Hurwitz Primes 521

Quaternions extend the complex numbers into four dimensions: a + bi + cj + dk, with three separate square roots of minus one. Hurwitz integers are the points of that space where all four coordinates are whole numbers, or all four are halves - a lattice denser than the obvious one, and the right setting for arithmetic.

A Hurwitz integer is prime when its norm - the sum of the squares of its four coordinates - is an ordinary prime. That single rule produces a result the Gaussian case does not. Lagrange proved in 1770 that every whole number is the sum of four squares, so every ordinary prime is a norm, and every ordinary prime therefore splits here. Nothing stays intact. Three, seven and eleven survive the move into the complex plane; none of them survives the move into the quaternions.

Multiplication is the price. Quaternions do not commute - ij and ji point in opposite directions - so factorisation into primes is unique only up to reordering and recombination. Arithmetic still works, but it stops behaving politely.

This print takes a two-dimensional slice through that four-dimensional lattice, holding the remaining coordinates at a fixed half-integer point, and marks the 44,489 primes in a 521 × 521 field. Colour follows distance from the origin, cycling through the spectrum as the shells move outward.

There are twenty-four units in this system - the Hurwitz integers of norm one - and they sit at the vertices of the 24-cell, one of the six regular polytopes in four dimensions. Those same twenty-four points generate the F4 root system, which is why this print and the F4 Coxeter Projection are two views of the same underlying object.

Part of the Signature series of mathematical art prints.

Signature is a collection of mathematical, scientific and infographic visualisations - prime number distributions, evolutionary trees, geological timescales, and other structures that become something unexpected when rendered at print scale.

Each piece begins as a data-driven computational process and is developed into a finished artwork through a combination of custom code and design refinement. The aim is not to illustrate mathematics but to make the underlying structures visible - to find the image that was always inside the data.

The series draws on fields including number theory, evolutionary biology, palaeontology, topology, and information design.

All prints are produced to order on 250gsm archival matte paper using pigment-based inks, chosen for colour accuracy and long-term stability.

Each print is rolled in acid-free tissue and shipped in a rigid cardboard tube, sealed for moisture protection, ready for framing on arrival.

Large · 70 × 70 cm · 28 × 28 in

XLarge · 100 × 100 cm · 40 × 40 in

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