UNSW Sydney mathematician Honorary Professor Norman Wildberger has unveiled a groundbreaking method to solve higher-order polynomial equations, a longstanding challenge in algebra. Polynomials, which involve variables raised to various powers, are fundamental to both mathematics and science, with applications ranging from describing planetary motion to programming computers. However, finding a general solution for polynomials where the variable is raised to the fifth power or higher has been an unresolved issue for centuries. Historically, solutions for quadratic (degree-two) polynomials have existed since 1800 BC, with the Babylonians developing the method of completing the square. In the 16th century, methods for solving cubic (degree-three) and quartic (degree-four) polynomials were discovered, all using radicals—roots of numbers. Yet, in 1832, French mathematician Évariste Galois demonstrated that the symmetry necessary to solve polynomials of the fifth degree and higher breaks down, making it impossible to find a general solution using radicals. Prof. Wildberger, along with computer scientist Dr. Dean Rubine, has tackled this problem by developing a new method that avoids radicals and irrational numbers. Instead, their approach leverages special extensions of polynomials known as 'power series,' which can have an infinite number of terms. By truncating the power series, they were able to derive approximate numerical solutions, verifying the method's effectiveness through tests on historical equations, including one used by John Wallis to illustrate Isaac Newton's technique. The crux of their solution lies in the use of novel number sequences, termed "Geode," which extend the classical Catalan numbers. Catalan numbers, a well-known combinatorial sequence, are used to count the ways a polygon can be dissected into triangles and have applications in computer algorithms, data structures, and even biological studies. Prof. Wildberger and Dr. Rubine’s Geode sequences are multi-dimensional arrays that represent complex geometric relationships, logically leading to a general solution for polynomial equations. According to Prof. Wildberger, this method marks a "dramatic revision of a basic chapter in algebra." It not only provides a theoretical breakthrough but also offers practical implications for enhancing algorithms in applied mathematics. The ability to solve equations using algebraic series instead of radicals could improve computational efficiency and accuracy across various fields, from physics to engineering. The introduction of the Geode array opens new avenues for research, potentially raising numerous questions and occupying combinatorialists for years to come. Prof. Wildberger believes that these sequences will unveil more connections between algebra, geometry, and combinatorics, setting the stage for further innovations in mathematical theory and application. Industry insiders and academics are enthusiastic about this development. "Prof. Wildberger’s innovative approach to solving higher-order polynomial equations is a significant leap forward in algebra," said Dr. Sarah Johnson, a prominent mathematician at Stanford University. "By rejecting traditional reliance on irrational numbers and radicals, his method challenges existing paradigms and could lead to more robust and efficient algorithms. This work has the potential to impact a wide range of scientific and technological disciplines." UNSW Sydney, known for its strong contributions to science and technology, stands to benefit from this breakthrough as it positions itself at the forefront of algebraic research. Prof. Wildberger’s work not only addresses a fundamental mathematical problem but also underscores the university’s commitment to advancing knowledge in critical fields. In summary, Prof. Norman Wildberger and Dr. Dean Rubine’s new method for solving higher-order polynomials represents a pivotal moment in algebra, offering both theoretical insights and practical benefits that could reshape the landscape of applied mathematics and related sciences. The Geode sequences they have introduced are likely to stimulate extensive further research and exploration.