This project verified its graphs in exact arithmetic, swept them with two independent solvers, counted their edges and conflicts — but had never looked at them. These are the first pictures: five unit-distance graphs, from the Moser spindle to Parts' record, by way of the first 5-chromatic graph built in-house. Every edge you see has length exactly 1. Drag to move, scroll to zoom, hover a point to see its neighbors.
Seven points, eleven unit edges, and the plane cannot get away with three colors. This is the mechanism everything else magnifies: two rigid rhombi glued together, rotated until their tips are at distance exactly 1.
Try it yourself: click a point to color it, click again to change color. A red edge is a conflict. With three colors you will always lose — not for lack of wit: it is a theorem.
The whole project is an attempt to redraw this picture one floor up: from 4 to 5 colors.
The graph is 4-colorable, but the two brass points — at (±4/3, 0), distance 8/3 — are forced to share a color in every 4-coloring. There is no way to separate them.
Distance 8/3 is not spindlable in the field (it would take √247): to use this spring, its length must first be changed.
A second copy of the 367, rotated around the forced vertex u by an angle with cos t = 23/32. Forcing is transitive: v ≡ u ≡ w, and the new pair (v, w) lands at distance exactly 2 — which is spindlable.
The chain plus its copy rotated by the Moser angle for distance 2 (cos = 7/8). Now w and w′ are both forced to v's color and sit at distance 1: the red edge is the contradiction. No 4-coloring exists.
This is the first 5-chromatic graph the project built rather than read from someone else's file. It does not beat the record — it chases it.
The smallest known 5-chromatic unit-distance graph (Parts, 2020). Vertex-critical: remove any one of its 509 points and it becomes 4-colorable again — verified in-house, point by point.
No exact symmetry, yet 93.7% of its points are invariant under 120°: minimization broke the hexagonal scaffolding and left its shadow. You can see it.
Parts' archive holds 43 intermediate graphs, from 136 to 510 points: the generations of the exact search that produced the record. In a row, from smallest to largest, you can watch a hexagonal shape densify, tilt, and lose its symmetry as every superfluous point is removed and every necessary one added. The last in line is the 509.