On Delaunay’s Theorem Classifying Coincidences of Parallelohedra at Faces of Codimension 3
https://doi.org/10.18255/1818-1015-2013-4-71-80
Abstract
In 1929 B.N. Delaunay obtained the complete classification of all possible combinatorial coincidence types of parallelohedra at their faces of codimension 3. It appeared that every such coincidence is dual to one of the following five three-dimensional polytopes: a tetrahedron, a quadrangular pyramid, an octahedron, a triangular prism, or a parallelepiped. The present paper contains a new combinatorial proof of this result based on Euler formula. Using the classification, we have obtained several further properties of faces of codimension 3 in parallelohedral tilings. For instance, we showed that the Dimension Conjecture holds for faces of codimension 3, i.e. if we take the affine hull of centers of all parallelohedra containing a particular face of codimension 3, this affine hull is three-dimensional. Finally, we proved that the set of centers of all parallelohedra sharing a face of codimension 3 spans a three-dimensional sublattice of index one.
About the Author
A. N. Magazinov
Steklov Mathematical Institute of RAS;
Yaroslavl State University
Russian Federation
Russian Federation
Gubkina street, 8, Moscow, 119991, Russia;
аспирант, B. N. Delaunay Laboratory «Discrete and Computational Geometry»,
Sovetskaya street, 14, Yaroslavl, 150000, Russia
References
1. Венков Б.А. Об одном классе эвклидовых многогранников // Вестник Ленинградского Университета. Сер. мат., физ., хим. 1954. 2. С. 11 – 31. (Venkov B.A. Ob odnom klasse evklidovykh mnogogrannikov // Vestnik Leningradskogo Universiteta. Ser. mat., fiz., khim. 1954. 2. P. 11 – 31 [in Russian]).
2. Долбилин Н.П. Свойства граней параллелоэдров // Геометрия, топология и математическая физика. II: Сборник статей. К 70-летию со дня рождения академика Сергея Петровича Новикова. Тр. МИАН. 2009. 266. С. 112 – 126. (English Translation: Dolbilin N.P. Properties of Faces of Parallelohedra // Proc. Steklov Inst. Math. 2009. 266. P. 105 – 119).
3. Долбилин Н.П. Параллелоэдры: ретроспектива и новые результаты // Труды ММО. 2012. 73:2. С. 259 – 276. (English Translation: Dolbilin N.P. Parallelohedra: A retrospective and new results // Trans. Moscow Math. Soc. 2012. 73. P. 207 – 220).
4. Фоменко А.Т., Фукс Д.Б. Курс гомотопической топологии. М.: Наука, 1989. (Fomenko A.T., Fuks D.B. Kurs gomotopicheskoy topologii. Moskva: Nauka, 1989 [in Russian]).
5. Danzer L., Grünbaum B. Über zwei Probleme bezüglich konvexer Körper von P. Erdös und von V. L. Klee // Math. Z. 1962. 79. P. 95 – 99.
6. Delaunay B.N. Sur la partition réguli`ere de l’espace à 4 dimensions // Izv. Acad. sci. of the USSR. Ser. VII. Sect. of phys. and math. sci. 1929. 1 – 2. P. 79 – 110, 147 – 164.
7. Dutour M. The six-dimensional Delaunay polytopes // European Journal of Combinatorics. 2004. 25. P. 535 – 548.
8. Minkowski H. Allgemeine Lehrsätze über die konvexe Polyeder. Nach. Ges. Wiss., Göttingen, 1897. P. 198 – 219.
9. Ordine A. Proof of the Voronoi conjecture on parallelotopes in a new special case: Ph.D. Thesis / Queen’s University, Ontario, 2005.
10. Ryshkov S.S., Rybnikov K.A. Jr. The theory of quality translations with applications to tilings // European Journal of Combinatorics. 1997. 18. P. 431 – 444.
11. Voronoi G. Nouvelles applications des paramétres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les paralléloèdres primitifs // J. Reine Angew. Math. 1908. 134. P. 198 – 287; 1909. 136. P. 67 – 178. Перевод: Вороной Г.Ф. Исследования о примитивных параллелоэдрах: Собр. соч. Т. 2. Киев: Изд-во АН УССР, 1952. С. 239 – 368.
12. Zitomirskij O.K. Versch¨arfung eines Satzes von Woronoi // J. Leningrad. Fiz.-Mat. Ob-va. 1929. 2. P. 131 – 151.
Review
For citations:
Magazinov A.N. On Delaunay’s Theorem Classifying Coincidences of Parallelohedra at Faces of Codimension 3. Modeling and Analysis of Information Systems. 2013;20(4):71-80. (In Russ.) https://doi.org/10.18255/1818-1015-2013-4-71-80
Views:
914
Email this article 