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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Junction of noncomposite polyhedra

Author: A. V. Timofeenko
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 21 (2009), nomer 3.
Journal: St. Petersburg Math. J. 21 (2010), 483-512
MSC (2000): Primary 52B10
Published electronically: March 2, 2010
Erratum: St. Petersburg Math. J. 23 (2012), 779--780
MathSciNet review: 2588767
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Abstract | References | Similar Articles | Additional Information

Abstract: All 3-dimensional convex regular-hedra are found, i.e., the convex polyhedra having positive curvature of each vertex and such that every face is either a regular polygon or is composed of two regular polygons. The algorithm for constructing such solids is based on calculation of the corresponding symmetry groups and gives a listing of all possible adjoins along entire faces of convex regular-hedra that cannot be cut by any plane into smaller regular-hedra.

References [Enhancements On Off] (What's this?)

  • 1. A. M. Gurin and V. A. Zalgaller, To the history of the study of convex polyhedra with regular faces and faces composed of regular, Trudy S.-Peterburg. Mat. Obshch. 14 (2008), 215-294. (Russian)
  • 2. V. A. Zalgaller, Convex polyhedra with regular faces, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 2 (1967), 220 pp. (Russian) MR 0227860 (37:3444)
  • 3. B. A. Ivanov, Polyhedra with faces that are composed of regular polygons, Ukrain. Geom. Sb. No. 10 (1971), 20-34. (Russian) MR 0301634 (46:790)
  • 4. Yu. A. Pryakhin, Convex polyhedra with regular faces, Ukrain. Geom. Sb. No. 14 (1973), 83-88. (Russian) MR 0338930 (49:3693)
  • 5. A. V. Timofeenko, On classification of convex polyhedra with regular faces, Algebra and Logic (Internat. Russian-China Sem.), Irkut. Gos. Ped. Univ., Irkutsk, 2007, pp. 103-108. (Russian)
  • 6. -, The non-Platonic and non-Archimedean noncomposite polyhedra, Fundam. i Prikl. Mat. 14 (2008), no. 2, 179-205; English transl., J. Math. Sci. 162 (2009), no. 5, 710-729. MR 2475600
  • 7. -, Convex regular polyhedra that are not cut by any plane into regular polyhedral parts, Mat. Tr. (Novosibirsk, Akad. Nauk) 11 (2008), no. 1, 132-152. (Russian) MR 2437485
  • 8. -, On a junction of noncomposite polyhedra, Proc. of the Internat. School-Sem. on Geometry and Analysis in Memory of N. V. Efimov, Rostov-on-Don, 2008, pp. 70-72. (Russian)
  • 9. A. V. Timofeenko and A. M. Gurin, On the theory of convex polyhedra with regular faces, Dokl. Akad. Nauk 419 (2008), no. 3, 320-323. (Russian) MR 2462091
  • 10. -, Algebraic and computer models of convex polyhedra with regular faces, and faces formed from regular polygons, Geometry in Odessa - 2008 (Internat. Conf., thesis) (V. V. Gol'dberg, V. M. Kuzakon', A. G. Kushner, and V. V. Lychagin, eds.), Odessa, 2008, pp. 131-132. (Russian)
  • 11. N. W. Johnson, Convex polyhedra with regular faces, Canad. J. Math. 18 (1966), 169-200. MR 0185507 (32:2973)
  • 12. V. A. Zalgaller, Convex polyhedra with regular faces, Sem. in Math. Steklov Math. Inst., Leningrad, vol. 2, Consultants Bureau, New York, 1969. MR 0240719 (39:2064)

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Additional Information

A. V. Timofeenko
Affiliation: Institute of Computational Modeling, Siberian Branch, Russian Academy of Sciences, Academgorodok 50/44, Krasnoyarsk 660036, Russia

Keywords: Regular-hedra, concomposite polyhedra, symmetry groups, superfundamental faces
Received by editor(s): August 31, 2008
Published electronically: March 2, 2010
Additional Notes: Supported by grant 09-09-1/NSh from the V. P. Astaf′ev Krasnoyarsk State Pedagogical University, and also by grants 09-01-00395-a and 09-01-00717-a from RFBR
Dedicated: To my son’s coming of age
Article copyright: © Copyright 2010 American Mathematical Society

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