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Re: getting all interesting sections of 7-d simplex
*To*: mathgroup at smc.vnet.net
*Subject*: [mg115118] Re: getting all interesting sections of 7-d simplex
*From*: Yaroslav Bulatov <yaroslavvb at gmail.com>
*Date*: Fri, 31 Dec 2010 05:18:55 -0500 (EST)
Just wanted to note that I finally solved this relying on some
feedback from mathgroup users and group theory functionality in
version 8
http://mathematica-bits.blogspot.com/2010/12/listing-interesting-sections-of-simplex.html
On Sep 30, 12:50 am, Yaroslav Bulatov <yarosla... at gmail.com> wrote:
> On Sep 21, 10:57 pm, Yaroslav Bulatov <yarosla... at gmail.com> wrote:
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> > On Sep 20, 11:03 pm, YaroslavBulatov<yarosla... at gmail.com> wrote:
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> > > I'm trying tovisualizeinteresting 3d sections of a 7d regular
> > >simplex. An interesting section is a 3d space that goes through
> > >simplexcentroid and 3 other points, each of which is a centroid of
> > > some non-empty set ofsimplexvertices.
>
> > > For instance, {{1},{1,2},{3}} defines a section that goes through
> > >simplexcenter 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-dsimplex. It
> > > uses the fact that Hadamard matrix gives a mapping between points of a
> > > regularsimplexin 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 thesimplexgive polytopes equivalent under rigid
> > transformations. Final goal is tovisualizeentropy of distributions
> > over 8 outcomes. Motivated by visualizing entropy of distributions
> > over 4 outcomes which can be done by taking 2d sections of 3dsimplex
> > like this --http://yaroslavvb.com/upload/simplex-sections3.png(only
> > need 2 sections tovisualizeit)
>
> 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 themhttp://mathoverflow.net/questions/39429/how-many-non-equivalent-secti...
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