Equiangular lines

In geometry, a set of lines in Euclidean space is called equiangular if every pair of lines makes the same angle.

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. The maximum in dimensions 1 through 13 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, ... (sequence A002853 in 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 the components of this. The dot product of two of these vectors is 8 if both have a component 3 in the same place or -8 otherwise. Thus, the lines through the origin containing these vectors are equiangular. Moreoever, 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 dilation, 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.^[1]

Notes

Reference J. J. Seidel `Discrete non-Euclidean geomtery' In Buekenhout (ed.), Handbook of Incidence Geometry, Elsevier, Amsterdam, The Nederlands (1995) claims, without proof, that the maximum number of equiangular lines in dimension 14 is 28. This is NOT known.

1. van Lint & Seidel 1966

References

• The On-Line Encyclopedia of Integer Sequences, Maximal size of a set of equiangular lines in n dimensions, sequence number A002853.
• Brouwer, A.E., Cohen, A.M., and Neumaier, A. Distance-Regular Graphs. Springer-Verlag, Berlin, 1989. Section 3.8.
• Godsil, Chris; Royle, Gordon (2001), Algebraic Graph Theory, Graduate Texts in Mathematics 207, New York: Springer-Verlag. (See Chapter 11.)
• Gosselin, S., Regular two-graphs and equiangular lines, Master's thesis, Mathematics Department, University of Waterloo, 2004.
• van Lint, J. H.; Seidel, J. J. (1966), "Equilateral point sets in elliptic geometry", Indagationes Mathematicae, Proc. Koninkl. Ned. Akad. Wetenschap. Ser. A 69 28: 335–348
• G. Greaves, J. H. Koolen, A. Munemasa, and F. Szollosi: Equiangular lines in Euclidean spaces, JCTA (2015), preprint.
• A. Barg and W.-H. Yu : New bounds for equiangular lines, in: Discrete Geometry and Algebraic Combinatorics, A. Barg and O. Musin, Editors, AMS Series: Contemporary Mathematics, vol. 625, 2014, pp.111--121, preprint.