If You Split Something into Two Parts, You Will Get Three Pieces: The Bilateral Binomial Theorem and its Consequences

Abstract

If a mathematical quantity is taken apart, the binomial expansion will result in three Pascal triangles. Thus a split into two parts generates a mathematical pattern of three pieces. This binomial expansion can now be generalized into a bilateral picture, resulting in an expansion of positive and negative powers as well. In so doing the Binomial Theorem will be generalized into the Bilateral Binomial Theorem, applying the intriguing mathematics of bilateral hypergeometric functions.And if a mathematical quantity is taken apart into two anti-commuting parts, the threefold pattern will triple again. Thus a ninefold symmetry appears. As an identical ninefold symmetry will appear if complex numbers and their conjugates are multiplied, complex conjugation must be seen as a strange, brutal, and illegitimate mathematical trick to model non-commutative structures by using commuting quantities.