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| In [[geometry]], a set of [[line (geometry)|line]]s in [[Euclidean space]] is called '''equiangular''' if every pair of lines makes the same angle.
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| 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 18 is listed in The On-Line Encyclopedia of Integer Sequences as follows:
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| 1, 3, 6, 6, 10, 16, 28, 28, 28, 28, 28, 28, 28, 28, 36, 40, 48, 48, ...
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| 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 <math>\mathbb{R}^8</math>, 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 <math> \mathbb{R}^8</math>, 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 [[3 21 polytope|3<sub>21</sub> polytope]]. In other words, they are the weight vectors of the 56-dimensional representation of the Lie group [[E7 (mathematics)|E<sub>7</sub>]].
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| 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 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 (mathematics)|origin]] of coordinates. Choose one unit vector in each line. Form the [[matrix (mathematics)|matrix]] ''M'' of [[inner product]]s. 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.
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| ==References==
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| * The On-Line Encyclopedia of Integer Sequences, [http://oeis.org/A002853 Maximal size of a set of equiangular lines in n dimensions], sequence number A002853.
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| * Gosselin, S., [http://ion.uwinnipeg.ca/~sgosseli/Master%27s_Thesis_Gosselin.pdf Regular two-graphs and equiangular lines], Master's thesis, Mathematics Department, University of Waterloo, 2004.
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| * van Lint, J. H., and Seidel, J. J. Equilateral point sets in elliptic geometry. ''Proc. Koninkl. Ned. Akad. Wetenschap. Ser. A'' '''69''' (= ''Indagationes Mathematicae'' '''28''') (1966), 335-348.
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| * [[Andries Brouwer|Brouwer, A.E.]], Cohen, A.M., and Neumaier, A. ''Distance-Regular Graphs.'' Springer-Verlag, Berlin, 1989. Section 3.8.
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| * {{citation|first1=Chris|last1=Godsil|authorlink1=Chris Godsil|first2=Gordon|last2=Royle|authorlink2=Gordon Royle|title=Algebraic Graph Theory|series=Graduate Texts in Mathematics|volume=207|publisher=Springer-Verlag|location=New York|year=2001}}. ''(See Chapter 11.)''
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| [[Category:Euclidean geometry]]
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| [[Category:Combinatorics]]
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