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)[1]
The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices. There are two chiral forms, with either clockwise or counterclockwise windings. Unlike any other stacking of Platonic solids, 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,[2] and in 4-dimensional space this helix repeats in rings of exactly 30 tetrahedral cells that tessellate the 3-sphere surface of the 600-cell, one of the six regular convex polychora.
A Boerdijk helical sphere packing has each sphere centered at a vertex of the Coxeter helix. Each sphere is in contact with 6 neighboring spheres.
Buckminster Fuller named it a tetrahelix and considered them with regular and irregular tetrahedral elements.[3]
The coordinates of vertices of Boerdijk–Coxeter helix composed of tetrahedrons with unit edge length can be written in the form
where , , and is an arbitrary integer. The two different values of correspond to two chiral forms. All vertices are located on the cylinder with radius along z-axis. Given how the tetrahedra alternate, this gives an apparent twist of every two tetrahedra. There is another inscribed cylinder with radius inside the helix.[4]
Higher-dimensional geometryedit
30 tetrahedral ring from 600-cell projection
The 600-cell partitions into 20 rings of 30 tetrahedra, each a Boerdijk–Coxeter helix.[5] When superimposed onto the 3-sphere 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 Hopf fibration.[6] While in 3 dimensions the edges are helices, in the imposed 3-sphere topology they are geodesics and have no torsion. They spiral around each other naturally due to the Hopf fibration.[7] The collective of edges forms another discrete Hopf fibration of 12 rings with 10 vertices each. These correspond to rings of 10 dodecahedrons in the dual 120-cell.
^Sadoc & Rivier 1999, 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 both left- and right-spiraling helices of linked edges.
Pugh, Anthony (1976). "5. Joining polyhedra §5.36 Tetrahelix". Polyhedra: A visual approach. University of California Press. p. 53. ISBN978-0-520-03056-5.
Sadler, Garrett; Fang, Fang; Kovacs, Julio; Klee, Irwin (2013). "Periodic modification of the Boerdijk-Coxeter helix (tetrahelix)". arXiv:1302.1174v1 [math.MG].
Lord, E.A.; Ranganathan, S. (2004). "The γ-brass structure and the Boerdijk–Coxeter helix" (PDF). Journal of Non-Crystalline Solids. 334–335: 123–5. Bibcode:2004JNCS..334..121L. doi:10.1016/j.jnoncrysol.2003.11.069.
Lord, Eric A.; Mackay, Alan L.; Ranganathan, S. (2006). "§4.5 The Boerdijk–Coxeter helix". New Geometries for New Materials. Cambridge University Press. p. 64. ISBN978-0-521-86104-5.
Banchoff, Thomas F. (1988). "Geometry of the Hopf Mapping and Pinkall's Tori of Given Conformal Type". In Tangora, Martin (ed.). Computers in Algebra. New York and Basel: Marcel Dekker. pp. 57–62.
Banchoff, Thomas F. (2013). "Torus Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie (ed.). Shaping Space. Springer New York. pp. 257–266. doi:10.1007/978-0-387-92714-5_20. ISBN978-0-387-92713-8.
Sadoc, J.F.; Rivier, N. (1999). "Boerdijk-Coxeter helix and biological helices". The European Physical Journal B. 12 (2): 309–318. Bibcode:1999EPJB...12..309S. doi:10.1007/s100510051009. S2CID 92684626.
Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
boerdijk, coxeter, helix, coxeter, helices, from, regular, tetrahedra, turningedges, colored, into, groups, main, helixes, cyan, with, concave, edges, forming, slow, forward, helix, magenta, backwards, helixes, yellow, orange, named, after, coxeter, boerdijk, . Coxeter helices from regular tetrahedra CCW and CW turningEdges 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 1 The Boerdijk Coxeter helix named after H S M Coxeter and A H Boerdijk is a linear stacking of regular tetrahedra arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices There are two chiral forms with either clockwise or counterclockwise windings Unlike any other stacking of Platonic solids 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 2 and in 4 dimensional space this helix repeats in rings of exactly 30 tetrahedral cells that tessellate the 3 sphere surface of the 600 cell one of the six regular convex polychora A Boerdijk helical sphere packing has each sphere centered at a vertex of the Coxeter helix Each sphere is in contact with 6 neighboring spheres Buckminster Fuller named it a tetrahelix and considered them with regular and irregular tetrahedral elements 3 Contents 1 Geometry 2 Higher dimensional geometry 3 Related polyhedral helixes 4 In architecture 5 See also 6 Notes 7 References 8 External linksGeometry editThe coordinates of vertices of Boerdijk Coxeter helix composed of tetrahedrons with unit edge length can be written