Modular Circles MC0003

Mark points evenly around a circle, number them, and join each one by a straight line to the point at three times its number, wrapping around when the count runs past the total. Every line is straight. No curve is drawn at any stage.

A nephroid emerges: two cusps, kidney-shaped, the curve traced by a point on a circle rolling around another circle exactly twice its size. It is the shape of the bright caustic that forms when parallel light reflects off the inside of a cylinder, which is why it turns up in a coffee cup and in the reflections inside a wedding ring.

The relationship between multiplier and curve is exact and simple: the number of cusps is always one less than the multiplier. Two gives a cardioid with one cusp, three gives this, four gives three cusps, and the sequence continues indefinitely. Each is an epicycloid, arrived at without anything rolling.

What the construction demonstrates is that curvature can be emergent. The cusps are not drawn, not calculated, and not present in any individual line. They are a property of the whole family of lines, and of nothing within it.

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

UK: Free · 3-5 working days

Europe: €8.50 · 3-7 working days · No customs charges

USA & Canada: $8.95 / $12.00 CAD · 5-10 working days

Australia: $14.00 AUD · 5-10 working days

Rest of World: £14.95 · 7-14 working days

All prints are produced to order and dispatched within 1-3 working days. Orders placed before 5pm GMT ship the same day. You'll receive tracking information by email once dispatched.

Orders outside Europe may be subject to local customs charges on delivery - these are the responsibility of the recipient.

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Returns accepted within 30 days. Email returns@axisophy.com with your order number and we'll provide return instructions.

Return postage is the customer's responsibility except where the print arrives damaged or there's been an error - in which case we'll arrange a replacement or refund immediately, no return needed.

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