  We take a tour of a set of equiangular lines in eight-dimensional Hilbert
space. This structure defines an informationally complete measurement, that is,
a way to represent all quantum states of three-qubit systems as probability
distributions. Investigating the shape of this representation of state space
yields a pattern of connections among a remarkable spread of mathematical
constructions. In particular, studying the Shannon entropy of probabilistic
representations of quantum states leads to an intriguing link between the
questions of real and of complex equiangular lines. Furthermore, we will find
relations between quantum information theory and mathematical topics like
octonionic integers and the 28 bitangents to a quartic curve.
