UNSW Mathematician Unveils New Method to Solve Higher Polynomial Equations Using Geode Number Sequences

UNSW Sydney Honorary Professor Norman Wildberger has unveiled a groundbreaking method to solve higher-order polynomial equations, a problem that has confounded mathematicians for centuries. Polynomials, expressions involving variables raised to various powers, are foundational in both mathematics and science, with applications ranging from planetary motion to computer programming. However, solving polynomials of degree five or higher has been particularly challenging due to their complexity and the limitations of existing methods. Historically, the method of completing the square, developed by the Babylonians in 1800 BC, provided solutions for quadratic (degree two) equations. This evolved into more sophisticated techniques in the 16th century, which could solve cubic and quartic (degrees three and four) equations using radicals—roots of numbers. However, in 1832, French mathematician Évariste Galois demonstrated that the mathematical symmetry required to solve lower-order polynomials breaks down for quintic and higher-order equations, proving that no general formula using radicals could solve them. Since then, approximate methods have been employed, but these do not fully adhere to the principles of pure algebra. Prof. Wildberger's new approach sidesteps the use of radicals and irrational numbers, which he argues are based on problematic concepts of infinity. Irrational numbers, such as the cube root of seven, are decimals that extend infinitely without repeating and cannot be expressed as simple fractions. Prof. Wildberger contends that assuming these numbers exist in a complete form is illogical and leads to inconsistencies in mathematics. Instead, his method leverages special extensions of polynomials known as power series, which can have an infinite number of terms. Power series are sequences of sums where each term is a coefficient multiplied by a variable raised to a power. By truncating these series, Prof. Wildberger and his collaborator, computer scientist Dr. Dean Rubine, were able to derive approximate solutions to polynomial equations, effectively demonstrating the method's practicality and accuracy. They tested their approach on a cubic equation used by John Wallis in the 17th century to showcase Newton's method, achieving excellent results. The crux of Prof. Wildberger's innovation lies in the development of new number sequences that represent complex geometric relationships. These sequences, which he and Dr. Rubine call the "Geode," extend the classical Catalan numbers from a one-dimensional to a multi-dimensional array. The Catalan numbers are well-known in combinatorics, a branch of mathematics that deals with the arrangement of discrete objects, and they have practical applications in fields ranging from computer algorithms and data structures to game theory and biology. By constructing the Geode array, Prof. Wildberger and Dr. Rubine have laid the groundwork for a general solution to polynomial equations. The Geode sequences are based on the number of ways polygons can be divided using non-intersecting lines, a concept that Prof. Wildberger believes is crucial for solving higher-degree equations. This method not only addresses the theoretical gap left by Galois's findings but also offers practical benefits, such as enhancing computational algorithms used in applied mathematics. The potential implications of this discovery are significant. Polynomial equations are central to many areas of applied mathematics and engineering, and a reliable algebraic method for solving them could streamline a variety of computational processes. For instance, computer programs that currently rely on approximate methods or numerical techniques could adopt this new approach, leading to more precise and efficient solutions. Industry insiders have lauded Prof. Wildberger's work, noting its potential to revolutionize certain aspects of algebra and combinatorics. His contributions to rational trigonometry and universal hyperbolic geometry, which similarly reject irrational numbers and radicals, have already garnered attention. The introduction of the Geode array opens up new avenues for research, raising numerous questions that will keep mathematicians and combinatorialists occupied for years to come. Prof. Wildberger's method represents a significant leap forward in algebraic theory, offering both theoretical depth and practical utility. Norman Wildberger is a prominent mathematician known for his innovative and often controversial views on the foundations of mathematics. His work challenges traditional concepts and proposes alternative frameworks that emphasize precision and logical consistency. Dr. Dean Rubine, a computer scientist, has contributed to various mathematical and computational projects, bringing a practical perspective to the theoretical work of Prof. Wildberger. Together, their collaboration has produced a method that could reshape how we approach and solve higher-order polynomial equations.