vertex figuresPick a vertex. Join the midpoints of the two edges that meet at this vertex; this is the vertex figure. Look at the image on the right; the vertex figure of a polygon is a line. What shapes do you get for regular polygons if you join all the vertex figures together? What are the lengths of their sides in relation to the original polygon? Perhaps you could investigate less regular polygons. Can you find a formula for the length of the sides of a vertex figure for an nsided polygon? The inradius of a polygon is the radius of the circle to the midpoints of the edges. What is the inradius of any regular polygon? 

tessellations
You could do something similar to our investigations on duality. What happens in different tessellations if you join the vertex figures together? Below are a couple I tried  I chose a pair of dual tessellations  and they created the same vertex figure tessellation... I wonder if this is always true?
The tessellations created here made me think of the ones created by the activity moving tessellations.
higher dimensions...
What is the vertex figure for a 3D shape? Join the midpoints of the edges meeting at a vertex.
Try it for a cube, or if you're feeling adventurous, go for the dodecahedron or icosahedron. What do you notice?
Investigate what happens for the nonregular polyhedra.
What is the vertex figure for a 4D shape?
schlafli symbolsSchlafli symbols are a way of describing polytopes (the general word for shapes!). For example, the Schlafli symbol for a square is {4}, meaning it has 4 edges. The Schlafli symbol for a cube is {4,3}, meaning it has 3 squares at each vertex. What are the Schlafli symbols for the regular 3D polyhedra? What do you notice? What about 4D polyhedra? Schlafli symbols aren't just limited to the 'closed' polygons. Investigate the Schlafli symbols for starpolygons and try the ideas described in beachball sequences. 
I wonder what the vertex figures of star polygons look like?
