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faces. When right triangles are glued together in the pattern of a disphenoid, they form a flat figure (a doubly-covered rectangle) that does not enclose any volume. When obtuse triangles are glued in this way, the resulting surface can be folded to form a disphenoid (by
1422:
609:
1193:, a paper toy that can rotate on 4 sets of faces in a hexagon. The rotation of the six disphenoids with opposite edges of length l, m and n (without loss of generality n≤l, n≤m) is physically realizable if and only if
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1169:. It is symmetrical about each of three mutually perpendicular diad axes of symmetry in all classes except the tetragonal-disphenoidal, in which the form is generated by an inverse tetrad axis of symmetry.
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1116:. The disphenoid whose four vertices are (-1, 0, 0), (1, 0, 0), (0, 1, 1), and (0, 1, -1) is such a disphenoid. Each of its four faces is an isosceles triangle with edges of lengths
742:
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1052:{\displaystyle {\tfrac {1}{2}}(l^{2}+m^{2}-n^{2}),\quad {\tfrac {1}{2}}(l^{2}-m^{2}+n^{2}),\quad {\tfrac {1}{2}}(-l^{2}+m^{2}+n^{2}).}
2084:
1165:. It has four triangular faces that are alike and that correspond in position to alternate faces of the tetragonal or orthorhombic
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258:) but one with acute triangle faces and with edges that in general do not lie along the edges of the given obtuse triangles.
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bisecting two opposite faces. Both have four equal edges going around the sides. The digonal has two pairs of congruent
255:
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1514:
1512:; Matsunaga, Kiyoko (2020), "An Algorithm for Folding a Conway Tile into an Isotetrahedron or a Rectangle Dihedron",
1494:
860:
1135:
1841:
Brown, B. H. (April 1926), "Theorem of Bang. Isosceles tetrahedra", Undergraduate
Mathematics Clubs: Club Topics,
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in the shape of an acute triangle, divided into four similar triangles by segments connecting the edge midpoints.
1780:
1417:{\displaystyle -8*(l^{2}-m^{2})^{2}*(l^{2}+m^{2})-5*n^{6}+11*(l^{2}-m^{2})^{2}*n^{2}+2*(l^{2}+m^{2})*n^{4}>=0}
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604:{\displaystyle V={\sqrt {\frac {(l^{2}+m^{2}-n^{2})(l^{2}-m^{2}+n^{2})(-l^{2}+m^{2}+n^{2})}{72}}}.}
1067:
If the four faces of a tetrahedron have the same perimeter, then the tetrahedron is a disphenoid.
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has faces with two different shapes, both isosceles triangles, with two faces of each shape. The
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759:. There is also the following interesting relation connecting the volume and the circumradius:
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1433:
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406:, the tetrahedra in which all four faces have the same area, and the tetrahedra in which the
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242:: as well as being congruent to each other, all of their faces are symmetric to each other.
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of a disphenoid are the same, and the sum of the face angles at each vertex is equal to two
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The
Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching
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acute-angled triangles. It can also be described as a tetrahedron in which every two
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The disphenoids are the only polyhedra having infinitely many non-self-intersecting
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Two more types of tetrahedron generalize the disphenoid and have similar names. The
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that are opposite each other have equal lengths. Other names for the same shape are
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Crystallography: An
Introduction for Earth Science (and other Solid State) Students
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dihedral symmetry. Unlike the tetragonal disphenoid, the rhombic disphenoid has no
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If the four faces of a tetrahedron have the same area, then it is a disphenoid.
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describes, it can be folded without cutting or overlaps from a single sheet of
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International
Journal of Mathematical Education in Science and Technology
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The disphenoids are the tetrahedra in which all four faces have the same
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Gibb, William (1990), "Paper patterns: solid shapes from metric paper",
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faces, while the tetragonal has four congruent isosceles triangle faces.
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Six tetragonal disphenoids attached end-to-end in a ring construct a
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A space-filling tetrahedral disphenoid inside a cube. Two edges have
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Leech, John (1950), "Some properties of the isosceles tetrahedron",
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399:. On a disphenoid, all closed geodesics are non-self-intersecting.
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We also have that a tetrahedron is a disphenoid if and only if the
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34:
849:{\displaystyle \displaystyle 16T^{2}R^{2}=l^{2}m^{2}n^{2}+9V^{2}.}
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Hajja, Mowaffaq; Walker, Peter (2001), "Equifacial tetrahedra",
75:. It has three sets of edge lengths, existing as opposite pairs.
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MATCH Communications in
Mathematical and in Computer Chemistry
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similarly has faces with two shapes of scalene triangles.
