Lévy Tapestry

Start with a square and replace each side with two equal lines that meet at a right angle, pointing away from the square. The old corners flatten into straight lines, so you get a diamond. Now repeat on all the lines, always putting the new corner on the same side as you travel around.¹

Each side of the square grows into a Lévy curve, so the tapestry is four copies of one curve, each turned a quarter turn from the last.

From Round 3 on, look for short whiskers pointing into the middle, where the pen goes out along a line and comes straight back along it. That is when the count of strokes starts to run ahead of the count of different edges.

By Round 12 the outline looks like an octagon, and the middle is still empty. Each curve bulges a long way out from its side, up to one full side length, but it only ever reaches about a quarter of a side length back in. So no line ever enters the central square, half a side wide.²

Further exploring

• Lévy Dragon — Larry Riddle
• Lévy Tapestry (Outside) — Larry Riddle

Sources and notes

1. Lévy Tapestry. ↩
2. Calculation. Measured on this page's curve up to Round 20: with a side from $(0, 0)$ to $(1, 0)$, the curve stays between $y = -0.999$ and $y = 0.249$, and these limits approach $-1$ and $1/4$. ↩