The Forgotten Curves: From Ancient Greek Angle-Trisectors to Theodorus' Spiral
Not every famous curve is a household name. Some were invented to solve a specific ancient problem and then forgotten. Others bear the names of mathematicians who barely studied them. Still others are so simple you wonder why they weren’t discovered earlier. Here are eight of the most intriguing lesser-known curves — each with a story worth telling.
The Quadratrix: The First Curve With a Mission
Around 420 BCE, Hippias of Elis set out to solve one of the three classical problems of Greek geometry: trisecting an angle. He invented what is probably the first curve in history defined not by a geometric construction but by a kinematic description — the simultaneous motion of two lines, one rotating and one translating.
The quadratrix is defined implicitly: a horizontal line moves downward at constant speed while a radius rotates clockwise at constant speed. Their intersection traces the curve. Once you have the quadratrix, you can trisect any angle — and even square the circle (hence the name).
It caused a philosophical scandal. Greek geometers insisted curves should be constructible with compass and straightedge; the quadratrix required a mechanical motion. It was a curve ahead of its time.
The Spiral of Theodorus: Building √n, One Triangle at a Time
Theodorus of Cyrene (Socrates’ math teacher) constructed this discrete spiral around 400 BCE. Start with a right triangle of legs 1 and 1 (hypotenuse √2). Build another right triangle using √2 as one leg and 1 as the other (hypotenuse √3). Continue: each step adds a triangle whose hypotenuse is √n.
The resulting spiral of points winds outward, each vertex at distance √n from the origin. Theodorus proved the irrationality of √3, √5, …, √17 using this construction — and then, mysteriously, stopped at √17. Why? Mathematicians still debate this. The spiral itself approximates an Archimedean spiral for large n, but its discrete nature gives it a unique jagged beauty.
The Semicubical Parabola: A Cusp and Nothing More
\(y^2 = x^3\) is one of the simplest curves with a cusp — a sharp point where the curve reverses direction. It has no loops, no asymptotes, no petals. Its very simplicity made it important: it was one of the first examples where mathematicians recognised that a curve could be smooth everywhere except at isolated singularities.
The Kappa Curve: Gergonne’s Greek Letter
Joseph Gergonne named this curve for its resemblance to κ (kappa). Given by \(y^2(a^2 - x^2) = a^2 x^2\), it has two branches that extend vertically, approaching the y-axis asymptotically but never touching it. Gergonne was the founding editor of the first purely mathematical journal, Annales de Mathématiques, in 1810. The kappa curve was one of dozens he catalogued.
Tschirnhausen’s Cubic: A Polynomial With Style
Ehrenfried von Tschirnhaus is mostly remembered for the Tschirnhaus transformation — a method for eliminating intermediate terms from polynomial equations. His cubic curve, \(27ay^2 = (a - x)(x + 3a)^2\), is a lesser legacy but a beautiful one, with a single cusp and a graceful loop.
The Evolute of an Ellipse
The evolute is the envelope of normals — or equivalently, the locus of all centres of curvature. The evolute of an ellipse is a stretched astroid with four cusps, each corresponding to a point of extreme curvature on the ellipse. Huygens used evolutes in his design of pendulum clocks: by making the pendulum bob follow a cycloidal evolute path, he made the period independent of amplitude — the first isochronous pendulum.
Try it: Click buttons to switch between curves. Move the slider to modify each curve’s defining parameter. Hit Tour All for a hands-free journey through every forgotten curve.