238:. Both tetragonal disphenoids and rhombic disphenoids are
1587:(1950), "Some properties of the isosceles tetrahedron",
684:{\displaystyle R={\sqrt {\frac {l^{2}+m^{2}+n^{2}}{8}}}}
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Tetrahedron § Isometries of irregular tetrahedra
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1149:"Disphenoid" is also used to describe two forms of
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of 90°, and four edges have dihedral angles of 60°.
2116:Mathematical Analysis of Disphenoid by H C Rajpoot
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245:It is not possible to construct a disphenoid with
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2054:, Cambridge University Press, pp. 363–366,
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1817:
430:of a disphenoid with opposite edges of length
165:, and its edges have three different lengths.
1176:arranged in pairs, constituting a tetragonal
1092:to the edges they connect and to each other.
1689:(2007), "Tile-makers and semi-tile-makers",
1485:(3rd ed.), Dover Publications, p.
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1755:, Cambridge University Press, p. 424,
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755:is the area of any face, which is given by
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1822:(2nd ed.), Birkhäuser, pp. 30–31
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71:faces, and can fit diagonally inside of a
2085:On-Line Encyclopedia of Integer Sequences
2049:
1778:
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161:, because, in general, its faces are not
16:Tetrahedron whose faces are all congruent
1994:(1981), "Which tetrahedra fill space?",
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1450:with 12 equilateral triangle faces and D
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318:Another characterization states that if
272:Disphenoids can also be seen as digonal
1978:
1883:"Closed geodesics on regular polyhedra"
1818:Andreescu, Titu; Gelca, Razvan (2009),
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1157:A wedge-shaped crystal form of the
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13:
2098:: CS1 maint: overridden setting (
859:The squares of the lengths of the
737:{\displaystyle r={\frac {3V}{4T}}}
421:
414:. They are the polyhedra having a
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1515:Journal of Information Processing
339:are the common perpendiculars of
179:If the faces of a disphenoid are
169:Special cases and generalizations
157:. However, a disphenoid is not a
1820:Mathematical Olympiad Challenges
1172:A crystal form bounded by eight
1136:disphenoid tetrahedral honeycomb
56:
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256:Alexandrov's uniqueness theorem
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1844:American Mathematical Monthly
1692:American Mathematical Monthly
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1781:"The most chiral disphenoid"
1752:Geometric Folding Algorithms
1554:Whittaker, E. J. W. (2013),
7:
1890:Moscow Mathematical Journal
1881:; Fuchs, Ekaterina (2007),
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410:of all four vertices equal
43:can be positioned inside a
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2075:Sloane, N. J. A.
1779:Petitjean, Michel (2015),
1458:Trirectangular tetrahedron
172:
136:almost regular tetrahedron
1654:10.1080/00207390110038231
219:as its faces is called a
1590:The Mathematical Gazette
1560:, Elsevier, p. 89,
1439:Orthocentric tetrahedron
98: 'wedgelike') is a
2150:"Isosceles tetrahedron"
2079:"Sequence A338336"
1096:Honeycombs and crystals
1529:10.2197/ipsjjip.28.750
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205:. In this case it has
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2040:Mathematics in School
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203:tetragonal disphenoid
181:equilateral triangles
173:Further information:
128:isosceles tetrahedron
1997:Mathematics Magazine
1938:Mathematical Gazette
1434:Irregular tetrahedra
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616:circumscribed sphere
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195:tetrahedral symmetry
1896:(2): 265–279, 350,
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1085:of the disphenoid.
1073:The centers in the
232:reflection symmetry
199:isosceles triangles
185:regular tetrahedron
2146:Weisstein, Eric W.
2127:Weisstein, Eric W.
1992:Senechal, Marjorie
1981:, pp. 71–72).
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267:phyllic disphenoid
263:digonal disphenoid
221:rhombic disphenoid
215:. A sphenoid with
159:regular polyhedron
65:rhombic disphenoid
49:isosceles triangle
2088:, OEIS Foundation
1762:978-0-521-71522-5
1482:Regular Polytopes
1477:Coxeter, H. S. M.
