getting all interesting sections of 7-d simplex

• To: mathgroup at smc.vnet.net
• Subject: [mg112579] getting all interesting sections of 7-d simplex
• From: Yaroslav Bulatov <yaroslavvb at gmail.com>
• Date: Tue, 21 Sep 2010 02:03:45 -0400 (EDT)

```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}}

```

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