58:
112:, the Boerdijk–Coxeter helix is not rotationally repetitive in 3-dimensional space. Even in an infinite string of stacked tetrahedra, no two tetrahedra will have the same orientation, because the helical pitch per cell is not a rational fraction of the circle. However, modified forms of this helix have been found which are rotationally repetitive, and in 4-dimensional space this helix repeats in rings of exactly 30 tetrahedral cells that tessellate the
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Edges can be colored into 6 groups, 3 main helixes (cyan), with the concave edges forming a slow forward helix (magenta), and two backwards helixes (yellow and orange)
405:
425:
385:
804:, p. 314, §4.2.2 The Boerdijk-Coxeter helix and the PPII helix; the helix of tetrahedra occurs in a left- or right-spiraling form, but each form contains
554:
curvature it becomes periodic, with a period of ten vertices, encompassing all 30 cells. The collective of such helices in the 600-cell represent a discrete
1120:
1076:
971:
257:
434:
1241:
981:
Sadler, Garrett; Fang, Fang; Kovacs, Julio; Klee, Irwin (2013). "Periodic modification of the
Boerdijk-Coxeter helix (tetrahelix)".
1086:
Banchoff, Thomas F. (1988). "Geometry of the Hopf
Mapping and Pinkall's Tori of Given Conformal Type". In Tangora, Martin (ed.).
142:
The coordinates of vertices of
Boerdijk–Coxeter helix composed of tetrahedrons with unit edge length can be written in the form
925:
570:. They spiral around each other naturally due to the Hopf fibration. The collective of edges forms another discrete Hopf
148:
1173:
1376:
1028:
Zhu, Yihan; He, Jiating; Shang, Cheng; Miao, Xiaohe; Huang, Jianfeng; Liu, Zhipan; Chen, Hongyu; Han, Yu (2014).
1095:
Banchoff, Thomas F. (2013). "Torus
Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie (ed.).
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86:
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forms, with either clockwise or counterclockwise windings. Unlike any other stacking of
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correspond to two chiral forms. All vertices are located on the cylinder with radius
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862:, pp. 577–578, §2.5 The 30/11 symmetry: an example of other kind of symmetries.
726:
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742:
672:
46:
1130:
Sadoc, J.F.; Rivier, N. (1999). "Boerdijk-Coxeter helix and biological helices".
1112:
935:
Boerdijk, A.H. (1952). "Some remarks concerning close-packing of equal spheres".
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can be chained with 3 vertex configurations, 3.3.5, 3.5.3.5, and 3.3.3.5.3.3.5:
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of the
Coxeter helix. Each sphere is in contact with 6 neighboring spheres.
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and considered them with regular and irregular tetrahedral elements.
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322:{\displaystyle \theta =\pm \cos ^{-1}(-2/3)\approx 131.81^{\circ }}
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56:
18:
476:{\displaystyle 2\theta -{\frac {4}{3}}\pi \approx 23.62^{\circ }}
427:
along z-axis. Given how the tetrahedra alternate, this gives an
1223:
962:
Pugh, Anthony (1976). "5. Joining polyhedra §5.36 Tetrahelix".
717:, 3.4.3.4 and 3.3.4.3.3.4. This helix exists as finite ring of
1169:"Helices and helix packings derived from the {3,3,5} polytope"
1030:"Chiral Gold Nanowires with Boerdijk–Coxeter–Bernal Structure"
550:, each a Boerdijk–Coxeter helix. When superimposed onto the
487:
tetrahedra. There is another inscribed cylinder with radius
1063:
Lord, Eric A.; Mackay, Alan L.; Ranganathan, S. (2006).
16:
Linear stacking of regular tetrahedra that form helices
998:"The γ-brass structure and the Boerdijk–Coxeter helix"
493:
437:
413:
393:
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is an arbitrary integer. The two different values of
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1090:. New York and Basel: Marcel Dekker. pp. 57–62.
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808:left- and right-spiraling helices of linked edges.
713:can also be chained together as a helix, with two
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246:
203:{\displaystyle (r\cos n\theta ,r\sin n\theta ,nh)}
202:
783:Skew apeirogon#Helical apeirogons in 3-dimensions
966:. University of California Press. p. 53.
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8:
534:30 tetrahedral ring from 600-cell projection
801:
574:with 10 vertices each. These correspond to
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1071:. Cambridge University Press. p. 64.
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20:Coxeter helices from regular tetrahedra
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719:30 pyramids in a 4-dimensional polytope
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758:is based on a Boerdijk–Coxeter helix.
859:
7:
996:Lord, E.A.; Ranganathan, S. (2004).
