Concept

Hyperbola

The conic section obtained by cutting a cone at an angle steeper than its own side, giving two separate branches. Its points are those whose distances to two foci differ by a fixed amount, and the whole-number points on one branch are what Pell's equation asks for.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Four conic sections from one cone. Circle, ellipse, parabola and hyperbola, produced by tilting a single cutting plane further and further.

One cone, four curves

The circle, the ellipse, the parabola and the hyperbola look like four separate objects with four separate equations. They are one object, cut at four angles.

geometry · conic sections

Whole-number points on x² − 2y² = 1. The branch of the hyperbola x² − 2y² = 1 in the first quadrant, with the whole-number points on it marked and labelled, and the lattice drawn faintly behind.

One solution that makes all the others

The equation x² − 2y² = 1 has infinitely many whole-number solutions, and every one of them is a power of the smallest. The multiplication that produces them is what multiplying two numbers of the form a + b√2 comes to when the √2 terms are collected — so an equation about a hyperbola turns out to carry a group.

number · pell

Named alongside it

The objects these essays reach for when they reach for this one.

Conic Continued fractions Convergent Dandelin spheres Degenerate conic Ellipse Focus Fundamental solution Lattice point Parabola Pell equation Projection

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