in the form r cos n 8 r sin n 8 n h displaystyle r cos n theta r sin n theta nh nbsp where r 3 3 10 displaystyle r 3 sqrt 3 10 nbsp 8 cos 1 2 3 131 81 displaystyle theta pm cos 1 2 3 approx 131 81 circ nbsp h 1 10 displaystyle h 1 sqrt 10 nbsp and n displaystyle n nbsp is an arbitrary integer The two different values of 8 displaystyle theta nbsp correspond to two chiral forms All vertices are located on the cylinder with radius r displaystyle r nbsp along z axis Given how the tetrahedra alternate this gives an apparent twist of 2 8 4 3 p 23 62 displaystyle 2 theta frac 4 3 pi approx 23 62 circ nbsp every two tetrahedra There is another inscribed cylinder with radius 3 2 20 displaystyle 3 sqrt 2 20 nbsp inside the helix 4 Higher dimensional geometry edit nbsp 30 tetrahedral ring from 600 cell projectionThe 600 cell partitions into 20 rings of 30 tetrahedra each a Boerdijk Coxeter helix 5 When superimposed onto the 3 sphere 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 Hopf fibration 6 While in 3 dimensions the edges are helices in the imposed 3 sphere topology they are geodesics and have no torsion They spiral around each other naturally due to the Hopf fibration 7 The collective of edges forms another discrete Hopf fibration of 12 rings with 10 vertices each These correspond to rings of 10 dodecahedrons in the dual 120 cell In addition the 16 cell partitions into two 8 tetrahedron rings four edges long and the 5 cell partitions into a single degenerate 5 tetrahedron ring 4 polytope Rings Tetrahedra ring Cycle lengths Net Projection600 cell 20 30 30 103 152 nbsp nbsp 16 cell 2 8 8 8 42 nbsp 5 cell 1 5 5 5 5 nbsp Related polyhedral helixes editEquilateral square pyramids can also be chained together as a helix with two vertex configurations 3 4 3 4 and 3 3 4 3 3 4 This helix exists as finite ring of 30 pyramids in a 4 dimensional polytope nbsp And equilateral pentagonal pyramids can be chained with 3 vertex configurations 3 3 5 3 5 3 5 and 3 3 3 5 3 3 5 nbsp In architecture editThe Art Tower Mito is based on a Boerdijk Coxeter helix See also editClifford parallel cell rings Toroidal polyhedron Line group Helical symmetry Skew apeirogon Helical apeirogons in 3 dimensionsNotes edit Sadoc amp Rivier 1999 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 both left and right spiraling helices of linked edges Sadler et al 2013 Fuller 1975 930 00 Tetrahelix Tetrahelix Data Sadoc 2001 pp 577 578 2 5 The 30 11 symmetry an example of other kind of symmetries Banchoff 2013 studied the decomposition of regular 4 polytopes into honeycombs of tori tiling the Clifford torus which correspond to Hopf fibrations Banchoff 1988 References editCoxeter H S M 1974 Regular Complex Polytopes Cambridge University Press ISBN 052120125X Boerdijk A H 1952 Some remarks concerning close packing of equal spheres Philips Res Rep 7 303 313 Fuller R Buckminster 1975 Applewhite E J ed Synergetics Macmillan Pugh Anthony 1976 5 Joining polyhedra 5 36 Tetrahelix Polyhedra A visual approach University of California Press p 53 ISBN 978 0 520 03056 5 Sadler Garrett Fang Fang Kovacs Julio Klee Irwin 2013 Periodic modification of the Boerdijk Coxeter helix tetrahelix arXiv 1302 1174v1 math MG Lord E A Ranganathan S 2004 The g brass structure and the Boerdijk Coxeter helix PDF Journal of Non Crystalline Solids 334 335 123 5 Bibcode 2004JNCS 334 121L doi 10 1016 j jnoncrysol 2003 11 069 Zhu Yihan He Jiating Shang Cheng Miao Xiaohe Huang Jianfeng Liu Zhipan Chen Hongyu Han Yu 2014 Chiral Gold Nanowires with Boerdijk Coxeter Bernal Structure J Am Chem Soc 136 36 12746 52 doi 10 1021 ja506554j PMID 25126894 Lord Eric A Mackay Alan L Ranganathan S 2006 4 5 The Boerdijk Coxeter helix New Geometries for New Materials Cambridge University Press p 64 ISBN 978 0 521 86104 5 Banchoff Thomas F 1988 Geometry of the Hopf Mapping and Pinkall s Tori of Given Conformal Type In Tangora Martin ed Computers in Algebra New York and Basel Marcel Dekker pp 57 62 Banchoff Thomas F 2013 Torus Decompostions of Regular Polytopes in 4 space In Senechal Marjorie ed Shaping Space Springer New York pp 257 266 doi 10 1007 978 0 387 92714 5 20 ISBN 978 0 387 92713 8 Sadoc J F Rivier N 1999 Boerdijk Coxeter helix and biological helices The European Physical Journal B 12 2 309 318 Bibcode 1999EPJB 12 309S doi 10 1007 s100510051009 S2CID 92684626 Sadoc Jean Francois 2001 Helices and helix packings derived from the 3 3 5 polytope European Physical Journal E 5 575 582 doi 10 1007 s101890170040 S2CID 121229939 External links editBoerdijk Coxeter helix animation http www rwgrayprojects com rbfnotes helix helix01 html Retrieved from https en wikipedia org w index php title Boerdijk Coxeter helix amp oldid 1171082784, wikipedia, wiki, book, books, library,
