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. It is known that the maximum number in any dimension n is less than or equal to \binom{n+1}{2}. The maximum in dimensions 1 through 16 is listed in the
On-Line Encyclopedia of Integer Sequences as follows: :1, 3, 6, 6, 10, 16, 28, 28, 28, 28, 28, 28, 28, 28, 36, 40, ... . In particular, the maximum number of equiangular lines in 7 dimensions is 28. We can obtain these lines as follows. Take the vector (−3,−3,1,1,1,1,1,1) in \mathbb{R}^8, and form all 28 vectors obtained by permuting its components. The
dot product of two of these vectors equals 8 if both have a component of -3 in the same place, and it equals −8 otherwise. Thus, the lines through the origin containing these vectors are equiangular. Moreover, all 28 vectors are orthogonal to the vector (1,1,1,1,1,1,1,1) in \mathbb{R}^8, so they lie in a 7-dimensional space. 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. 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
Seidel adjacency matrix of a two-graph. Conversely, every two-graph can be represented as a set of equiangular lines. 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. The answer is expressed in spectral graph theoretic terms. Let N_\alpha(d) denote the maximum number of lines through the origin in d dimensions with common pairwise angle \arccos\alpha. Let k denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly (1-\alpha)/(2\alpha). If k is finite, then N_\alpha(d) = \lfloor k(d-1)/(k-1) \rfloor for all sufficiently large dimensions d (here the "sufficiently large" may depend on \alpha). If no k exists, then N_\alpha(d) = d+ o(d). == Equiangular lines in complex vector space ==