Seeing the forest in the tree: applying VRML to mathematical problems in number theory

摘要

Abstract: Hamming claimed 'the purpose of computing is insight,not numbers.' In a variant of that aphorism, we showhow the Virtual Reality Modeling Language (VRML) canprovide powerful insight into the mathematicalproperties of numbers. The mathematical problem weconsider is the relatively recent conjecturecolloquially known as the '3x $PLU 1 problem'. Itrefers to an iterative integer function that also canbe though of as a digraph rooted at unity with theother numbers in any iteration sequence locate atseemingly randomized positions throughout the tree. Themathematical conjecture states that there is a uniquecycle at unity. So far, a proof for this otherwisesimple function has remained intractable. Manydifficult problems in number theory, however, have beencracked with the aid of geometrical representations.Here, we show that any arbitrary portion of the 3x $PLU1 digraph can be constructed by iterative applicationof a unique subgraph called the G-cell generator -similar in concept to a fractal geometry generator. Wedescribe the G-cell generator and present some examplesof the VRML worlds developed programmatically with it.Perhaps surprisingly, this seem to be one of the fewattempts to apply VRML to problems in number theory.!14