# Polyhedra: Truncations

Truncation is the act of cutting the corners from the faces of a shape around its vertices. When truncating a regular polyhedron, each vertex will create a new face. This new face will have the same amount of edges as the number of faces meeting at the vertex. Therefore, when truncating a tetrahedron which has three triangle meeting at each of its vertices, a new triangle is formed at the vertex.

Now, what happens when a regular polyhedron is truncated to the point where there are no edges from the original shape? In this case, the only edges of the original shape would be of those that were cut form the corners. The number of faces on the new shape would therefore be dependent on the number of vertices on the old shape. A tetrahedron, for example, has four vertices. Therefore, the new shape would have four faces. These faces, as mentioned previously, would be triangles. However, notice that the tetrahedron has four triangular faces already. Therefore, a fully truncated tetrahedron produces a new tetrahedron!

Next we will explore what happens when we truncate a cube. As stated before, the number of faces will be dependent on the number of vertices. With a cube, there are eight vertices. After we truncate the cube to the point where there are no more faces of the regular shape, we should expect to see a new polyhedron with eight faces. These new faces would have the shape of a triangle. Our new polyhedron will be an octahedron. We see a relation between these two polyhedra. The octahedron has eight faces and six vertices, and the cube has six faces and eight vertices. Both of these polyhedra when truncated completely will create one another.

The final two regular polyhedra are the dodecahedron and the icosahedron. It follows the same formula as before with relation to the vertices and faces. Once you truncate the dodecahedron completely it ends up as the icosahedron. Similar to the cube and octahedron, the icosahedron truncates fully to the dodecahedron.

This is the truncation of the dodecahedron to the icosahedron:

If you need a visual on these shapes truncating, the site http://www.vandeveen.nl/Wiskunde/Diversen/Archimedean.html can be very useful.\

With all of our regular polyhedra, we see that when you truncate them completely you will end up with another regular polyhedra. The tetrahedra truncates to another tetrahedra. A cube to an octahedron and vice versa. Finally, the dodecahedron will truncate completely into the icosahedron and, similar to the cube and octahedron, the icosahedron truncates to the dodecahedron.