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- Equiangular_lines abstract "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 18 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, 48, 48, ...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 , 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 , 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 related 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 adjacency matrix of a two-graph.".
- Equiangular_lines wikiPageExternalLink Master%27s_Thesis_Gosselin.pdf.
- Equiangular_lines wikiPageExternalLink A002853.
- Equiangular_lines wikiPageID "2706034".
- Equiangular_lines wikiPageRevisionID "585844080".
- Equiangular_lines hasPhotoCollection Equiangular_lines.
- Equiangular_lines subject Category:Combinatorics.
- Equiangular_lines subject Category:Euclidean_geometry.
- Equiangular_lines comment "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.".
- Equiangular_lines label "Equiangular lines".
- Equiangular_lines sameAs m.07z5cd.
- Equiangular_lines sameAs Q5384433.
- Equiangular_lines sameAs Q5384433.
- Equiangular_lines wasDerivedFrom Equiangular_lines?oldid=585844080.
- Equiangular_lines isPrimaryTopicOf Equiangular_lines.