# Truncation

Wed Jul 01 2015

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.

We start with a polygon — let's say a triangle. This triangle has 3 vertices. In order to truncate we need to split each vertex into two. Now the triangle is made up of 6 vertices. Or rather 3 pairs of vertices. Each pair occupies the same position as the original vertex it is split from.

The vertices are used to define the sides of a triangle. Therefore, each vertex has a direct relation to a side. The process of truncation involves moving each vertex along it's line towards the mid-point of the line. Fig 1.0: Types of truncation on a square (src)

Once all the vertices reach the mid-points this is known as Complete Truncation. If we keep moving beyond the mid-point then it is known as Hypertruncation. As we execute Hypertruncation there is a point where the vertex reaches the starting location of the vertex directly opposite to itself. If we keep going beyond that location then we are executing Quasitruncation. Lastly, we can also go away from the mid-points. This is known as Antitruncation.