Octavian Primes 521

The octonions are the last stop. Real numbers, then complex, then quaternions, then octonions - eight dimensions - and then nothing. Hurwitz proved in 1898 that these four are the only number systems in which division works at all, so the sequence does not continue.

Each extension costs something. The complex numbers give up ordering; the quaternions give up commutativity; the octonions give up associativity, which means that for octonions a, b and c, the products (ab)c and a(bc) are generally different numbers. Arithmetic survives this, but only just.

The octavian integers - Coxeter described them in 1946, building on Dickson's earlier work - form a lattice within that eight-dimensional space, and one of their points is prime when the sum of the squares of its eight coordinates is an ordinary prime. This print takes a two-dimensional slice through the lattice, holding the remaining coordinates at a fixed point, and marks the 22,572 primes in a 521 × 521 field. Colour follows distance from the origin, cycling through the spectrum as the shells move outward.

It is the sparsest of the three prime lattices, at 8.3 per cent against the Gaussian 12.4 and the Hurwitz 16.4, and the difference is visible at a glance when the three are hung together.

There are 240 units in this system, and they are the 240 roots of E8 - the largest of the five exceptional root systems and the densest sphere packing possible in eight dimensions. This print and the E8 Coxeter Projection describe the same lattice from different directions: one shows its symmetry, the other its primes.

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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