105. [M1. 8.] The Equiangular Spiral

Tremain equiangular tight frames

We combine Steiner systems with Hadamard matrices to produce a new class of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph.

Steiner equiangular tight frames

We provide a new method for constructing equiangular tight frames (ETFs). The construction is valid in both the real and complex settings, and shows that many of the few previously-known examples of ETFs are but the first representatives of infinite families of such frames. It provides great freedom in terms of the frame’s size and redundancy. This method also explicitly constructs the frame ve...

Equiangular subspaces in Euclidean spaces

A set of lines through the origin is called equiangular if every pair of lines defines the same angle, and the maximum size of an equiangular set of lines in R was studied extensively for the last 70 years. In this paper, we study analogous questions for k-dimensional subspaces. We discuss natural ways of defining the angle between k-dimensional subspaces and correspondingly study the maximum s...

Equiangular lines in Euclidean spaces

Lattices from tight equiangular frames

We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular (k, n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a latti...