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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


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