6.5310: Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Spring 2025)

Prof. Erik Demaine TAs Josh Brunner, Jenny Diomidova

[+] Polyhedron refolding: Dissection-like open problem, regular tetrahedron to box, Platonic solids to tetrahedra, box to box, polycubes, orthogonal unfoldings with nonorthogonal foldings. Smooth polyhedron folding: Smooth Alexandrov, D-forms, ribbon curves. Smooth polyhedron unfolding: Smooth prismatoids. Smooth origami: wrapping smooth surfaces with flat paper, Mozartkugel, contractive mapping, Burago-Zalgaller Theorem (crinkling/crumpling), stretched path, stretched wrapping, source wrapping, strip wrapping, petal wrapping, comb wrapping, Pareto curve.
This lecture is a big collection of fun results related to unfolding, refolding, and smooth folding: common unfolding of a regular tetrahedron and near-cube common unfoldings of Platonic solids and near-regular tetrahedra common unfoldings of boxes of different sizes common unfoldings of many polycubes orthogonal unfoldings with nonorthgonal foldings smooth Alexandrov's theorem, applied to smooth convex shapes (D-forms) unfolding smooth convex polyhedra (prismatoids) wrapping (paper folding) smooth surfaces like spheres, using a new definition of origami, where distances can shrink instead of necessarily staying the same The last section has practical applications to computational confectionery, reducing the material usage in wrappings of spherical chocolates such as Mozartkugel. Yum!