Solutions to degree-two polynomials have been around since 1800 BC, thanks to the Babylonians' "method of completing the square," which evolved into the quadratic formula familiar to many high school math students. This approach, using roots of numbers called "radicals," was later extended to solve three- and four-degree polynomials in the 16th century.

Then, in 1832, French mathematician Évariste Galois showed how the mathematical symmetry behind the methods used to resolve lower-order polynomials became impossible for degree five and higher polynomials. Therefore, he figured, no general formula could solve them.

Prof. Wildberger's rejection of radicals inspired his best-known contributions to mathematics, rational trigonometry and universal hyperbolic geometry. Both approaches rely on mathematical functions like squaring, adding, or multiplying, rather than irrational numbers, radicals, or functions like sine and cosine.

His new method to solve polynomials also avoids radicals and irrational numbers, relying instead on special extensions of polynomials called "power series," which can have an infinite number of terms with the powers of x.

His method uses novel sequences of numbers that represent complex geometric relationships. These sequences belong to combinatorics, a branch of mathematics that deals with number patterns in sets of elements.

The most famous combinatorics sequence, called the Catalan numbers, describes the number of ways you can dissect a polygon, which is any shape with three or more sides, into triangles.

"The Catalan numbers are understood to be intimately connected with the quadratic equation. Our innovation lies in the idea that if we want to solve higher equations, we should look for higher analogs of the Catalan numbers."

Prof. Wildberger's work extends these Catalan numbers from a one-dimensional to multi-dimensional array based on the number of ways a polygon can be divided using non-intersecting lines.