Through the Interesting Esoterica postings on Mathstodon I learned of this neat post. Joseph O’Rourke published this year Pop-Up Geometry: The Mathematics Behind Pop-Up Cards. I haven’t got the book (yet), but O’Rourke has a page with animated GIFs showing how basic shapes work. The animations, even without narrative, are eye-opening, revealing how to make complicated and curved motions with a single rotating plane and fixed-length attachments. It isn’t properly origami but the subject is related.
Interesting Esoterica has an abstract to this entry here. An advantage to searching there is the archive of interesting topics, searchable by tags. These sprawl considerably over difficulty range: Under the tag ‘things to make and do’ are this piece on Pop-up Geometry, but also on group theory as it applies to laying model train tracks, or a 1959 essay describing how to build a computer out of paper. Or, if you’re looking for a more advanced project, Fibbinary Zippers in a Monoid of Toroidal Hamiltonian Cycles that Generate Hilbert-Style Square-Filling Curves. (This one is closer to the train tracks paper than you imagine, and you can follow its point from looking at the pictures.) You’re likely to find something delightful there.
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