Gaussian Primes 521

Ordinary primes live on a line. Gaussian primes live on a plane. Take the complex integers - every number of the form a + bi where a and b are whole - and the question of which are prime has a two-dimensional answer.

The rules are not what you would expect. Two is no longer prime here; nor is five, or thirteen, or any prime that is one more than a multiple of four. Each of those splits into a pair of Gaussian primes, because each can be written as the sum of two squares. What survives intact are the primes three more than a multiple of four - 3, 7, 11, 19 - which cannot be split and stay prime in the complex plane.

This print maps every complex integer from -260 to +260 on both axes, 271,441 of them, and marks the 33,648 that are prime. Only 116 sit on the axes themselves. Colour follows distance from the origin, cycling through the spectrum as the shells of primes move outward, so the eye reads the radial structure before it reads the individual points.

The symmetry is exact and unavoidable: multiplying a Gaussian prime by -1, i or -i gives another Gaussian prime, so the whole field repeats eightfold about the origin. Whatever appears in one octant appears in all of them.

One question about this picture remains open. If you could step only on Gaussian primes, taking strides of some fixed maximum length, could you walk from the origin out to infinity? Nobody knows. It is called the Gaussian moat problem, and computer searches have got some way out before running into water.

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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