Computing the maximum number of equiangular lines in n-dimensional Euclidean space is a difficult problem, and unsolved in general, though bounds are known.
The maximal number of equiangular lines in 2-dimensional Euclidean space is 3: we can take the lines through opposite vertices of a regular hexagon, each at an angle 120 degrees from the other two.
The maximum in 3 dimensions is 6: we can take lines through opposite vertices of an icosahedron.
[1] This upper bound is tight up to a constant factor to a construction by de Caen.
[2] The maximum in dimensions 1 through 16 is listed in the On-Line Encyclopedia of Integer Sequences as follows: In particular, the maximum number of equiangular lines in 7 dimensions is 28.
, and form all 28 vectors obtained by permuting its components.
Thus, the lines through the origin containing these vectors are equiangular.
In fact, these 28 vectors and their negatives are, up to rotation and dilatation, the 56 vertices of the 321 polytope.
In other words, they are the weight vectors of the 56-dimensional representation of the Lie group E7.
Equiangular lines are equivalent 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.
Conversely, every two-graph can be represented as a set of equiangular lines.
[3] The problem of determining the maximum number of equiangular lines with a fixed angle in sufficiently high dimensions was solved by Jiang, Tidor, Yao, Zhang, and Zhao.
[4] The answer is expressed in spectral graph theoretic terms.
denote the maximum number of lines through the origin in
dimensions with common pairwise angle
denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly
It is known that an upper bound for the number of complex equiangular lines in any dimension
Unlike the real case described above, it is possible that this bound is attained in every dimension
The conjecture that this holds true was proposed by Zauner[5] and verified analytically or numerically up to
[6] A maximal set of complex equiangular lines is also known as a SIC or SIC-POVM.