# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 11: Self-Dual Polytopes

The dual structure of an $n$-dimensional  polytope is the result of exchanging each $(k-1)$-face for an $(n - k)$-face for $k = 1 \dots n-1$, while maintaining the original connectivity between these. If this resulting structure is similar to the original polytope, the polytope is said to be self-dual.

Examples of self-dual polytopes include all regular 2-polytopes (regular polygons), and all regular simplices (equilateral triangles, regular tetrahedrons, etc.).