Novel Visualisations of Prime Numbers
Prime numbers are one of the oldest and most fundamental objects in mathematics, as they compose the basis for the construction of numbers themselves. Understanding their properties and behaviour can be illuminating both in terms of their pure number theoretic roots, as well as the vast number of fields predicated on them. Prime numbers are used in many aspects of modern life – one of the most prominent of which being the cryptographic systems that allow the existence of a large part of the internet. Despite their importance, there is a surprising amount that is not understood about this set of numbers. As such, they require further study. One method of investigating mathematical relationships is using visualisations. Mathematical visualisation is a method that generates images that may elucidate patterns. Various properties of prime numbers can be investigated using computer-assisted visualisations. Two of the most famous prime number visualisations are the Ulam spiral and Klauber triangle. Our project investigates novel visualisations of the prime numbers through the manipulation of structure, and the different integer sequences within them. The intention of this project is to produce and investigate visualisations that may uncover patterns within the distribution of the primes – to the ultimate end of better understanding this set of numbers. The irrational linear flow on the torus was used to plot the prime numbers and explore possible geometric or numerical relationships. This model has possible wide-reaching consequences that may add to numerous fields of study – both mathematical or otherwise.