The geometry of structural equilibrium

Building on a long tradition from Maxwell, Rankine, Klein and others, this paper puts forward a geometrical description of structural equilibrium which contains a procedure for the graphic analysis of stress resultants within general three-dimensional frames. The method is a natural generalization of Rankine’s reciprocal diagrams for three-dimensional trusses. The vertices and edges of dual abstract 4-polytopes are embedded within dual four-dimensional vector spaces, wherein the oriented area of generalized polygons give all six components (axial and shear forces with torsion and bending moments) of the stress resultants. The relevant quantities may be readily calculated using four-dimensional Clifford algebra. As well as giving access to frame analysis and design, the description resolves a number of long-standing problems with the incompleteness of Rankine’s description of three-dimensional trusses. Examples are given of how the procedure may be applied to structures of engineering interest, including an outline of a two-stage procedure for addressing the equilibrium of loaded gridshell rooves.

The Euclidean distortion of generalized polygons

We determine the Euclidean distortion of finite generalized polygons by means of semidefinite programing.

Integral Hecke Algebras for Finite Generalized Polygons

Suppose that (P,B,F) is a triple consisting of the points, blocks and flags of a generalized m-gon, and H(F) the associated rank-2 Iwahori–Hecke algebra. H(F) acts naturally on the integral standard module ZF based on F. This work gives arithmetic conditions on a subring R, where R contains the integers and is contained in the rationals, that insure the associated R-ary Iwahori–Hecke algebra to be completely reducible on RF. The constituent multiplicities are related to the R-normal form of the incidence matrix of (P,B,F).