The Fifty-Nine Icosahedra - Authors' Contributions - Miller's Rules

Miller's Rules

Although Miller did not contribute to the book directly, he was a close colleague of Coxeter and Petrie. His contribution is immortalised in his set of rules for defining which stellation forms should be considered "properly significant and distinct":

(i) The faces must lie in twenty planes, viz., the bounding planes of the regular icosahedron.
(ii) All parts composing the faces must be the same in each plane, although they may be quite disconnected.
(iii) The parts included in any one plane must have trigonal symmetry, without or with reflection. This secures icosahedral symmetry for the whole solid.
(iv) The parts included in any plane must all be "accessible" in the completed solid (i.e. they must be on the "outside". In certain cases we should require models of enormous size in order to see all the outside. With a model of ordinary size, some parts of the "outside" could only be explored by a crawling insect).
(v) We exclude from consideration cases where the parts can be divided into two sets, each giving a solid with as much symmetry as the whole figure. But we allow the combination of an enantiomorphous pair having no common part (which actually occurs in just one case).

Rules (i) to (iii) are symmetry requirements for the face planes. Rule (iv) excludes buried holes, to ensure that no two stellations look outwardly identical. Rule (v) prevents any disconnected compound of simpler stellations.

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