Английская Википедия:Boerdijk–Coxeter helix
| Файл:Coxeter helix 3 colors.png Файл:Coxeter helix 3 colors cw.png |
| CCW and CW turning |
| Файл:Coxeter helix edges.png 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)Шаблон:Sfn |
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,Шаблон:Sfn 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.
Buckminster Fuller named it a tetrahelix and considered them with regular and irregular tetrahedral elements.Шаблон:Sfn
Geometry
The coordinates of vertices of Boerdijk–Coxeter helix composed of tetrahedrons with unit edge length can be written in the form
- <math>(r\cos n\theta,r\sin n\theta,n h)</math>
where <math>r=3\sqrt{3}/10</math>, <math>\theta=\pm\cos^{-1}(-2/3) \approx 131.81^\circ</math>, <math>h=1/\sqrt{10}</math> and <math>n</math> is an arbitrary integer. The two different values of <math>\theta</math> correspond to two chiral forms. All vertices are located on the cylinder with radius <math> r </math> along z-axis. Given how the tetrahedra alternate, this gives an apparent twist of <math>2\theta - \frac{4}{3}\pi \approx 23.62^\circ</math> every two tetrahedra. There is another inscribed cylinder with radius <math>3\sqrt{2}/20</math> inside the helix.[1]
Higher-dimensional geometry
The 600-cell partitions into 20 rings of 30 tetrahedra, each a Boerdijk–Coxeter helix.Шаблон:Sfn 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.Шаблон:Sfn 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.Шаблон:Sfn 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. Шаблон:-
| 4-polytope | Rings | Tetrahedra/ring | Cycle lengths | Net | Projection |
|---|---|---|---|---|---|
| 600-cell | 20 | 30 | 30, 103, 152 | Файл:Coxeter helix 600-cell net.png | Файл:600-cell Coxeter helix-ring.png |
| 16-cell | 2 | 8 | 8, 8, 42 | Файл:16-cell 8-ring net4.png | |
| 5-cell | 1 | 5 | (5, 5), 5 | Файл:5-cell 5-ring net.png | |
Related polyhedral helixes
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:
In architecture
The Art Tower Mito is based on a Boerdijk–Coxeter helix.
See also
- Clifford parallel cell rings
- Toroidal polyhedron
- Line group#Helical symmetry
- Skew apeirogon#Helical apeirogons in 3-dimensions
Notes
References
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite arXiv
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite journal
External links
