{"product_id":"modular-circles-mc0002","title":"Modular Circles MC0002","description":"\u003cp\u003eMark a number of points evenly around a circle and label them 0, 1, 2, and so on. From each point, draw a straight line to the point with twice its number, wrapping around when the count exceeds the total. That is the entire construction: no curves are drawn, only straight lines between marked points.\u003c\/p\u003e\n\u003cp\u003eA cardioid appears. Not an approximation of one - the envelope of those straight lines is exactly the heart-shaped curve that forms in the reflected light inside a coffee cup, and which bounds the main body of the Mandelbrot set. Nothing in the instructions suggests a curve should emerge at all.\u003c\/p\u003e\n\u003cp\u003eThe rule generalises. A multiplier of three produces a nephroid with two cusps, four produces three, and so on: the number of cusps is always one less than the multiplier. Each is an epicycloid, the curve traced by a point on a circle rolling around another circle, arrived at here by a route that involves no rolling and no circles beyond the first.\u003c\/p\u003e\n\u003cp\u003eThe construction is old enough to have been a schoolroom exercise with pins and thread, and simple enough to explain in a sentence. What it demonstrates is that curvature can be an emergent property of straight lines - that a shape can be present in a rule without being present in any of its parts.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46883481649313,"sku":"AXS-MODC-0002-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46883481682081,"sku":"AXS-MODC-0002-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/axisophy-modular-circles-mc0002-art-print-700mm-mockup.jpg?v=1786707341","url":"https:\/\/axisophy.com\/products\/modular-circles-mc0002","provider":"Axisophy","version":"1.0","type":"link"}