ILLUSTRATING MATHEMATICS USING 3D PRINTERS

Abstract. 3D printing technology can help to visualize proofs in mathematics. In this document we aim to illustrate how 3D printing can help to visualize concepts and mathematical proofs. As already known to educators in ancient Greece, models allow to bring mathematics closer to the public. The new 3D printing technology makes the realization of such tools more accessible than ever. This is an updated version of a paper included in [62]. 1. Visualization Visualization has always been an important ingredient for communicating mathematics [80]. Figures and models have helped to express ideas even before formal mathematical language was able to describe the structures. Numbers have been recorded as marks on bones, represented with pebbles, then painted onto stone, inscribed into clay, woven into talking knots, written onto papyrus or paper, then printed on paper or displayed on computer screens. While figures extend language and pictures allow to visualize concepts, realizing objects in space has kept its value. Already in ancient Greece, wooden models of Apollonian cones were used to teach conic sections. Early research in mathematics was often visual: figures on Babylonian Clay tablets illustrate Pythagorean triples, the Moscow mathematical papyrus features a picture which helps to derive the volume formula for a frustum. Al-Khwarizmi drew figures to solve the quadratic equation. Visualization is not only illustrative, educational or heuristic, it has practical value: Pythagorean triangles realized by ropes helped measuring and dividing up of land in Babylonia