Overview

• Norman Wildberger and Dean Rubine's method extends Catalan numbers into a new Geode array, forming the basis for solving quintic and higher-degree polynomials.
• The approach, published in *The American Mathematical Monthly*, employs truncated power series to avoid radicals and irrational numbers, challenging long-held algebraic conventions.
• The method was successfully validated by solving a historic 17th-century cubic equation by John Wallis, demonstrating its practical applicability.
• This development revisits Évariste Galois's 1832 conclusion that no general radical-based formula exists for higher-degree polynomials, offering a combinatorial alternative instead.
• Wildberger anticipates the method's integration into computational tools, potentially improving algorithms in applied mathematics and inspiring further combinatorial research.