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# Miller's Rules

Miller's rules, originally devised to restrict the number of icosahedron stellations to avoid, for example, the occurrence of models that appear identical but have different internal structures, state:

1. The faces must lie in the twenty bounding planes of the icosahedron.

2. The parts of the faces in the twenty planes must be congruent, but those parts lying in one plane may be disconnected.

3. The parts lying in one plane must have threefold rotational symmetry with or without reflections.

4. All parts must be accessible, i.e., lie on the outside of the solid.

5. Compounds are excluded that can be divided into two sets, each of which has the full symmetry of the whole.

These rules can easily be extended for finding stellations of any polyhedron (Webb).

Rule 1 essentially just defines the process of stellation. Rules 2 and 3 stipulate that a valid stellation should have the same full symmetry (but possibly without reflection) as the original polyhedron. Rule 4 requires that the vertices in the cell diagram be connected (i.e., the cell types be connected to each other). Finally, rule 5 requires that all unused vertices of the cell diagram are also connected (with a single exception which is the subject some debate). Coxeter stated in 2001 that he could not remember what Miller intended by his fifth rule, so several different interpretations have been used by various authors (Webb).

Fully Supported Stellation, Icosahedron Stellations, Stellation

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## References

Coxeter, H. S. M.; Du Val, P.; Flather, H. T.; and Petrie, J. F. The Fifty-Nine Icosahedra. Stradbroke, England: Tarquin Publications, 1999.Webb, R. "Miller's Fifth Rule." http://www.software3d.com/Millers5th.php.Webb, R. "Stella's Polyhedral Glossary: Miller's Rules." http://www.software3d.com/Glossary.php#miller.

Miller's Rules

## Cite this as:

Weisstein, Eric W. "Miller's Rules." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MillersRules.html