Three mutually perpendicular congruent rectangles, one blue, one outlined in green and one in purple, all of fixed width AB, are centered at the origin C and aligned with the axes of Euclidean 3-space. Three orthogonal segments through C are colored according to the rectangles whose widths they span. (It is an artifact of isometric projection that these segments appear also to span other rectangles.)
Red skew edges join nearest-neighbor vertices on distinct rectangles. These edges together with 6 orthogonal edges (rectangle edges of length AB) outline an icosahedron.
Pull point D toward or away from B to vary the length of the rectangles. When D coincides with B, the length of skew edges is AB/√2. As D moves away, the skew edges lengthen without limit. By continuity, at some position of D all edges attain length AB. The fact that all 12 vertices lie on a sphere completes the proof of regularity.
Doyle offered an alternate explanation: drape an icosahedron about three Borromean rectangles and adjust the rectangle lengths to make all the icosahedron edges equal--a lovely illustration of Wilhelm Magnus's dictum that one really understands a theorem only when one can explain it to a colleague over drinks without aid of writing. This page is a heavy gloss on Doyle's lucid formulation.