Calculating fractal sets in n ≥ 1 embedding dimensions without the use of truncation

The theme of this research is an examination of the traditional multiplication operator, in the case of fractal sets in $n \geq 1$ embedding dimensions. After this examination, an alternative, new multiplication operator is introduced. The traditional multiplication operator has a time complexity of $O(n^2)$, whereas the new multiplication operator has a time complexity of $O(n)$. Taking amortized costs into account, it is found that the new multiplication operator is more time-efficient where $n \geq 32$, using an optimizing C++ compiler. There is no problem, other than concerns about time complexity, with the traditional multiplication operator. It was hypothesized that there would be differences between the two multiplication operators. These differences, and some similarities, between the two multiplication operators are visualized in terms of fractal sets, via OpenGL. The C++ and Python code is available upon request.