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Concept

Conjugate

The function recording, for each slope, how far the line of that slope must be pushed to touch a given curve. It is convex whatever it came from, and applying the construction twice returns the original exactly when the original was convex.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

The quaternion multiplication table, from i² = j² = k² = ijk = −1. A four-by-four multiplication table of the quaternion units, with the row giving the left factor, every entry computed from Hamilton's rule, and the pair that differs between the two orders marked.

A multiplication that remembers the order

Four characters — i² = j² = k² = ijk = −1 — define a multiplication in which ab and ba are different numbers. Everything follows from them, including the fact that a rotation of a solid body has two names, and that two full turns are needed to get one of them home.

algebra · Quaternions x² and its conjugate. Two panels: the curve x² with tangent lines of several slopes, and the conjugate function plotted against slope, whose value at each slope is the intercept of the corresponding tangent.

The function seen from its tangents

A convex function is the upper envelope of its own tangent lines, so it can be described by giving, for each slope, how far the line of that slope has to be pushed down. That description is a second function, and applying the construction twice returns the original.

analysis · Convexity The first 60 powers of (3 + 4i)/5. The powers of (3 + 4i)/5 marked on the unit circle, each a further turn by the same angle, with the power that comes closest to returning to 1 marked.

On the circle and never home

Every root of unity lies on the unit circle, and so does the point (3 + 4i)/5 — yet no power of it ever returns to 1. A root of a whole-number polynomial of degree ten does the same. What forces a point home is a condition on the numbers its polynomial ties it to, and how far one of them may stray is a question open since 1933.

algebra · Roots of unity The states the continued fraction of √61 can be in. A grid of whole-number pairs with the band of reduced states shaded, the pairs that qualify marked, and the cycle of states visited by the expansion of the square root numbered in order.

Why the expansion has to repeat

The continued fraction of √61 runs 7; 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14 and then starts again. It must: each step's state is a pair of whole numbers trapped in a small band, and only 14 pairs fit. The expansion of √61 visits 11 of them in a cycle, the other 3 form a cycle of their own, and the period reads the same backwards before its last term, which is twice the first.

number · Pell Seven powers of √2 + ∛3 in a space of six. A table of the powers 1 to (√2 + ∛3)⁶ as coordinate vectors over a six-element basis, with the coefficients of the dependency among them: the minimal polynomial x⁶ − 6x⁴ − 6x³ + 12x² − 36x + 1.

Seven powers in a space of six

Is √2 + ∛3 a root of some polynomial with whole-number coefficients? It lives in a field of dimension six, so its first seven powers are seven vectors in a six-dimensional space and must be dependent — and the dependency, solved exactly, is the polynomial. The same count shows every sum, product and quotient of algebraic numbers is algebraic, without ever needing a formula.

algebra · Field extensions How often each factor pattern occurs, against the Galois group. Bars for four polynomials — x³ − 3x + 1, x³ − 2, x⁴ − 10x² + 1, x⁴ − 2 — giving the share of primes up to 20000 with each factorisation pattern, beside the predicted share from each Galois group.

How a polynomial breaks modulo the primes

Reduce x³ − 2 modulo a prime and it factors: into three linear pieces for some primes, one linear and one quadratic for others, not at all for the rest. Over the primes up to twenty thousand those three patterns occur a sixth, a half and a third of the time — exactly the shares of the identity, the flips and the rotations in the symmetry group of a triangle, the group that permutes the three cube roots of 2. A polynomial's factorisations modulo primes are a census of its Galois group.

algebra · Field extensions The spiral of Theodorus, √2 to √17, built from square corners and a unit length. 16 right triangles with legs √k and 1 arranged in a spiral around a common corner; their long sides have lengths √2 to √17, and together they turn through 351.2 degrees.

The lengths dividers cannot reach

A pair of dividers carries a length from one place to another and draws nothing. With a straightedge it finds midpoints, parallels and right angles, and it draws the regular 17-gon. It cannot draw a segment of length √(1 + √2) — and the reason is not on the page at all, but in the other root of the equation that number solves.

computation · Compass-only

Named alongside it

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

Field extensionDimensionMinimal polynomialPigeonhole principleAlgebraic integerAlgebraic numberAssociativityBasisCompassComplex numbersConstructible numberConstruction

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