|
13
|
|
edited Dec 22 2010 at 19:14 |
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.
|
|
|
|
|
12
|
|
edited Dec 22 2010 at 4:52 |
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
|
|
|
|
|
11
|
|
edited Sep 22 2010 at 4:33 |
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.
|
|
|
|
10
|
|
edited Sep 22 2010 at 4:01 |
How many non-equivalent sections of 7-d a regular simplex7-simplex?
|
|
|
|
|
9
|
|
edited Sep 22 2010 at 3:53 |
Suppose we have a regular 7-simplex in 7 dimensions centered at $\mathbf{0}$. \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 defined by of $\mathbf{0}$ \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 their intersections with the simplex forms polytopes equivalent they are identical spaces under rigid transformationspermutation 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 2d sections 2-sections of a 3d simplex3-simplex in 4 dimensions. There seems to be 4 non-equivalent sections, out of which only 2 are interesting (ie, section is not degenerate)has dimension 2). This solves the problem of visualizing entropy (contour lines) of distributions over 4 outcomes, and I'd like extend it to 8 outcomes.
|
|
|
|
|
8
|
|
edited Sep 21 2010 at 4:51 |
Suppose we have a regular simplex in 7 dimensions centered at $\mathbf{0}$. A section is a 3-dimensional subspace defined by $\mathbf{0}$ and three other points, each of which is a centroid of a non-empty set of simplex vertices. Two sections are equivalent if their intersections with the simplex forms polytopes that are equivalent under permutation of verticesrigid transformations. How many non-equivalent sections are there? Is there an efficient way to enumerate them?
Motivation: visualizing symmetric priors over distributions over 8 outcomes
Here's example of 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 2d sections of a 3d simplex. There seems to be 4 non-equivalent sections, out of which only 2 are interesting (ie, section is not degenerate). This solves the problem of visualizing entropy (contour lines) of distributions over 4 outcomes, and I'd like extend it to 8 outcomes.
|
|
|
|
|
7
|
|
edited Sep 21 2010 at 4:44 |
Suppose we have a regular simplex in 7 dimensions centered at $\mathbf{0}$. A section is a 3-dimensional subspace defined by $\mathbf{0}$ and three other points, each of which is a centroid of a non-empty set of simplex vertices. Two sections are equivalent if they their intersections with the simplex forms polytopes that are sets of points equal equivalent under some permutation of coordinatesvertices. How many non-equivalent sections are there? Is there an efficient way to enumerate them?
Motivation: visualizing symmetric priors over distributions over 8 outcomes
Here's example of 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 2d sections of a 3d simplex. There seems to be 4 non-equivalent sections, out of which only 2 are interesting (ie, section is not degenerate). This solves the problem of visualizing entropy (contour lines) of distributions over 4 outcomes, and I'd like extend it to 8 outcomes.
|
|
|
|
|
6
|
|
edited Sep 20 2010 at 21:49 |
Suppose we have a regular simplex in 7 dimensions . centered at $\mathbf{0}$. A section is a 3-dimensional subspace defined by simplex centroid $\mathbf{0}$ 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 sets of points equal under some permutation of coordinates. How many non-equivalent sections are there? Is there an efficient way to enumerate them?
Motivation: visualizing symmetric priors over distributions over 8 outcomes
Here's example of 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 2d sections of a 3d simplex. There seems to be 4 non-equivalent sections, out of which only 2 are interesting (ie, section is not degenerate). This solves the problem of visualizing entropy (contour lines) of distributions over 4 outcomes, and I'd like extend it to 8 outcomes.
|
|
|
|
|
5
|
|
edited Sep 20 2010 at 21:40 |
Suppose we have a regular simplex in 7 dimensions. A section is a 3-dimensional subspace defined by 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 define the same space are sets of points equal under some permutation of coordinates. How many non-equivalent sections are there? Is there an efficient way to enumerate them?
Motivation: visualizing symmetric priors over distributions over 8 outcomes
Here's example of 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 2d sections of a 3d simplex. There seems to be 4 non-equivalent sections, out of which only 2 are interesting (ie, section is not degenerate). This solves the problem of visualizing entropy (contour lines) of distributions over 4 outcomes, and I'd like extend it to 8 outcomes.
|
|
|
|
|
4
|
|
edited Sep 20 2010 at 21:16 |
Suppose we have a regular simplex in 7 dimensions. A section is a 3-dimensional subspace defined by 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 define the same space under permutation of coordinates. How many non-equivalent sections are there? Is there an efficient way to enumerate them?
Motivation: visualizing symmetric priors over distributions over 8 outcomes
Here's example of 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 2d sections of a 3d simplex, where section contour lines correspond to entropy. There seems to be 4 non-equivalent sections, out of which only 2 are interesting (ie, section is not degeneratedegenerate). This solves the problem of visualizing entropy (contour lines) of distributions over 4 outcomes, and I'd like extend it to 8 outcomes.
|
|
|
|
|
3
|
|
edited Sep 20 2010 at 21:08 |
Suppose we have a regular simplex in 7 dimensions. A section is a 3-dimensional subspace defined by 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 define the same space under permutation of coordinates. How many non-equivalent sections are there? Is there an efficient way to enumerate them?
Motivation: visualizing symmetric priors over distributions over 8 outcomes
Here's example of 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 2d sections of a 3d simplex, where density section contour lines correspond to be visualized is entropy. There seems to be 4 non-equivalent sections, out of which only 2 are interesting (ie, section is 2 dimensionalnot degenerate)
|
|
|
|
|
2
|
|
edited Sep 20 2010 at 21:01 |
Suppose we have a regular simplex in 7 dimensions. A section is a 3-dimensional subspace defined by 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 define the same space under permutation of coordinates. How many non-equivalent sections are there? Is there an efficient way to enumerate them?
Motivation: visualizing symmetric priors over distributions over 8 outcomes
Here's example of 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 2d sections of a 3d simplex, where density to be visualized is entropy. There seems to be 4 non-equivalent sections, out of which only 2 are interesting (section is 2 dimensional)
|
|
|
|
|
1
|
|
asked Sep 20 2010 at 20:17 |
How many non-equivalent sections of 7-d regular simplex?
Suppose we have a regular simplex in 7 dimensions. A section is a 3-dimensional subspace defined by 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 define the same space under permutation of coordinates. How many non-equivalent sections are there? Is there an efficient way to enumerate them?
Motivation: visualizing symmetric priors over distributions over 8 outcomes
Here's example of 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}}
|
|
|