597:partitions into a single degenerate
96:, is a linear stacking of regular
14:
1065:"§4.5 The Boerdijk–Coxeter helix"
1005:Journal of Non-Crystalline Solids
247:{\displaystyle r=3{\sqrt {3}}/10}
1301:
1210:Boerdijk-Coxeter helix animation
1069:New Geometries for New Materials
1021:10.1016/j.jnoncrysol.2003.11.069
951:(1975). Applewhite, E.J. (ed.).
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360:{\displaystyle h=1/{\sqrt {10}}}
120:, one of the six regular convex
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1132:The European Physical Journal B
515:{\displaystyle 3{\sqrt {2}}/20}
1103:. Springer New York. pp.
920:. Cambridge University Press.
303:
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1:
1167:Sadoc, Jean-Francois (2001).
1113:10.1007/978-0-387-92714-5_20
964:Polyhedra: A visual approach
768:Clifford parallel cell rings
678:
655:
625:
43:
38:
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1174:European Physical Journal E
778:Line group#Helical symmetry
593:, four edges long, and the
526:Higher-dimensional geometry
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705:Related polyhedral helixes
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916:Regular Complex Polytopes
693:
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576:rings of 10 dodecahedrons
802:Sadoc & Rivier 1999
400:{\displaystyle \theta }
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87:Arie Hendrick Boerdijk
79:Boerdijk–Coxeter helix
74:
1187:10.1007/s101890170040
1152:10.1007/s100510051009
949:Fuller, R.Buckminster
715:vertex configurations
572:fibration of 12 rings
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1088:Computers in Algebra
878:which correspond to
589:partitions into two
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1144:1999EPJB...12..309S
1013:2004JNCS..334..121L
773:Toroidal polyhedron
735:pentagonal pyramids
591:8-tetrahedron rings
61:A Boerdijk helical
40:CCW and CW turning
21:
1007:. 334–335: 123–5.
818:Sadler et al. 2013
599:5-tetrahedron ring
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522:inside the helix.
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128:Buckminster Fuller
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1122:978-0-387-92713-8
1078:978-0-521-86104-5
1047:10.1021/ja506554j
973:978-0-520-03056-5
910:Coxeter, H. S. M.
847:"Tetrahelix Data"
834:930.00 Tetrahelix
702:
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585:In addition, the
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420:{\displaystyle r}
380:{\displaystyle n}
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83:H. S. M. Coxeter
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1215:Tetrahelix Data
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750:In architecture
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110:Platonic solids
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1099:Shaping Space
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616:Cycle lengths
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957:. Macmillan.
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709:Equilateral
708:
584:
578:in the dual
566:and have no
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131:
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78:
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1500:Pitch angle
1476:Logarithmic
1424:Archimedean
1387:Polyproline
1181:: 575–582.
988:1302.1174v1
954:Synergetics
830:Fuller 1975
622:Projection
130:named it a
90: [
1530:Categories
1489:On Spirals
1439:Hyperbolic
943:: 303–313.
927:052120125X
902:References
860:Sadoc 2001
638:30, 10, 15
607:4-polytope
548:tetrahedra
132:tetrahelix
98:tetrahedra
1541:Polyhedra
1510:Spirangle
1505:Theodorus
1444:Poinsot's
1434:Epispiral
1278:Curvature
1273:Algebraic
1195:121229939
691:(5, 5), 5
564:geodesics
562:they are
469:∘
461:≈
458:π
445:−
442:θ
431:twist of
395:θ
315:∘
307:≈
290:−
284:
276:−
268:±
262:θ
186:θ
180:
168:θ
162:
122:polychora
65:has each
1466:Involute
1461:Fermat's
1402:Collagen
1338:Symmetry
1160:92684626
1056:25126894
912:(1974).
762:See also
628:600-cell
580:120-cell
560:topology
552:3-sphere
544:20 rings
540:600-cell
429:apparent
138:Geometry
118:600-cell
114:3-sphere
1536:Helices
1495:Padovan
1429:Cotes's
1417:Spirals
1323:Antenna
1311:Helices
1283:Gallery
1259:helices
1251:Spirals
1140:Bibcode
1009:Bibcode
668:8, 8, 4
658:16-cell
587:16-cell
568:torsion
102:helices
1481:Golden
1397:Triple
1377:Double
1343:Triple
1293:Topics
1266:Curves
1255:curves
1193:
1158:
1119:
1107:–266.
1075:
1054:
970:
924:
681:5-cell
595:5-cell
546:of 30
483:every
311:131.81
213:where
106:chiral
71:vertex
67:sphere
1456:Euler
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