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

Abstract

AbstractBuckminster 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).