Title: 3D turtle geometry: artwork, theory, program equivalence and symmetry
Author: Tom Verhoeff
Address: Faculty of Mathematics and Computer Science, Eindhoven University of Technology, Den Dolech 2, Eindhoven 5612 AZ, Netherlands
Abstract: We define a 3D variant of turtle graphics and present the theoretical foundations of 3D turtle geometry. This theory enables one to reason about open and closed 3D polygonal paths by means of algebraic calculations. In particular, we introduce several equivalence relations on turtle programs and theorems that define corresponding standard forms. Also we express the relationship between the symmetries of a 3D polygonal path and the symmetries of a generating turtle program in a suitable standard form. Finally, we discuss software tool support for 3D turtle geometry. Along the way, we present some artworks designed through 3D turtle graphics. These artworks have never been described in the literature before.
Keywords: 3D turtle geometry; turtle programs; program equivalence; symmetry; mathematical art; turtle graphics; 3D polygonal paths; algebraic calculations.
Int. J. of Arts and Technology, 2010 Vol.3, No.2/3, pp.288 - 319
Available online: 07 Apr 2010