:
254:
186:
126:rhombic hexahedron
2345:
2344:
2264:
2263:
2101:snub dodecahedron
2076:icosidodecahedron
1845:"Vector Addition"
1829:Lines, L (1965).
1729:More accurately,
1636:{\displaystyle z}
1604:
1566:
1562:
1549:
1497:
1487:
1473:
1428:
1380:{\displaystyle a}
1360:{\displaystyle h}
1336:
1291:
1287:
1281:
1254:
1232:
1196:
1192:
1161:{\displaystyle V}
1131:
1127:
1063:
994:
912:
893:{\displaystyle a}
873:{\displaystyle V}
751:
738:
682:
456:
437:
436:
229:
228:
114:
113:
88:, , (×), order 2
2375:
2223:
2222:
2219:Dihedral uniform
2194:Dihedral regular
2117:
2033:
1982:
1958:
1951:
1944:
1935:
1934:
1925:
1924:
1897:
1895:
1863:
1857:
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1852:
1841:
1835:
1834:
1826:
1820:
1813:
1807:
1806:
1789:(491): 213–219.
1778:
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492:
491:
466:
464:
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454:
391:
389:
388:
383:
381:
380:
347:√2 Rhombohedron
336:
335:
329:
327:
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321:
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263:
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2378:
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2373:
2372:
2348:
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2341:
2330:
2269:Dihedral others
2260:
2239:
2214:
2189:
2118:
2115:
2114:
2105:
2034:
2023:
2022:
2013:
1976:
1974:Platonic solids
1968:
1962:
1906:
1901:
1900:
1885:10.2307/2300415
1864:
1860:
1850:
1848:
1843:
1842:
1838:
1827:
1823:
1814:
1810:
1795:10.2307/3619198
1779:
1775:
1752:
1748:
1743:
1738:
1737:
1728:
1724:
1719:
1711:Lists of shapes
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174:
124:(also called a
109:parallelohedron
86:
17:
12:
11:
5:
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2240:
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2205:
2197:
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2177:
2172:
2167:
2162:
2157:
2152:
2147:
2142:
2137:
2132:
2127:
2121:
2119:
2112:Catalan solids
2110:
2107:
2106:
2104:
2103:
2098:
2093:
2088:
2083:
2078:
2073:
2068:
2063:
2058:
2053:
2051:truncated cube
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2018:
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1996:
1991:
1985:
1983:
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1969:
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1960:
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1938:
1932:
1931:
1926:
1915:"Rhombohedron"
1905:
1904:External links
1902:
1899:
1898:
1879:(8): 499–502,
1858:
1836:
1821:
1815:Coxeter, HSM.
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1682:Main article:
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925:parallelepiped
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440:Solid geometry
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134:parallelepiped
112:
111:
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74:Symmetry group
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66:
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58:
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47:
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37:
33:
32:
24:
23:
15:
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6:
4:
3:
2:
2380:
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2211:
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2206:
2204:
2203:
2199:
2198:
2196:
2192:
2186:
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2181:
2178:
2176:
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2168:
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2158:
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2099:
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2079:
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2072:
2069:
2067:
2064:
2062:
2059:
2057:
2054:
2052:
2049:
2047:
2046:cuboctahedron
2044:
2042:
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2031:
2027:
2021:
2016:
2010:
2007:
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1467:
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1458:
1455:
1447:
1444:
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1438:
1435:
1423:
1420:
1417:
1410:
1409:
1408:
1394:
1374:
1354:
1331:
1328:
1325:
1320:
1316:
1292:
1283:
1278:
1275:
1270:
1266:
1261:
1257:
1251:
1248:
1243:
1240:
1234:
1229:
1226:
1221:
1217:
1212:
1208:
1202:
1198:
1189:
1184:
1181:
1178:
1171:
1170:
1169:
1155:
1132:
1124:
1121:
1116:
1112:
1108:
1105:
1102:
1099:
1094:
1090:
1086:
1083:
1080:
1073:
1069:
1065:
1057:
1054:
1051:
1048:
1045:
1042:
1034:
1026:
1023:
1020:
1017:
1014:
1004:
1000:
996:
991:
988:
985:
982:
979:
976:
968:
965:
962:
959:
956:
948:
944:
940:
937:
930:
929:
928:
926:
907:
887:
867:
858:
856:
855:
847:
846:
838:
837:
829:
828:
820:
819:
811:
810:
802:
801:
793:
792:
784:
783:
760:
747:
744:
741:
735:
732:
727:
723:
719:
716:
713:
710:
705:
701:
697:
694:
691:
684:
678:
675:
672:
667:
664:
659:
655:
651:
648:
645:
642:
636:
633:
630:
627:
612:
611:
604:
603:
588:
578:
575:
572:
569:
566:
563:
560:
557:
554:
539:
538:
531:
530:
515:
505:
502:
499:
496:
493:
478:
477:
470:
469:
468:
451:
433:
429:
427:
423:
421:
420:Regular solid
418:
415:
411:
408:
405:
402:
399:
395:
393:
377:
373:
369:
366:
359:
354:
349:
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344:
341:
338:
337:
334:
331:
315:
311:
307:
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282:
278:
274:
271:
249:
245:
241:
238:
225:
220:
216:
212:
208:
205:
201:
200:
197:
183:
172:Special cases
169:
167:
163:
159:
155:
151:
147:
143:
139:
135:
131:
127:
123:
119:
110:
106:
102:
98:
95:
91:
87:
85:
81:
77:
75:
71:
67:
63:
59:
55:
52:
48:
44:
41:
38:
34:
30:
25:
22:Rhombohedron
20:
2337:
2256:trapezohedra
2207:
2200:
2004:dodecahedron
1918:
1876:
1870:
1867:Court, N. A.
