Re: getting all interesting sections of 7-d simplex

*To*: mathgroup at smc.vnet.net*Subject*: [mg112767] Re: getting all interesting sections of 7-d simplex*From*: Yaroslav Bulatov <yaroslavvb at gmail.com>*Date*: Thu, 30 Sep 2010 04:49:19 -0400 (EDT)*References*: <i79hrv$of5$1@smc.vnet.net> <i7c5t6$5fe$1@smc.vnet.net>

On Sep 21, 10:57 pm, Yaroslav Bulatov <yarosla... at gmail.com> wrote: > On Sep 20, 11:03 pm, YaroslavBulatov<yarosla... at gmail.com> wrote: > > > > > > > I'm trying to visualize interesting 3d sections of a 7d regular > > simplex. An interesting section is a 3d space that goes through > > simplex centroid and 3 other points, each of which is a centroid of > > some non-empty set of simplex vertices. > > > For instance, {{1},{1,2},{3}} defines a section that goes through > > simplex center vertex 1, vertex 3 and centroid of vertices 1,2. Two > > sections are equivalent if they define the same space under some > > permutation of coordinates. There is a lot of sections, but seems to > > be a much smaller number of equivalence classes. I tried enumerating > > them by checking all permutations of 7 coordinates, but this is is > > quite slow, can anyone can see a practical way to do this in > > Mathematica? > > > Code below plots random interesting section of the 7-d simplex. It > > uses the fact that Hadamard matrix gives a mapping between points of a > > regular simplex in 7 dimensions and probability distributions over 8 > > outcomes. > > > hadamard = KroneckerProduct @@ Table[{{1, 1}, {1, -1}}, {3}]; > > invHad = Inverse[hadamard]; > > vs = Range[8]; > > m = mm /@ Range[8]; > > sectionAnchors = Subsets[vs, {1, 7}]; > > randomSection := > > Mean[hadamard[[#]] & /@ #] & /@ > > Prepend[RandomChoice[sectionAnchors, 3], vs]; {p0, p1, p2, > > p3} = randomSection; > > section = > > Thread[m -> > > p0 + {x, y, > > z}.Orthogonalize[{p1 - p0, p2 - p0, > > p3 - p0}]]; RegionPlot3D @@ {And @@ > > Thread[invHad.m > 0 /. section], {x, -3, 3}, {y, -3, 3}, {z, -3, > > 3}} > > Correction: I'd like to get a representative set of interesting > sections (hopefully all) where intersections of sections from the set > with the simplex give polytopes equivalent under rigid > transformations. Final goal is to visualize entropy of distributions > over 8 outcomes. Motivated by visualizing entropy of distributions > over 4 outcomes which can be done by taking 2d sections of 3d simplex > like this --http://yaroslavvb.com/upload/simplex-sections3.png(only > need 2 sections to visualize it) Still not sure about the best way to use Mathematica for this, but Peter Shor enumerated non-equivalent sections by hand, and it looks like there are 49 of them http://mathoverflow.net/questions/39429/how-many-non-equivalent-sections-of-a-regular-7-simplex