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A verf, or a vertex figure, is one interesting feature of a shape that you might have heard about. This blog post is an attempt to demystify these verfs and allow you begin to understand other things that follow from these.

So first up: what is a verf?

A verf is the shape that you get when you draw a small hypersphere around any vertex of a uniform polytope. For example, take a square. Picture a small circle around one of the sides of the square. There is an arc where the edge of the circle is inside the area of the square. Notice that as the circle moves further from the centre away from the vertex, the place where it is inside the square becomes more and more linear, until eventually it becomes completely flat.

This can also be applied to a cube, and you should find that the verf of a cube is a triangle. The verf of an octahedron is a square (try it!) and the verf of an isosahedron is pentagon.

It should be simple to demonstrate that for regular polyhedra, a shape with a schlaefi symbol of ${\displaystyle \{p, q\}}$ has a verf with a schaefi symbol of ${\displaystyle \{q\}}$.

That's all for this post. Comment with any questions or feedback! There'll be more on verfs next post.

Holomanga (talk) 22:18, October 1, 2014 (UTC)