The twentieth century

towards the axiomatization, operationalization and algebraization of the fold

pp. 271-354

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

Abstract

The last chapter ended with the marginalization of folding in kindergartens and with the success of Row's book, positing Fröbelian folding on a new mathematical level, though almost completely ignoring, albeit unintentionally, Fröbel's influence. Hence, it is already questionable as to how 'successful" this success was, if its sources are almost completely hidden or ignored. The current chapter aims to survey the influence of Row's book in the twentieth century that prompted several attempts to discover basic folding operations (Sect. 5.1), culminating in the work of Beloch (Sect. 5.2). The ideas presented in Row's book were further developed and eventually re-conceptualized in Europe and the United States in various ways: either emphasizing its operative character, pointing to its lack of or need for an axiomatically sound basis, or focusing on the algebraic consequences. As a result, folding-based geometry now had the potential to transform into a full-blown mathematical discipline, being stronger mathematically than the well-known compass and straightedge-based geometry and its constructions. While Beloch's work in the 1930s proved the strength of this geometry, socially speaking, the focus of research geometry, however, no longer lay in the constructive aspects of two-dimensional plane geometry. In addition, whereas the axiomatizations offered by Hilbert's Euclidean geometry and later Tarski's (among others) were easily generalizable to an n-dimensional space, the same could not be said of folding-based geometry: not only were the attempts to propose an axiomatic base incomplete, they were also restricted to plane geometry. In this sense, it is essential to emphasize that, while Row's results were re-conceptualized in the various existing mathematical traditions, they never—until 1989, as we will see in Chap.  6—constituted a field of research on their own. As a result, one can see that after the 1934–1936 discoveries of Beloch, her results were completely forgotten; neither her nor Row's resulted in further discoveries concerning n-dimensional spaces.