Alec Macher joined Wolfram Research in 2018. Alec is pursuing a bachelor of science at University of British Columbia as a combined major in mathematics and computer science. He has been independently researching prime numbers and their relation to the sum of quotients function and other functions since 2016 as well as the connection between abstract algebra and cellular automata. He also participated in the Wolfram Summer School: Peru 2018 where he implemented a hashing algorithm based on elementary cellular automata. These various opportunities have exposed him to various topics in computer science, philosophy and higher mathematics.
Computational Essay: Computational Text Frequency Analysis on Caesar Cipher
Project: Hypercomplex Numbers Package
Real numbers, complex numbers and quaternions have shown an unparalleled use in various areas of humanity. Further abstractions of these so-called hypercomplex numbers, which include octonions, sedenions and so on (2^n-ions), are showing promising applications as well in areas such as quantum physics. The goal is to provide a design-conscious implementation of a Hypercomplex Package that allows for intuitive manipulation, visualization and analysis.
Summary of Results
Hypercomplex numbers are defined using the Cayley–Dickson construction; we use this construction to develop a design-conscious package that supports symbolic manipulation of any hypercomplex number. This package includes new heads such as Quaternion, Octonion, Sedenion, Hypercomplex companioned by compatibility with existing basic mathematical functions and operations. We also provide some non-basic functionality: RotationQuaternion, RotationQuaternionMatrix and HypercomplexCayleyTable.
Now that hypercomplex numbers are housed in Mathematica, I could start studying what I define as “ultracomplex numbers”. These are numbers that are formed by applying the Cayle–Dickson construction indefinitely, and they have not been studied at all, though isomorphic structures might exist. Otherwise, 32-ions and higher have not been studied as extensively as R,C,H,O,S. Studying these might reveal some interesting properties, and I could extend the package to include split-complex and quasi-quaternions and the like, that are built by a different construction than the Cayley–Dickson.