Simplices are n-dimensional analogues of triangles. Their omnitruncated siblings have all n-2 facets truncated. Cut vertices off triangles to form hexagons, vertices & edges off tetrahedra for a truncated octahedra. These are also called permutohedra: https://en.wikipedia.org/wiki/Permutohedron … 2/17
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In the case of the hexagon, consider a cube from [1,1,1] to [3,3,3] intersecting a plane through its center perpendicular to opposing corners. The vertices of the resulting hex can be thought of as occupying all 3! (6) permutations of the coordinate set {1,2,3} in 3-space. 3/17pic.twitter.com/ef9IT5VMeo
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These permutations constitute the symmetric group S₃, isomorphic to the symmetry group of an equilateral triangle. (Neither here nor there but we can think of an edge between 2 hex vertices as the equivalent of flipping & rotating a triangle 120°.) https://en.wikipedia.org/wiki/Symmetric_group … 4/17
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What's interesting of course is that vertices of the truncated octahedron can be mapped to every permutation of {1,2,3,4} in 4-space. And the omnitruncated 5-cell (or 4-simplex) permutes {1,2,3,4,5}, &c. We get a shape with n! vertices embedded in every n-dimensional space. 5/17pic.twitter.com/dTC9VJB5AX
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Every vertex is connected to n-1 neighbors, where each neighbor has 2 adjacent coordinate numbers flipped — e.g. [1,2,3] and [2,1,3]. Thus, we have n!(n-1)/2 total edges, and each edge is √2 long (starting w/ the order-2 permutohedron, a diagonal line from [1,2] to [2,1]). 6/17pic.twitter.com/nvhnaxltgd
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Since the permutation of any three adjacent coordinates forms a hexagon, all higher-dimensional permutohedra have hexagonal faces. E.g. we can permute 4, 5, and 6 in the coordinate [4, 3, 2, 1, 5, 6] to form a hexagon embedded in the 1st, 5th, and 6th dimension. 7/17
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We also have square faces in higher-dimensional permutohedra (as in the truncated octahedron), since permuting 2 separate pairs of coordinates gives us 2 pairs of lines at 90° angles. Other edge-paths, as we see in the truncated octahedron, don't lie in planes as polygons. 8/17
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All permutohedra are embedded in n-1 dimensional hyperplanes where every point sums to n(n-1)/2 — the sum of permuted coordinates. In the case of the hexagon, 6. We can find translation vectors, each summing to 0, that allow us to move the hexagon anywhere in this plane. 9/17
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E.g., [1,-1,0], [0,1,-1], [-1,0,1]. Since each of these vectors represents a neighboring coordinate flip in our original hexagon, we can imagine each of them, and their inverses, moving the center of a hexagon to one of its vertices. 10/17pic.twitter.com/h5PsiEaoA9
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We can then imagine moving moving the hex to another spot in a uniform tiling, e.g. by moving [1,-1,0] + [0,-1,1] = [1,-2,1], which gives us the coordinates {[2,0,4], [3,-1,4], [4,-1,3], [4,0,2], [3,1,2], [2,1,3]}. The last two are shared with the original hex's position. 11/17pic.twitter.com/ZwnYpdVTq5
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We moved principally in the -y direction, but also a bit in x and z. In this way we can tessellate 2-space along 3 axes, using these whole-number coordinates summing to zero. A similar technique is described by
@redblobgames in his hex grid article: https://www.redblobgames.com/grids/hexagons/ 12/17Prikaži ovu nit -
This works with our truncated octahedron as well. E.g., add [1,1,1,-3] to get shared coordinates [2,3,4,1], [3,2,4,1], [3,4,2,1], [4,3,2,1], [4,2,3,1], [2,4,3,1] along a hexagonal face. Or add [2,2,-2,-2] to share a square face: [4,3,1,2], [4,3,2,1], [3,4,2,1], [3,4,1,2]. 13/17
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For an example of a truncated octahedral or if you will bitruncated cubic tiling, please see this interesting model I printed last year, available on Thingiverse: https://www.thingiverse.com/thing:3971087 14/17pic.twitter.com/D8tVBPeAFa
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With the hex tiling, we have an equilateral triangular "vertex figure" (the shape resulting from cutting out a corner of the tiling). Higher-dimensional permutohedra also have simplectic vertex figures, but they are irregular, meeting both square and hexagonal faces. 15/17
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Thus, we see that the
#hexagon is truly unique among this amazing class of shapes — it is both the only regular permutohedron, and the only one whose tiling gives a regular vertex figure. It is also the 2-facet backbone upon which all higher permutohedra are based. 16/17Prikaži ovu nit -
Beyond being amazing, permutohedra have many potential applications in fields like AI and image processing, and will no doubt find more uses going forward. They're an excellent example of why raising
#hexagonalawareness is so critical for our civilization at this moment. 17/17Prikaži ovu nit
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