The non-platonic and non-Archimedean noncomposite polyhedra
- 32 Downloads
If a convex polyhedron with regular faces cannot be divided by any plane into two polyhedra with regular faces, then it is said to be noncomposite. We indicate the exact coordinates of the vertices of noncomposite polyhedra that are neither regular (Platonic), nor semiregular (Archimedean), nor their parts cut by no more than three planes. Such a description allows one to obtain a short proof of the existence of each of the eight such polyhedra (denoted by M 8, M 20–M 25, M 28) and to obtain other applications.
KeywordsComputer Algebra System Convex Polyhedron Common Edge Identical Transformation Triangular Face
- 1.A. Brønsted, An Introduction to Convex Polytopes, Springer, New York (1983).Google Scholar
- 3.A. Hume, Exact Descriptions of Regular and Semi-Regular Polyhedra and Their Duals, Comp. Sci. Tech. Rep., No. 130, AT & T Bell Laboratories, Murray Hill (1986).Google Scholar
- 5.Mathematical Encyclopaedic Dictionary [in Russian], Sovetskaya Entsiklopediya, Moscow (1988).Google Scholar
- 6.G. Prohorov, V. Kolbeev, K. Zhelnov, and M. Ledenev, Mathematical Package Maple V Release 4: The Manual, http://www.exponenta.ru/soft/maple/kaluga/1.asp.
- 8.A. V. Timofeenko, “Computer models of groups and polyhedra,” in: Selected Papers Int. Conf. Math. Mech., Tomsk State Univ. [in Russian], Izd. TGU, Tomsk (2003), pp. 31–38.Google Scholar
- 9.A. V. Timofeenko, “On the theory of convex polyhedra with regular faces,” in: Proc. Int. School-Seminar Geom. Anal. Dedicated to the Memory of N. V. Efimov, Abrau-Durso, September 5–11, 2004 [in Russian], TSVVR, Rostov-na-Donu (2004), pp. 58–60.Google Scholar
- 10.A. V. Timofeenko, O. V. Golovanova, and I. V. Timofeev, “The beauty of a formula is embodied in the beauty of a spatial form. I,” in: Bull. Krasnoyarsk Architecture and Construction Academy, Proc. Conf. “Siberia—New Technologies in Architecture, Construction, and Housing-and-Municipal Facilities,” No. 8, Krasnoyarsk (2005), pp. 287–293.Google Scholar
- 11.A. V. Timofeenko, O. V. Golovanova, and I. V. Timofeev, “The beauty of a formula is embodied in the beauty of a spatial form. II,” in: Bull. Krasnoyarsk Architecture and Construction Academy, Proc. Conf. “Siberia—New Technologies in Architecture, Construction, and Housing-and-Municipal Facilities,” No. 8, Krasnoyarsk (2005), pp. 293–297.Google Scholar
- 12.A. V. Timofeenko and S. G. Stukalov, “Orbits of some groups in the three-dimensional space simulated by computer algebra systems,” in: Proc. Int. Conf. Math. Mech., Tomsk State Univ., Tomsk (2003), p. 58.Google Scholar
- 13.E. W. Weisstein, Johnson Solid, http://mathworld.wolfram.com/JohnsonSolid.html.
- 14.V. A. Zalgaller, Convex Polyhedra with Regular Faces, Consultants Bureau, New York (1969).Google Scholar