A new dimension

I am on a path to learn Blender, the open-source computer graphics software. This way, I can now start creating 3d visuals, rather than the plain old 2-dimensional imagery.

Since I have just started (and since I’m extremely busy with exams), I’ll do something really quick and small.

I recently read about permutohedrons, and one particular thing that struck me was that the Permutohedron of the groupĀ {\mathbf{S}_3}, which is the symmetric group of order 3, is a hexagon.

I’ll briefly describe what I’m talking about. A permutahedron of a symmetric group of order {n} (denoted by {\mathbf{S}_n}), which is nothing but the group of all permutations of {n} distinct items, is an {n-1} dimensional polyhedron in the space {\mathbb{R}^n} whose vertices are formed by the set {\{(\sigma(1),\sigma(2),\sigma(3)\dots \sigma(n))\}_{\sigma \in \mathbf{S}_n}}.

In other words, take the coordinate {(1,2,3\dots n) \in \mathbb{R}^n} and shuffle the numbers 1,2,3 etc. among themselves to get the vertices of the polyhedron.

Now when {n=3}, this results in a hexagon inside {\mathbb{R}^3}. This fact is quite exciting to think about, so I made a little visualization of it. Here’s the output:





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