1861:
1849:. Retrieved
1839:
1830:
1824:
1816:
1811:
1786:
1782:
1776:
1762:(1): 18–24.
1759:
1755:
1749:
1725:
1687:
1669:
1660:
1650:
1649:
1644:
1619:
1582:
1407:is given by
1307:
1147:
859:
850:
849:
841:
840:
832:
831:
823:
822:
814:
813:
805:
804:
796:
795:
787:
786:
778:
777:
775:
606:
605:
533:
532:
472:
471:
443:
410:Golden ratio
357:constraints
332:
230:
223:
218:
175:
157:
153:
129:
125:
122:rhombohedron
121:
115:
83:
79:
2026:semiregular
2009:icosahedron
1989:tetrahedron
860:The volume
416:Occurrence
101:equilateral
2352:Categories
2321:prismatoid
2251:bipyramids
2235:antiprisms
2209:hosohedron
1999:octahedron
1741:References
105:zonohedron
93:Properties
2368:Zonohedra
2316:birotunda
2306:bifrustum
2071:snub cube
1966:polyhedra
1920:MathWorld
1559:θ
1556:
1547:θ
1544:
1525:θ
1522:
1506:−
1484:θ
1481:
1471:θ
1468:
1448:θ
1445:
1439:−
1395:θ
1332:θ
1329:
1276:θ
1267:
1244:−
1227:θ
1218:
1125:θ
1122:
1103:θ
1100:
1084:−
1058:θ
1055:
1027:θ
1024:
1018:−
992:θ
989:
969:θ
966:
960:−
908:θ
748:θ
745:
736:θ
733:
714:θ
711:
695:−
679:θ
676:
668:θ
665:
652:−
649:θ
646:
634:θ
631:
573:θ
570:
561:θ
558:
452:θ
378:∘
367:θ
316:∘
305:θ
283:∘
272:θ
250:∘
239:θ
184:θ
146:honeycomb
2296:bicupola
2276:pyramids
2202:dihedron
1768:30214564
1731:rhomboid
1705:See also
1623:, where
613: :
540: :
479: :
130:rhomboid
118:geometry
65:Vertices
2338:italics
2326:scutoid
2311:rotunda
2301:frustum
2030:uniform
1979:regular
1964:Convex
1893:2300415
1803:3619198
2291:cupola
2244:duals:
2230:prisms
1891:
1851:17 May
1801:
1766:
1603:
1583:Note:
1565:
1496:
1427:
1335:
1290:
1195:
1130:
911:
830:, and
455:
297:. For
166:square
150:apices
138:rhombi
97:convex
51:rhombi
1889:JSTOR
1799:JSTOR
1764:JSTOR
1717:Notes
355:Angle
339:Form
57:Edges
46:Faces
40:prism
1994:cube
1853:2016
343:Cube
275:<
242:>
162:cube
144:, a
120:, a
36:Type
2028:or
1881:doi
1791:doi
1553:sin
1535:cos
1513:cos
1478:sin
1465:cos
1442:cos
1326:sin
1258:sin
1209:sin
1113:cos
1091:cos
1052:cos
1021:cos
986:cos
963:cos
742:sin
724:cos
702:cos
673:sin
656:cos
643:cos
628:cos
567:sin
555:cos
406:√2
116:In
60:12
2354::
1917:.
1887:,
1877:41
1875:,
1797:.
1787:81
1785:.
1760:18
1758:.
1692::
857:.
848:+
839:+
821:+
812:,
803:+
794:,
785:+
403:1
374:90
312:90
279:90
246:90
168:.
107:,
103:,
99:,
68:8
49:6
2340:.
2032:)
2024:(
1981:)
1977:(
1957:e
1950:t
1943:v
1923:.
1896:.
1883::
1855:.
1805:.
1793::
1770:.
1657:.
1653:3
1651:e
1645:3
1631:z
1620:3
1606:z
1600:a
1597:=
1594:h
1568:.
1539:3
1531:2
1528:+
1517:2
1509:3
1503:1
1493:a
1490:=
1462:2
1459:+
1456:1
1451:)
1436:1
1433:(
1424:a
1421:=
1418:h
1375:a
1355:h
1321:2
1317:a
1293:.
1284:)
1279:2
1271:(
1262:2
1252:3
1249:4
1241:1
1235:)
1230:2
1222:(
1213:2
1203:3
1199:a
1190:3
1185:2
1182:=
1179:V
1156:V
1133:.
1117:3
1109:2
1106:+
1095:2
1087:3
1081:1
1074:3
1070:a
1066:=
1061:)
1049:2
1046:+
1043:1
1040:(
1035:2
1031:)
1015:1
1012:(
1005:3
1001:a
997:=
983:2
980:+
977:1
972:)
957:1
954:(
949:3
945:a
941:=
938:V
888:a
868:V
853:3
851:e
844:2
842:e
835:1
833:e
826:3
824:e
817:2
815:e
808:3
806:e
799:1
797:e
790:2
788:e
781:1
779:e
761:.
756:)
728:3
720:2
717:+
706:2
698:3
692:1
685:,
660:2
637:,
623:(
609:3
607:e
589:,
584:)
579:0
576:,
564:,
550:(
536:2
534:e
516:,
511:)
506:0
503:,
500:0
497:,
494:1
489:(
475:1
473:e
370:=
308:=
84:i
80:C
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