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1567:9781483285566
1563:
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1498:
1496:0-486-61480-8
1492:
1488:
1484:
1483:
1478:
1472:
1468:
1459:
1456:
1449:
1448:Johnson solid
1445:
1442:
1440:
1437:
1435:
1432:
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1411:
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1399:
1395:
1387:
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1225:
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1216:
1209:
1206:
1203:
1196:
1195:
1194:
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1179:
1178:scalenohedron
1175:
1171:
1168:
1164:
1160:
1156:
1155:
1154:
1152:
1147:
1145:
1141:
1137:
1133:
1115:
1107:
1102:
1093:
1091:
1090:perpendicular
1086:
1084:
1080:
1076:
1075:circumscribed
1071:
1068:
1046:
1038:
1034:
1030:
1025:
1021:
1017:
1012:
1008:
1004:
995:
992:
985:
977:
973:
969:
964:
960:
956:
951:
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937:
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919:
915:
911:
906:
902:
898:
893:
889:
879:
876:
866:
865:
864:
862:
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837:
833:
829:
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817:
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784:
778:
774:
770:
762:
761:
760:
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669:
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621:
620:
619:
617:
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575:
570:
566:
562:
557:
553:
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538:
534:
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525:
521:
517:
512:
508:
496:
492:
488:
483:
479:
475:
470:
466:
455:
452:
445:
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443:
442:is given by:
441:
437:
433:
429:
419:
417:
409:
405:
400:
398:
393:
391:
390:perpendicular
388:are pairwise
387:
380:
373:
366:
362:
358:
354:
350:
346:
342:
338:
331:
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316:
314:
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306:
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299:
295:
285:
283:
279:
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226:
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176:
166:
164:
160:
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148:
143:
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137:
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125:
121:
117:
113:
109:
105:
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97:
93:
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85:
74:
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42:
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28:
19:
2153:
2134:
2131:"Disphenoid"
2120:Academia.edu
2082:
2069:
2051:
2043:
2039:
2033:
2001:
1995:
1986:
1974:
1941:
1937:
1893:
1889:
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1848:
1842:
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1791:
1787:
1751:
1737:
1696:
1690:
1687:Akiyama, Jin
1645:
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1556:
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1510:Akiyama, Jin
1504:
1480:
1471:
1191:kaleidocycle
1188:
1148:
1111:
1087:
1072:
1069:
1066:
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752:
748:
746:
698:has radius:
693:
613:
439:
435:
431:
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401:
394:
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319:
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220:
206:
202:
188:
178:
155:right angles
147:solid angles
144:
139:
135:
131:
127:
123:
119:
115:
95:
87:
81:
64:
33:
18:
1585:Leech, John
1140:Gibb (1990)
234:, so it is
223:and it has
102:whose four
100:tetrahedron
96:sphenoeides
41:disphenoids
2173:Tetrahedra
1464:References
1185:Other uses
1159:tetragonal
1132:tessellate
1114:honeycombs
315:coincide.
278:alternated
274:antiprisms
183:, it is a
124:bisphenoid
90:(from
88:disphenoid
35:tetragonal
2155:MathWorld
2136:MathWorld
1966:125145099
1670:218495301
1627:125145099
1538:230108666
1454:symmetry.
1396:∗
1364:∗
1345:∗
1322:−
1306:∗
1287:∗
1281:−
1249:∗
1226:−
1210:∗
1204:−
1167:dipyramid
1005:−
957:−
912:−
861:bimedians
550:−
518:−
489:−
404:perimeter
108:congruent
2167:Category
2094:cite web
2046:(3): 2–4
1749:(2007),
1729:32897155
1713:27642275
1479:(1973),
1428:See also
1144:a4 paper
1083:centroid
694:and the
311:and the
240:isohedra
145:All the
120:sphenoid
84:geometry
2077:(ed.),
2026:0644075
2018:2689983
1958:3611029
1910:2337883
1865:2299548
1800:3242747
1721:2341323
1662:1847966
1619:0038667
1611:3611029
1151:crystal
1125:√
1118:√
307:in the
39:digonal
2058:
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2016:
1964:
1956:
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1759:
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1493:
747:where
428:volume
355:; and
305:center
282:prisms
276:or as
236:chiral
138:, and
73:cuboid
45:cuboid
2118:from
2014:JSTOR
1962:S2CID
1954:JSTOR
1886:(PDF)
1861:JSTOR
1784:(PDF)
1725:S2CID
1709:JSTOR
1666:S2CID
1623:S2CID
1607:JSTOR
1534:S2CID
1409:>=
1138:. As
863:are:
187:with
112:edges
104:faces
94:
92:Greek
2100:link
2083:The
2056:ISBN
1757:ISBN
1562:ISBN
1491:ISBN
1446:- A
1077:and
614:The
438:and
426:The
381:and
365:ABCD
359:and
351:and
343:and
332:and
149:and
106:are
86:, a
37:and
32:The
2006:doi
1946:doi
1898:doi
1853:doi
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830:9
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794:=
789:2
785:R
779:2
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753:T
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729:T
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718:3
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709:r
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385:3
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329:2
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322:1
320:d
228:2
225:D
207:D
192:d
189:T
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