Tuesday, December 21, 2010

Visualizing 7 dimensional simplex

One way to visualize a 7 dimensional simplex is to plot sections of it. You could restrict attention to (hyper)plane sections that go through points defined by averaging some set of vertices, and discard geometrically identical ones. There are only two such sections for 3d simplex

For 7d simplex, there are 49 such sections, and the most symmetrical one is the octahedron, which in the notebook below was found as the set of all 8 dimensional unit vectors with non-negative components orthogonal to (0, 3, 1, 0, -5, 0, 0, 1)

Here are all 49 distinct sections

Complete code is here.
An outline of how they were generated:

  • Use Peter Shor's observation that there's a homomorphism between distinct sections and ways to assign 8 identical objects to vertices of the cube
  • Find such permutations by breaking space of all permutations of 8 objects into orbits under action of group of cube rotations and taking one representative from each orbit, using PermutationGroup, GroupElements and Permute, implemented as nonEquivalentPermutations
  • Use BooleanCounting function and SatisfiabilityInstances to find assignment of vertices to centroids consistent with given cube representation, implemented as findAssignment
  • Use NullSpace, Reduce and FindInstances to find 0-dimensional intersections of hyperplane defining the section with some subset of 8 simplex cells, implemented as getExtremePoints and getCons
  • Those point sets define vertices of the section, use SingularValueDecomposition to find vertex sets with distinct sets of four non-zero singular values, implemented as singularCert and distinctPoses
  • Project distinct point sets onto hyperplane defining the section and use ComputationalGeometry`Methods`ConvexHull3D to get description of the shape in terms of polygons, implemented as getVerts3D and getPoly



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