Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T15:40:46.990Z Has data issue: false hasContentIssue false
  • English
  • Français

The Simplicial Helix and the Equation tan nθ = n tan θ

Published online by Cambridge University Press:  20 November 2018

H. S. M. Coxeter
H. S. M. Coxeter*
Affiliation:
University of Toronto, Toronto, Ontario, CanadaM5S 1A1
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Buckminster Fuller has coined the name tetrahelix for a column of regular tetrahedra, each sharing two faces with neighbours, one 'below' and one 'above' [A. H. Boerdijk, Philips Research Reports 7 (1952), p. 309]. Such a column could well be employed in architecture, because it is both strong and attractive. The (n — 1)-dimensional analogue is based on a skew polygon such that every n consecutive vertices belong to a regular simplex. The generalized twist which shifts this polygon one step along itself is found to have the characteristic equation

(λ - 1)2{(n - 1)λn-2 + 2(n - 2)λn-3 + 3(n - 3)λn-4 + . . . + (n - 2)2λ + (n - 1)} = 0,

which can be derived from tan nθ = n tan θ by setting λ = exp (2θi).

Keywords

S1N20S1M2010A32Chebyshev polynomialPetrie polygonSimplexTetrahelixCross polytopeRegular polytopeSpherical spaceTwist
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 4 , 01 December 1985 , pp. 385 - 393
Copyright
Copyright © Canadian Mathematical Society 1985

References

0. Boerdijk, A.H., Some remarks concerning close-packing of equal spheres, Philips Research Reports 7 (1952), pp. 303–313.Google Scholar
1. Coxeter, H.S.M., Introduction to Geometry (2nd ed.), Wiley, New York, 1969.Google Scholar
2. Coxeter, H.S.M., Regular Polytopes (3rd ed.), Dover, New York, 1973.Google Scholar
3. Coxeter, H.S.M., Regular Complex Polytopes, Cambridge University Press, 1974.Google Scholar
4. Buckminster Fuller, R., Synergetics, Macmillan, New York, 1975.Google Scholar
5. Hobson, E.W., A Treatise on Plane Trigonometry (6th ed.), Cambridge University Press, 1925.Google Scholar
6. Hurley, A.C., Some helical structures generated by reflexions, Australian J. Physics, 38 (1985), 299-310.Google Scholar