Equiangular lines
Encyclopedia
In geometry
, a set of line
s in Euclidean space
is called equiangular if every pair of lines makes the same angle.
Equiangular lines are related to two-graph
s.
Given a set of equiangular lines, let c be the cosine of the common angle. We assume that 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.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a set of line
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
s in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
is called equiangular if every pair of lines makes the same angle.
Equiangular lines are related to two-graph
Two-graph
In mathematics, a two-graph is a set of triples chosen from a finite vertex set X, such that every quadruple from X contains an even number of triples of the two-graph. A regular two-graph has the property that every pair of vertices lies in the same number of triples of the two-graph...
s.
Given a set of equiangular lines, let c be the cosine of the common angle. We assume that 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
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
of coordinates. Choose one unit vector in each line. Form the matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
M of inner products. This matrix has 1 on the diagonal and ±c everywhere else, and it is symmetric. Subtracting the identity matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
I and dividing by c, we have a symmetric matrix with zero diagonal and ±1 off the diagonal. This is the adjacency matrix
Adjacency matrix
In mathematics and computer science, an adjacency matrix is a means of representing which vertices of a graph are adjacent to which other vertices...
of a two-graph.