Seeing the Forest in the Tree: Applying VRML to Mathematical Problems in Number Theory

Hamming claimed Òthe purpose of computing is insight, not numbers.Ó In a variant of that aphorism, we show how the Virtual Reality Modeling Language (VRML) can provide powerful insight into the mathematical properties of numbers. The mathematical problem we consider is the relatively recent conjecture colloquially known as the "3x + 1 problem". It refers to an iterative integer function that also can be thought of as a digraph (or tree) rooted at unity with the other numbers in any iteration sequence located at seemingly randomized positions throughout the tree. The mathematical conjecture states that there is a unique cycle at unity. So far, a proof for this otherwise simple function has remained intractable. Many difficult problems in number theory, however, have been cracked with the aid of (often just 2-dimensional) geometrical representations. Here, we show that any arbitrary portion of the 3x+1 digraph can be constructed by iterative application of a unique subgraph called the G-cell generatorÐÐsimilar in concept to a fractal geometry generator. We describe the G-cell generator and present some examples of the VRML worlds developed programmatically with it. Perhaps surprisingly, this seems to be one of the few attempts to apply VRML to problems in number theory.