A mathematician has cracked a 200-year-old problem that once defeated history’s greatest mathematical minds, upending centuries of conventional wisdom about what’s possible in algebra.
UNSW Sydney’s Norman Wildberger, working with computer scientist Dean Rubine, has developed a revolutionary approach to solving polynomial equations of degree five and higher – something the legendary French mathematician Évariste Galois proved “impossible” back in 1832.
“Our solution reopens a previously closed book in mathematics history,” Wildberger explains in his groundbreaking paper published in the American Mathematical Monthly.
The key innovation lies in rejecting traditional approaches that rely on irrational numbers – those never-ending, non-repeating decimals that can’t be precisely calculated. Instead, Wildberger developed special extensions of polynomials called power series and a fascinating new sequence of numbers they’ve named “the Geode.”
This sequence extends the famous Catalan numbers, which count the ways to subdivide polygons into triangles, into a multi-dimensional array based on more complex polygon divisions. The approach reveals surprising connections between geometry, combinatorics, and algebra that have remained hidden for centuries.
“We’ve found these extensions, and shown how, logically, they lead to a general solution to polynomial equations. This is a dramatic revision of a basic chapter in algebra,” Wildberger states.
When tested on the famous cubic equation used by mathematician John Wallis in the 17th century to demonstrate Newton’s method, Wildberger’s solution “worked beautifully.” Beyond theoretical interest, the approach promises practical advances for computational mathematics across numerous fields.
The innovation goes beyond just solving equations. The newly discovered “Geode” array of numbers appears to underlie the classical Catalan sequence itself, suggesting a deeper mathematical structure that has eluded mathematicians until now.
“We expect that the study of this new Geode array will raise many new questions and keep combinatorialists busy for years,” predicts Wildberger. “Really, there are so many other possibilities. This is only the start.”
For a field where major breakthroughs can be centuries apart, Wildberger’s work represents a remarkable achievement – demonstrating that even in our oldest and most established branches of mathematics, revolutionary discoveries remain possible.
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