Abstract: A set of lines through the origin is equiangular if
the angles between any pair of lines are equal. In this talk I shall discuss
techniques for constructing such lines and related open problems.

A polytope in R^d is equipartite if it has 2n vertices and
for every choice of n vertices there is a symmetry of the polytope
that maps these vertices onto the other n vertices. Squares, rectangles,
two types of hexagons are examples in R^2. Rectangular boxes,
certain tetrahedra, rectangular bi-pyramid (cross polytope) are
examples of equipartite polytopes in R^3. We prove that the maximum number of
vertices of an equipartite polytope in R^d is 2d and show how to construct
all equipartite polytopes in R^d.