Abstract
An arbitrary precision solution of uniform polyhedra and their duals is presented. The solution is uniform for all polyhedra given by their kaleidoscopic construction, with no need to ‘examine’ each polyhedron separately.
Similar content being viewed by others
References
Ball, W. W. R. and Coxeter, H. S. M.,Mathematical Recreations and Essays (13th edn), Dover, New York, 1987.
Coxeter, H. S. M.,Regular Polytopes (3rd edn), Dover, New York, 1973.
Coxeter, H. S. M., Longuet-Higgins, M. S. and Miller, J. C. P., ‘Uniform polyhedra’,Phil. Trans. Roy. Soc. London, Ser. A 246 (1953), 401–409.
Cundy, H. M. and Rollett, A. P.,Mathematical Models (3rd edn), Tarquin, Stradbroke, 1981.
Grünbaum, B., Miller, J. C. P. and Shephard, G. C., ‘Uniform tilings with hollow tiles’, inThe Geometric Vein — The Coxeter Festschrift (eds C. Daviset al.), Springer-Verlag, New York, 1982, pp. 17–64.
Grünbaum, B. and Shephard, G. C.,Tilings and Patterns, Freeman, New York, 1987.
Har'El, Z.,Kaleido — Kaleidoscopic Construction of Uniform Polyhedra, Department of Mathematics, Technion, Haifa, 1992.
Hume, A., ‘Exact descriptions of regular and Semi-regular polyhedra and their duals’, Computing Science Technical Report No. 130, AT&T Bell Laboratories, Murray Hill, 1986.
Huybers, P., ‘Uniform polyhedra for building structures’,Bull. Internat. Ass. Shells Spatial Struct. 21 (1980), 27–38.
Johnson, N. W., ‘Convex polyhedra with regular faces’,Canad. J. Math. 18 (1966), 169–200.
Lines, L.,Solid Geometry, Dover, New York, 1961.
Moritz, R., ‘On Napier's fundamental theorem relating to right triangles’,Amer. Math. Monthly 22 (1915), 220–222.
Skilling, J., ‘The complete set of uniform polyhedra’,Phil. Trans. Roy. Soc. London, Ser. A 278 (1975), 111–135.
Wenninger, M. J.,Polyhedron Models, Cambridge University Press, Cambridge, 1971.
Wenninger, M. J.,Dual Models, Cambridge University Press, Cambridge, 1984.
Author information
Authors and Affiliations
Additional information
In memoriam of my father, Gershon Har'El, who introduced me to spatial structures.
Rights and permissions
About this article
Cite this article
Har'el, Z. Uniform solution for uniform polyhedra. Geom Dedicata 47, 57–110 (1993). https://doi.org/10.1007/BF01263494
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01263494