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Suppose we have a regular 7-simplex in $\mathbb{R}^8$ defined by vertices <1,0,0,...,0>, <0,1,0,..,0>,...,<0,...,0,1>. A section is a 3-dimensional linear subspace of $\mathbb{R}^8$ that contains simplex centroid and three other points, each of which is a centroid of a non-empty set of simplex vertices. Two sections are equivalent if they are identical spaces under permutation of coordinates. In other words, when some permutation of coordinates is the bijection between two spaces. How many non-equivalent sections are there? Is there an efficient way to enumerate them?

Motivation: visualizing symmetric priors over distributions over 8 outcomes

Update 12/09 I tried an automatic search and got 49 sections, same as Peter Shor below. Here they are. Note that grouping is a bit different since I group sections with or without unexpected centroids together.

No empty vertices:

One empty vertex:

Two empty vertices:

Three empty vertices:

Four empty vertices:

old stuff Here's an illustration of solving this problem for 2-sections of a 3-simplex in 4 dimensions. There seems to be only 2 non-equivalent 2-sections (triangle and square). This solves the problem of visualizing entropy (contour lines) of distributions over 4 outcomes, and I'd like extend it to 8 outcomes.

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Here's example of simplex visualized in a random sectionhadamard = KroneckerProduct @@ Table[{{1, 1}, {1, -1}}, {3}];invHad = Inverse[hadamard];vs = Range[8];m = mm /@ vs;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

Update 12/09I tried an automatic search and got 49 sections, p3 - p0}]];RegionPlot3D @@ {And @@ Thread[invHad.m >= 0 /same as Peter Shor below. section], {x, -3, 3}, {y, -3, 3}, {z, -3,3}}Here they are. Note that grouping is a bit different since I group sections with or without unexpected centroids together.

No empty vertices:

One empty vertex:

Two empty vertices:

Three empty vertices:

Four empty vertices:

old stuff

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Suppose we have a regular 7-simplex in $\mathbb{R}^8$ defined by vertices <1,0,0,...,0>, <0,1,0,..,0>,...,<0,...,0,1>. A section is a 3-dimensional linear subspace of $\mathbb{R}^8$ that contains simplex centroid and three other points, each of which is a centroid of a non-empty set of simplex vertices. Two sections are equivalent if they are identical spaces under permutation of coordinates. In other words, when some permutation of coordinates is the bijection between two spaces. How many non-equivalent sections are there? How many of them have dimension 3? Is there an efficient way to enumerate them?

Motivation: visualizing symmetric priors over distributions over 8 outcomes

Here's example of simplex visualized in a random section

hadamard = KroneckerProduct @@ Table[{{1, 1}, {1, -1}}, {3}];
invHad = Inverse[hadamard];vs = Range[8];m = mm /@ vs;
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}}

Here's an illustration of solving this problem for 2-sections of a 3-simplex in 4 dimensions. There seems to be 4 non-equivalent sections, out of which only 2 are interesting non-equivalent 2-sections (ie, section has dimension 2)triangle and square). This solves the problem of visualizing entropy (contour lines) of distributions over 4 outcomes, and I'd like extend it to 8 outcomes.

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