# 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 group ${\mathbf{S}_3}$, 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: