# Equiangular lines

In geometry, a set of lines is called equiangular if all the lines intersect at a single point, and every pair of lines makes the same angle.

## Equiangular lines in Euclidean space

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 ((n+1)/2){\textstyle {\binom {n+1}{2}}}. This upper bound is tight up to a constant factor to a construction by de Caen. 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, ... (sequence A002853 in the OEIS).
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 3$_{21}$ polytope. In other words, they are the weight vectors of the 56-dimensional representation of the Lie group E$_7$.
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

In a complex vector space equipped with an inner product, we can define the angle between unit vectors ${\hat {a}}$ and ${\hat {b}}$ by the relation $\cos(\theta )=|({\hat {a}},{\hat {b}})|$. It is known that an upper bound for the number of complex equiangular lines in any dimension $n$ is $n^{2}$. Unlike the real case described above, it is possible that this bound is attained in every dimension $n$. The conjecture that this holds true was proposed by Zauner and verified analytically or numerically up to $n=67$ by Scott and Grassl. A maximal set of complex equiangular lines is also known as a SIC or SIC-POVM.