E.A. Elsayed gave a talk about reliability at UW-Madison last week. The most interesting topic he talked about was the mathematics of folding flat paper into intricate designs (origami!). Determining whether a series of creases can be flattened is NP-complete through a reduction from the Not-All-Equal 3-SAT*

There are some basic mathematical principles that describe whether folding is possible. For example, the angles formed at a corner must sum to 360 degrees or it cannot be flattened. For example, the pyramids of Egypt cannot be flattened.

There are four mathematical rules for producing flat-foldable origami crease patterns:^{[7]}

The regions between the creases can be colored with two colors, differing across each crease. Equivalently every vertex has an even number of creases.

Maekawa’s theorem: at any vertex the number of valley and mountain folds always differ by two in either direction.

Kawasaki’s theorem: at any vertex, the sum of all the odd angles adds up to 180 degrees, as do the even.

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This made my day! <3
“It was the classes I took, the mentoring from Professor Laura Albert, the Badger Bracketolog… twitter.com/i/web/status/9…1 day ago

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