Mathematical experts once deemed a certain math problem as impossible. In a recent study, two mathematicians have introduced a novel geometric approach to tackle an age-old algebraic issue. While we are taught in school to multiply and factor polynomial equations like (x² – 1) or (x² + 2x + 1), the complexity of real-world equations escalates rapidly. Typically, mathematicians resort to approximating solutions for higher-degree polynomials.
However, the authors of this study propose a new method utilizing Catalan numbers, a geometric metric, to derive exact solutions for higher-degree polynomials. These Catalan numbers emerge naturally from various mathematical scenarios and are obtained using methods such as Pascal’s triangle of polynomial coefficients. They aid graph theorists and computer scientists in organizing tree-like data structures and determining the number of possible tree arrangements within specific constraints.
The driving force behind this research, mathematician Norman “N.J.” Wildberger, an honorary professor at the University of New South Wales, challenges conventional mathematical concepts by advocating against the use of infinity or irrational numbers in certain mathematical realms. This unconventional stance is pivotal in their findings.
Collaborating with computer scientist Dean Rubine, the paper navigates through intricate mathematical concepts to present the ‘hyper-Catalan’ array, which extends the classic Catalan numbers to solve polynomial equations. The authors meticulously define terms and construct arguments to unveil the Geode array, encapsulating the hyper-Catalan number series.
In conclusion, the study emphasizes the importance of formal power series as concrete alternatives to functions that are difficult to evaluate, advocating for a shift towards a more logical and efficient approach to mathematical problem-solving.
The mathematical landscape is currently filled with various complexities. This particular work, authored by an aging iconoclast and a seasoned quantitative executive, may face challenges in gaining widespread recognition. It is published in the peer-reviewed American Mathematical Monthly, a journal with broad appeal associated with the Mathematical Association of America. The journal offers advertising opportunities, paid editing services, and an option for authors to pay for open access publication, typically costing a few thousand dollars or more. Unfortunately, this pay-to-publish model has become the norm for open access publishing.
This unconventional approach could be attributed to the subject matter not being widely recognized, but it aligns with Wildberger’s lifelong goal of streamlining mathematical concepts and presenting them in a clear and accessible manner to a broad audience. On the tech forum Hacker News, Rubine shared his close observation of Wildberger’s work on a mathematical problem that started in 2021 when Wildberger announced his intention to solve it on his YouTube channel. Rubine followed Wildberger’s progress through a series of videos where he taught amateurs how to conduct mathematical research. Despite skepticism, Wildberger successfully tackled the problem after 41 videos. Rubine eventually drafted a paper based on Wildberger’s work, highlighting their collaborative effort.
Wildberger’s unwavering determination and inclusive approach to mathematical thinking make him a formidable force in the field. The paper he co-authored with Rubine presents intriguing questions that could lead to further exploration. It remains to be seen if others in the mathematics community will delve into these questions. The anticipation for new breakthroughs builds, as waiting for 41 more videos can be quite a lengthy process.
