# Octahedron unfoldings

## Puzzle

It is well know that cube has 11 different unfoldings. How many different unfoldings does octahedron have?

## History

At about 1992 I wrote a Pascal program that searched for different unfoldings of polycubes. To test it I fed it with monocube (=cube) and it came up with 11 -- well known result.

I tried to modify my program so it would search for unfoldings of another platonic solid. Tetrahedron was too trivial, Icosahedron was beyond possibilities of that program (executed on Z80 microprocessor), so octahedron was a natural choice. To my utter surprise the program came out with the same number -- 11.

## Solution

Indeed, since cube is dual to octahedron, each unfolding of a cube corresponds to a unique unfolding of octahedron. Edges of solids correspond to each other and cut edge of a cube unfolding corresponds to uncut edge of octahedron unfolding and vice versa. Same is true for any pair of dual polyhedra (with properly defined notion of unfolding, to allow self-intersection).

http://mathworld.wolfram.com/Net.html mentions this fact. It also lists 43380 as number of unfoldings for icosahedron.

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