Algebraic Curiosities: Twelve Strange Curves and the Mathematicians Who Found Them
Between the 17th and 19th centuries, mathematicians discovered a menagerie of curves that defied intuition: loops that cross themselves, shapes that resemble leaves and witches’ hats, and one curve so versatile it gave us the modern office building. These twelve algebraic curves — each defined by a polynomial equation — reveal how the search for geometric beauty drove centuries of discovery.
The Lemniscate: Symbol of Infinity
Jacob Bernoulli discovered the lemniscate in 1694 while studying elastic curves. Its figure-eight shape became the symbol for infinity (∞). The equation \((x^2 + y^2)^2 = 2a^2(x^2 - y^2)\) produces a curve where the product of distances to two fixed points is constant — making it a special case of the Cassini oval.
Bernoulli was so enchanted by it that he had it engraved on his tombstone with the inscription “Eadem mutata resurgo” — “Though changed, I shall arise the same.”
The Folium: Descartes’ Leaf
In 1638, Descartes challenged Fermat to find the tangent to \(x^3 + y^3 = 3axy\). Fermat succeeded, humiliating Descartes — who retaliated by claiming Fermat’s method wasn’t general enough. The resulting “leaf” has a self-intersection at the origin, and its area was one of the first non-trivial successes of integral calculus.
The Witch That Wasn’t
Maria Gaetana Agnesi published the first comprehensive calculus textbook by a woman in 1748. The curve \(y = a^3/(x^2 + a^2)\) was called versiera in Italian. When John Colson translated her work into English, he mistook versiera for avversiera (“witch”). The name stuck — a permanent monument to a translation error, and to Agnesi’s remarkable achievement.
The Superellipse: From Math to Architecture
| Gabriel Lamé generalised the ellipse in 1818: \( | x/a | ^n + | y/b | ^n = 1\). For \(n = 2\) you get an ellipse; for larger \(n\), the shape becomes increasingly rectangular. In 1959, Piet Hein applied the superellipse to urban design — the Sergels Torg roundabout in Stockholm is a superellipse. He also designed the “superegg,” a three-dimensional superellipse that stands upright on a flat surface, defying intuition. |
Use the slider to vary each curve’s defining parameter and watch the shape transform. Click Auto-Tour to sit back and let the curves introduce themselves.