In geometry, a set of lines in Euclidean space is called equiangular if every pair of lines makes the same angle.
Equiangular lines are related to two-graphs. Given a set of equiangular lines, let c be the cosine of the common angle. We assume that the angle is not 90°, since that case is trivial (i.e., not interesting, because the lines are just coordinate axes); thus, c is nonzero. We may move the lines so they all pass through the origin of coordinates. Choose one unit vector in each line. Form the matrix M of inner products. This matrix has 1 on the diagonal and ±c everywhere else, and it is symmetric. Subtracting the identity matrix I and dividing by c, we have a symmetric matrix with zero diagonal and ±1 off the diagonal. This is the adjacency matrix of a two-graph.
Famous quotes containing the word lines:
“It is the Late city that first defies the land, contradicts Nature in the lines of its silhouette, denies all Nature. It wants to be something different from and higher than Nature. These high-pitched gables, these Baroque cupolas, spires, and pinnacles, neither are, nor desire to be, related with anything in Nature. And then begins the gigantic megalopolis, the city-as-world, which suffers nothing beside itself and sets about annihilating the country picture.”
—Oswald Spengler (1880–1936)