The Lévy C curve begins with one harmless straight line. Then every segment bends into two equal arms meeting at a right angle. Here it meets its own mirror: two C curves rotate toward one another by equal and opposite angles, staying perfectly symmetric while their crossings grow increasingly feral from one instruction.
How to play with it
Colour the arms
Choose one colour for each of the first two branches. Their descendants keep the family colour all the way down. Where the arms crowd together, the glow mixes visually, but the mathematical ancestry stays separate.
Turn the collision
At zero degrees the two C curves share their endpoints and close around one another like a fractal lens. Increase the angle and they rotate in opposite directions. The picture stays reflected across its middle, while the places where the paths meet reorganise into gates, knots and bright little accidental rooms.
Put a C inside the C collision
The middle button adds a third, smaller Lévy C centred in the gap. At its default quarter-turn, its symmetry agrees with the horizontal mirror of the outer pair. Then you may drag it with a mouse or finger, rotate it through a full circle, enlarge it, shrink it and give it a colour that belongs to neither outer arm. “Centre it again” resets only its position; the other choices stay yours.
Connect a beauty to the beast
The beauty is five thinner, differently coloured copies of the very same Lévy C, rotated around one centre until they open like readable petals. The so-called beast moves to one side; one more shallow copy stretches between them as a bridge. No new drawing rule is introduced. Repetition makes both bodies, and viewpoint gives them their names.
The historical record
Nothing was ever enough for this blonde woman. She saw two mirrored fractals, requested a third one inside them, and then quite reasonably demanded that the third one should rotate, grow and choose its own coat. Mathematics survived. It may even have improved.
Follow the whole path
Switch to “rainbow path” and colour becomes a trail from one endpoint to the other. It lets your eye walk the actual order of the line through a picture that otherwise looks like a crowded two-dimensional body.
Take it away from the screen
The download is a scalable SVG sized for A4 paper. It stays sharp when printed large. The black outlines form hundreds of narrow cells; colour neighbouring cells, alternate the two arms, invent a gradient from the tips inward, or ignore the mathematics and let the creature wear whatever coat it wants.
The peculiar arithmetic
Every round doubles the number of segments and multiplies the total path length by √2. The endpoints never move. So the curve becomes infinitely long while remaining trapped between the same two endpoints — a very disciplined way of becoming enormous.
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