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Plato’s fourth solid and the “pyritohedron”

  • Paul Stephenson (a1)


Michael Deakin’s article (“A simple proof of the Beijing theorem” 76 pp251-4) will have set readers thoughts running in a number of directions. Mine turned backwards in time, towards rational approximations to the golden ratio and crystallography. The symmetries of the crystal lattice preclude the appearance of a regular pentagonal face or true 5-axis. However fair rational approximations are attained with surprising ease.



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1. Coxeter, H.S.M., Introduction to geometry, Wiley (1969) for Barlow’s short proof by contradiction.
2. Weyl, H., Symmetry, Princeton (1952), p 74.
3. Haeckel, E. Report on the scientific results of the voyage of ? M.S. Challenger, H.M.S.O. (1887), vol 18, pl. 117. No. 5 is our example.
4. Phillips, F.C., An introduction to crystallography, Longmans (1963), p 60.
5. Phillips, F.C., op. cit., same page.
6. Westaway, F.W., Craftsmanship in the teaching of elementary mathematics, Blackie (1931), pp. 578–80.

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Plato’s fourth solid and the “pyritohedron”

  • Paul Stephenson (a1)


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