The Boerdijk–Coxeter helix, named after
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
( 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
30 tetrahedral ring from
The
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 This polyhedron-related article is a stub. You can help by expanding it. |