What if the problem wasn't the equations, but the tools?
Por casi dos siglos, las matemáticas aceptaron como ley inapelable que las ecuaciones de quinto grado y superiores no podían resolverse con fórmulas exactas —una verdad establecida por el joven Galois en 1832. El matemático australiano Norman Wildberger no desafió esa conclusión de frente, sino que cuestionó algo más profundo: los instrumentos mismos con los que la humanidad había intentado resolver el problema. Al reemplazar los números irracionales con series de potencias exactas construidas sobre patrones geométricos, Wildberger y su colaborador Dean Rubine sugieren que algunos imposibles matemáticos no eran imposibles en sí mismos, sino imposibles dentro de un marco equivocado.
- Durante 200 años, la prueba de Galois funcionó como un techo conceptual: las ecuaciones de quinto grado y superiores simplemente no tenían solución algebraica exacta, y las matemáticas aprendieron a convivir con esa limitación.
- Wildberger identificó la raíz del problema no en las ecuaciones, sino en los números irracionales —infinitos, no repetibles, siempre aproximados— que constituían la base misma de los intentos de solución.
- Junto a Dean Rubine, desarrolló el sistema de números 'Geoda': matrices multidimensionales derivadas de los números de Catalan que permiten resolver ecuaciones complejas usando solo operaciones básicas, sin aproximaciones.
- El método fue validado con una ecuación cúbica del siglo XVII que Newton y Wallis habían utilizado históricamente, resolviéndola con precisión perfecta y sin estimaciones numéricas.
- El trabajo, publicado en The American Mathematical Monthly en 2025, abre la posibilidad de que campos como la ingeniería, la física y la computación accedan a soluciones exactas donde antes solo existían aproximaciones.
En 1832, Évariste Galois demostró que las ecuaciones de quinto grado y superiores no podían resolverse mediante fórmulas basadas en radicales. Durante casi dos siglos, esa conclusión fue aceptada como una frontera infranqueable de las matemáticas. Los científicos aprendieron a trabajar con aproximaciones —estimaciones numéricas suficientemente precisas para la práctica, pero nunca exactas.
Norman Wildberger, matemático de la Universidad de Nueva Gales del Sur en Australia, decidió cuestionar no las ecuaciones, sino las herramientas. Trabajando junto al informático Dean Rubine, propuso eliminar por completo los números irracionales —como la raíz cuadrada de dos o el número pi— argumentando que su naturaleza infinita e incalculable introduce una complejidad artificial. En su lugar, construyeron soluciones a partir de series de potencias exactas.
El mecanismo central del método descansa en los números de Catalan, una secuencia matemática conocida por sus aplicaciones en geometría combinatoria y ciencias de la computación. Wildberger y Rubine descubrieron que al expandir estas secuencias unidimensionales en matrices multidimensionales —a las que llamaron números 'Geoda'— obtenían una base lógica suficiente para resolver ecuaciones algebraicas complejas usando únicamente operaciones elementales.
Para validar su enfoque, aplicaron el método a una ecuación cúbica del siglo XVII, la misma que John Wallis había empleado para ilustrar el método de Newton. El resultado fue una solución perfecta, sin aproximaciones. El trabajo fue publicado en 2025 en The American Mathematical Monthly.
Más allá del hallazgo técnico, la propuesta de Wildberger tiene una dimensión filosófica: sugiere que ciertos imposibles matemáticos no eran imposibles en términos absolutos, sino imposibles dentro de un marco conceptual particular. Si las ecuaciones complejas pueden resolverse con exactitud mediante patrones geométricos y lógica combinatoria, las implicaciones para las matemáticas computacionales, la ingeniería y la física podrían ser considerables.
In 1832, a young French mathematician named Évariste Galois proved something that would reshape algebra forever: equations of the fifth degree and higher could not be solved using a single formula based on radicals—those square roots and cube roots that had worked so elegantly for simpler problems. For nearly two centuries, mathematicians accepted this as mathematical law. They learned to live with approximations, with numerical estimates close enough for practical work but never exact.
Then Norman Wildberger, a mathematician at the University of New South Wales in Australia, asked a different question. What if the problem wasn't the equations themselves, but the tools we'd chosen to solve them? Working with computer scientist Dean Rubine, Wildberger developed a method that sidesteps the entire framework Galois had worked within. Instead of using irrational numbers—those infinite, non-repeating decimals like the square root of two or pi—they replaced them with exact power series, mathematical sequences that could be computed with perfect precision.
The core insight was deceptively simple. Irrational numbers, Wildberger argued, create artificial complexity. They're incalculable in any practical sense; we can only approximate them. But what if you didn't need them at all? What if you could solve these supposedly impossible equations using only the basic operations of mathematics—squaring, adding, multiplying—arranged in the right sequence?
To make this work, Wildberger and Rubine turned to Catalan numbers, a famous sequence in mathematics used for everything from counting how to divide polygons into triangles to analyzing tree structures in computer science. The researchers discovered something unexpected: if you took these one-dimensional sequences and expanded them into multidimensional matrices, they could serve as the logical foundation for solving complex algebraic equations. They called this new system "Geoda" numbers, named for the geometric patterns they exploit.
The breakthrough was tested on a cubic equation from the seventeenth century—the same equation the mathematician John Wallis had used to demonstrate Newton's method centuries earlier. Wildberger and Rubine's approach solved it perfectly, without approximation, without numerical guessing. Their work was published in 2025 in The American Mathematical Monthly, a peer-reviewed journal that has documented mathematical progress for over a century.
What makes this significant isn't just that an old problem found a new solution. It's that Wildberger's method suggests a different way of thinking about what mathematics can do. By abandoning irrational numbers entirely and building solutions from geometric patterns and combinatorial logic, he's proposed that some of what we thought was impossible might have simply been impossible using the wrong approach. The implications ripple outward: computational mathematics, engineering, physics—any field that relies on solving complex equations—might now have access to exact solutions where only approximations existed before.
Citações Notáveis
One of the equations we tested was a famous cubic equation used by Wallis in the seventeenth century to demonstrate Newton's method. Our solution worked perfectly.— Norman Wildberger
A Conversa do Hearth Outra perspectiva sobre a história
So Galois proved these equations couldn't be solved. What exactly did he prove?
He showed that there's no single formula—no algebraic recipe using radicals—that works for all fifth-degree equations and higher. It was a proof of impossibility, which is actually quite rare and powerful in mathematics.
And Wildberger just... ignored that proof?
Not ignored. He worked around it. Galois's proof applies to a specific framework—using irrational numbers and radicals. Wildberger changed the framework entirely. He asked: what if we don't use irrational numbers at all?
But irrational numbers are fundamental to mathematics, aren't they?
They're useful, yes. But Wildberger argues they're not necessary for solving these equations. They just add complexity. His method uses exact sequences instead—no approximation, no infinite decimals.
What are these Geoda numbers, really?
They're multidimensional expansions of Catalan numbers—a sequence that describes geometric patterns. By arranging them in the right way, you can structure an exact solution to equations that seemed unsolvable.
Has anyone else verified this works?
The test case was a seventeenth-century equation Wallis used. Wildberger's method solved it perfectly. But it's still early. The mathematical community will need time to examine whether this holds up across different types of equations.