Abstract
This note contains a simple proof of the following theorem, which was (very nearly) stated by R. Buckminster Fuller (Marks, 1960, p. 35):
When congruent balls are arranged in cubic close-packing so as to fill a tetrahedron, square pyramid, octahedron, cuboctahedron, truncated tetrahedron or truncated octahedron, with n+1 balls along each edge, the total number of peripheral balls is \[bn2\, + \,2,\] \( b{{n}^{2}}{\mkern 1mu} + {\mkern 1mu} 2, \) where b = 2, 3, 4, 10, 14 or 30, respectively.
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Bibliography
Ball, W. W. R., Mathematical Recreations and Essays, Macmillan, London, 1967.
Coxeter, H. S. M., ‘An Upper Bound for the Number of Equal Non-Overlapping Spheres that Can Touch Another of the Same Size’, Proceedings of Symposia in Pure Mathematics (American Mathematical Society) 7 (1963) 53–71.
Hilbert, D., and Cohn-Vossen, S., Geometry and the Imagination, Chelsea, New York, 1952.
Kepler, Johannes, The Six-Cornered Snow flake, Oxford 1966.
Leech, John, ‘The Problem of the Thirteen Spheres’, Math.Gazette 14 (1956) 22–23.
Marks, R. W., The Dymaxion World of Buckminster Fuller, Southern Illinois University Press, Carbondale, 1960.
Needham, Joseph, Science and Civilisation in China, Vol. 3: Mathematics and the Sciences of the Heavens and the Earth, Cambridge University Press, 1959.
Steinhaus, Hugo, Mathematical Snapshots, Oxford University Press, 1950.
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© 1974 D. Reidel Publishing Company, Dordrecht-Holland
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Coxeter, H.S.M. (1974). Polyhedral Numbers. In: Cohen, R.S., Stachel, J.J., Wartofsky, M.W. (eds) For Dirk Struik. Boston Studies in the Philosophy of Science, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2115-9_4
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DOI: https://doi.org/10.1007/978-94-010-2115-9_4
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