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The Boerdijk–Coxeter helix, named after H. S. M. Coxeter
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. Contrary to any other stacking of Platonic solids, the Boerdijk–Coxeter helix
Boerdijk–Coxeter helix
is not rotationally repetitive. Even in an infinite string of stacked tetrahedra, no two tetrahedra will have the same orientation. This is because the helical pitch per cell is not a rational fraction of the circle. Buckminster Fuller
Buckminster Fuller
named it a tetrahelix and considered them with regular and irregular tetrahedral elements.[1]

Contents

1 Geometry 2 Architecture 3 Higher-dimensional geometry 4 Related polyhedral helixes 5 See also 6 Notes 7 References 8 External links

Geometry[edit] The coordinates of vertices of Boerdijk–Coxeter helix
Boerdijk–Coxeter helix
composed of tetrahedrons with unit edge length can be written in the form

( r cos ⁡ n θ , r sin ⁡ n θ , n h )

displaystyle (rcos ntheta ,rsin ntheta ,nh)

where

r = 3

3

/

10

displaystyle r=3 sqrt 3 /10

,

θ = ±

cos

− 1

⁡ ( − 2

/

3 )

displaystyle theta =pm cos ^ -1 (-2/3)

,

h = 1

/

10

displaystyle h=1/ sqrt 10

and

n

displaystyle n

is an arbitrary integer. The two different values of

θ

displaystyle theta

correspond to two chiral forms. All vertices are located on the cylinder with radius

r

displaystyle r

along z-axis. There is another inscribed cylinder with radius

3

2

/

20

displaystyle 3 sqrt 2 /20

inside the helix.[2] Architecture[edit] The Art Tower Mito
Art Tower Mito
is based on a Boerdijk-Coxeter helix. Higher-dimensional geometry[edit]

30 tetrahedral ring from 600-cell
600-cell
projection

The 600-cell
600-cell
partitions into 20 rings of 30 tetrahedra, each a Boerdijk–Coxeter helix. 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
600-cell
represent a discrete Hopf fibration. While in 3 dimensions the edges are helices, in the imposed 3-sphere
3-sphere
topology they are geodesics and have no torsion. They spiral around each other naturally due to the Hopf fibration. In addition, the 16-cell
16-cell
partitions into two 8-tetrahedron rings, four edges long, and the 5-cell
5-cell
partitions into a single degenerate 5-tetrahedron ring.

4-polytope Rings Tetrahedra/ring Cycle lengths Net Projection

600-cell 20 30 30, 103, 152

16-cell 2 8 8, 8, 42

5-cell 1 5 (5, 5), 5

Related polyhedral helixes[edit] Equilateral 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.

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:

See also[edit]

Toroidal polyhedron Line group#Helical symmetry Skew apeirogon#Helical apeirogons in 3-dimensions

Notes[edit]

^ http://www.rwgrayprojects.com/synergetics/s09/p3000.html ^ http://www.rwgrayprojects.com/rbfnotes/helix/helix01.html

References[edit]

H.S.M. Coxeter, Regular Complex Polytopes, Cambridge University, 1974. A.H. Boerdijk, Philips Res. Rep. 7 (1952) 30 The c-brass structure and the Boerdijk–Coxeter helix, E.A. Lord, S. Ranganathan, 2004, pp. 123–125 [1] Chiral Gold Nanowires with Boerdijk–Coxeter–Bernal Structure, Yihan Zhu, Jiating He, Cheng Shang, Xiaohe Miao, Jianfeng Huang, Zhipan Liu, Hongyu Chen and Yu Han, J. Am. Chem. Soc., 2014, 136 (36), pp 12746–12752 [2] Eric A. Lord, Alan Lindsay Mackay, Srinivasa Ranganathan, New geometries for new materials, p 64, sec 4.5 The Boerdijk–Coxeter helix J.F. Sadoc and N. Rivier, Boerdijk-Coxeter helix and biological helices The European Physical Journal B - Condensed Matter and Complex Systems, Volume 12, Number 2, 309-318, doi:10.1007/s100510051009 [3] Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7.  Chapter 5: Joining polyhedra, 5.36 Tetrahelix p. 53

External links[edit]

Boerdijk-Coxeter helix animation http://www.rwgrayprojects.com/rbfnotes/helix/helix01.html

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