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Concept

Elliptic curve

A smooth curve of degree three, such as y² = x³ + ax + b, whose points can be added: a line through two of them meets the curve at a third. That addition turns collinearity into arithmetic, and it governs counts of solutions modulo primes.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The cubic curves over GF(43) with the most and the fewest points. The solutions of two equations y squared equals x cubed plus ax plus b over the field with 43 elements, drawn as dots on a square grid: the curve with the most points and the curve with the fewest.

Give or take twice the square root

A cubic curve over the integers mod 43 should have about 44 points — one for each value of x, on average, and one at infinity. No curve misses by more than 13, the largest whole number below 2√43, and every count from 31 to 57 belongs to some curve. The first fact is Hasse's theorem, the second Deuring's, and the way the counts spread between the limits is a semicircle.

computation · Finite fields 9 trees on a cubic, 10 rows of three. The curve y = 1/(1 + x²) with 8 marked points on it and a 9th at infinity. Every line through three of them is drawn: 10 in all.

Rows of three, planted on a cubic

Nine trees can be planted in ten rows of three, and the arrangement that does it is not a grid or a star but nine points on a cubic curve. On the curve three points are in line exactly when their angles add up to a right angle, so choosing the points as a cyclic group turns collinearity into addition — and the count of rows it produces is the number Green and Tao proved is the most any planting can reach.

geometry · Ordinary lines Five triangles between the same two circles, every one closing. An outer circle of radius 1 and an inner of radius 0.38 at Euler's distance 0.4899; five triangles inscribed in the first and circumscribed about the second, started from different points.

A triangle that fits once fits everywhere

Put one circle inside another and try to fit a triangle between them, its corners on the outer circle and its sides touching the inner. Usually no triangle fits. But if one does, then one fits starting from every point of the outer circle — and whether it does is decided by a single equation in the two radii and the distance between the centres.

geometry · Triangle centres

Named alongside it

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

Counting argumentCircumcircleConjectureCyclic groupEuler relationExtremal configurationFinite fieldGenusIncidenceIncircleInvariant measureLegendre symbol

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