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Re: getting all interesting sections of 7-d simplex
*To*: mathgroup at smc.vnet.net
*Subject*: [mg112599] Re: getting all interesting sections of 7-d simplex
*From*: Yaroslav Bulatov <yaroslavvb at gmail.com>
*Date*: Wed, 22 Sep 2010 01:57:39 -0400 (EDT)
*References*: <i79hrv$of5$1@smc.vnet.net>
On Sep 20, 11:03 pm, Yaroslav Bulatov <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)
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