On Delaunay’s Theorem Classifying Coincidences of Parallelohedra at Faces of Codimension 3
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 pyrami...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2013-08-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/185 |
| _version_ | 1797877906802737152 |
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| author | A. N. Magazinov |
| author_facet | A. N. Magazinov |
| author_sort | A. N. Magazinov |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-10T02:25:24Z |
| format | Article |
| id | doaj.art-d83288d4921a489199002cecde28f0fb |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2024-04-10T02:25:24Z |
| publishDate | 2013-08-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-d83288d4921a489199002cecde28f0fb2023-03-13T08:07:32ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172013-08-01204718010.18255/1818-1015-2013-4-71-80179On Delaunay’s Theorem Classifying Coincidences of Parallelohedra at Faces of Codimension 3A. N. Magazinov0Математический институт им. В.А. Стеклова РАН; ЯрГУ им. П.Г. ДемидоваIn 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.https://www.mais-journal.ru/jour/article/view/185параллелоэдррешетчатое разбиениедуальная клетка |
| spellingShingle | A. N. Magazinov On Delaunay’s Theorem Classifying Coincidences of Parallelohedra at Faces of Codimension 3 Моделирование и анализ информационных систем параллелоэдр решетчатое разбиение дуальная клетка |
| title | On Delaunay’s Theorem Classifying Coincidences of Parallelohedra at Faces of Codimension 3 |
| title_full | On Delaunay’s Theorem Classifying Coincidences of Parallelohedra at Faces of Codimension 3 |
| title_fullStr | On Delaunay’s Theorem Classifying Coincidences of Parallelohedra at Faces of Codimension 3 |
| title_full_unstemmed | On Delaunay’s Theorem Classifying Coincidences of Parallelohedra at Faces of Codimension 3 |
| title_short | On Delaunay’s Theorem Classifying Coincidences of Parallelohedra at Faces of Codimension 3 |
| title_sort | on delaunay s theorem classifying coincidences of parallelohedra at faces of codimension 3 |
| topic | параллелоэдр решетчатое разбиение дуальная клетка |
| url | https://www.mais-journal.ru/jour/article/view/185 |
| work_keys_str_mv | AT anmagazinov ondelaunaystheoremclassifyingcoincidencesofparallelohedraatfacesofcodimension3 |