How to Fold It: The Mathematics of Linkages, Origami, and Polyhedra (2011, Cambridge University Press) by Joseph O'Rourke is a technical book that's also highly readable. There's deep math in it, but most of the discussion only requires high-school algebra and geometry to follow. Among the fascinating questions that O'Rourke addresses are:

• Linkages (idealized rigid bars connected at rotational joints)
  • what places can a multi-jointed "robot arm" (linear linkage) reach?
  • what angles do the joints of a robot arm have to be set to, in order to reach a chosen point?
  • how can a (non-linear, cross-connected) linkage be designed to change circular motion to straight-line motion, or to other shapes, or to scale one shape up or down?
  • how do angle constraints on a linkage (e.g., in an idealized protein molecule) affect the shape(s) the linkage can take?
  • how do "pop-up books" based on linkages work?
• Origami (idealized paper-folding)
  • what are the constraints required for folds to produce a flat configuration (like a folded map)?
  • what shapes can be made by folding a sheet of paper and then making a single straight-line cut?
  • what "rigid origami" shapes can be created when the faces between folds are not allowed to bend?
• Polyhedra (solid shapes with flat surfaces)
  • what polyhedra can be unfolded – to make flat (non-overlapping, one-piece) shapes – by cutting along some of their edges?
  • what flat shapes can be folded into what polyhedra?

Delightful discussions, grounded in physical reality and reminiscent of Z. A. Melzak's Companion to Concrete Mathematics.

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