Coda

1989—the axiomatization(s) of the fold

Martin Middeke

Christoph Reinfandt

pp. 355-375

in: Michael Friedman, A history of folding in mathematics, Basel, Birkhäuser, 2018

Abstract

The final chapter is, of course, not a final conclusion concerning the "fruition" of the mathematics of paper folding: this domain is in a process of continuous development, and its interweaving with computer sciences and computer modeling becomes more and more apparent and prominent. This, of course, raises the question as to how materiality, which earlier functioned partially as a hindering element, is considered; one may wonder, for example, as to the epistemological implications and differences between the twenty-first century mathematical computer models and the nineteenth century mathematical material folded models, but this discussion is beyond the scope of this chapter. What this chapter will attempt to describe is that which can be called the first step of this new, modern movement concerning mathematics and folding-based geometry: the complete axiomatization of this geometry. Though not the single modern point of view with which the mathematics of paper folding has been considered, the attempt to find axioms or basic operations for this geometry, started at the beginning of the twentieth century, bore fruit only towards the end of the 1980s. If Beloch's interest lay in proving that folding-based geometry is at least as powerful as straightedge and compass geometry, from an algebraic point of view as well as construction-wise, during the mid-1980s, one was interested in finding the basic, fundamental operations from which a folded figure was composed, and by this means, giving this geometry a logical sound basis, the same that straightedge and compass geometry had for centuries.