# Symmetric Stamp Foldings

If you’re not tired of stamp foldings yet, a type that some people seem to care about is symmetric stamp foldings. For a stack made from an even number of stamps, it’s top-to-bottom symmetry. For an odd number of stamps, it’s 180° rotational symmetry.

A stack of one stamp is trivially symmetric:

Both foldings of two stamps are symmetric:

But of the six foldings of three stamps, only two are symmetric:

And of the sixteen foldings of four stamps, only four are symmetric:

Five stamps, six symmetric foldings:

Six stamps, eight symmetric foldings:

Seven stamps, eighteen symmetric foldings:

Eight stamps, twenty symmetric foldings:

Is “foldings” even a word? Anyway, this is OEIS sequence A001010. Naturally, the counts are all even numbers (apart from the one-stamp stack, which I wish I’d never brought up) because the reverse of a symmetric folding is also symmetric.

If you kind of liked this, you’ll kind of love labeled stamp foldings, unlabeled stamp foldings, and map foldings.

See Martin Gardner, Wheels, Life and Other Mathematical Amusements, pp. 60–61, 1983.

Figures created with Mathematica 10.