# Transforming shapes (keep it short!)

Following the same logic as the first three rows, what shape should replace the question mark?

The question mark should be

an octahedron

because

each shape is the dual of the previous shape.

Hey, you said keep it short!

Okay, fine, here's a more detailed explanation:

Each shape is being converted to its dual, in which each vertex is converted to an edge and each edge is being converted to a vertex. Perhaps an easier way to think about it is that the vertices are being trimmed off as much as possible.

For the first three examples:

The triangle has each of its vertices trimmed down to the middle of each edge, leaving just one point on each edge and three brand new edges connecting them. The resulting shape is also an equilateral triangle, as regular polygons are self-dual. The same happens with the square. The circle, of course, has no corners, so it stays the same.

Finally, for the cube,

each vertex is instead converted into a face, and vice versa; the corners are cut off down to the middle of each edge, resulting in a point in the middle of each face. The resulting shape is an octahedron, as you can see here (image from Wikipedia):

• How did you draw the picture?
– Simd
Commented Jun 23, 2023 at 14:09
• Well done! I didn't even know it had a name, let alone was called rot13(n qhny). The "keep it short" referred to the fact that rot13(vs lbh fgneg sebz rnpu funcrf pragre naq pubbfr gur fubegrfg qvfgnaprf lbhyy unir fbzr cbvagf jurerol lbh pbaarppg gur nqwnprag cbvagf.) Commented Jun 23, 2023 at 14:37
• @Simd I didn't, I stole it from wikipedia. I'll add a link to the page I found it on Commented Jun 23, 2023 at 15:12
• @Prim3numbah that makes sense. I couldn't quite figure out that part of the title but I think I turned it into a pretty good joke lol. if I didn't already know the name of this property (shoutout jan misali) I might've connected the dots ;) Commented Jun 23, 2023 at 15:14