| Title: | Classification of convex polyhedra by their rotational orbit Euler characteristic |
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| Authors: | ID Kovič, Jurij (Author) |
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| Files: |  RAZ_Kovic_Jurij_i2017.pdf (272,96 KB)
MD5: 34C55610C9518FB2AA931FEDFA9E96B9
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| Language: | English |
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| Work type: | Unknown |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | ZUP - University of Primorska Press
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| Abstract: | Let ▫$\mathcal P$▫ be a polyhedron whose boundary consists of flat polygonal faces on some compact surface ▫$S(\mathcal P)$▫ (not necessarily homeomorphic to the sphere ▫$S^{2}$)▫. Let ▫$vo_{R}(\mathcal P), eo_{R}(\mathcal P)$▫, ▫$ fo_{R}(\mathcal P)$▫ be the numbers of rotational orbits of vertices, edges and faces, respectively, determined by the group ▫$G = G_{R}(P)$▫ of all the rotations of the Euclidean space ▫$E^{3}$▫ preserving ▫$\mathcal P$▫. We define the ''rotational orbit Euler characteristic'' of ▫$\mathcal P$▫ as the number ▫$Eo_{R}(\mathcal P) = vo_{R}(\mathcal P) - eo_{R}(\mathcal P) + fo_{R}(\mathcal P)$▫. Using the Burnside lemma we obtain the lower and the upper bound for ▫$Eo_{R}(\mathcal P)$▫ in terms of the genus of the surface ▫$S(P)$▫. We prove that ▫$Eo_{R} \in \lbrace 2,1,0,-1\rbrace $▫ for any convex polyhedron ▫$\mathcal P$▫. In the non-convex case ▫$Eo_{R}$▫ may be arbitrarily large or small. |
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| Keywords: | polyhedron, rotational orbit, Euler characteristic |
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| Year of publishing: | 2017 |
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| Number of pages: | str. 23-30 |
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| Numbering: | Vol. 13, no. 1 |
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| PID: | 20.500.12556/RUP-17627  |
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| UDC: | 514.113.5:519.1 |
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| ISSN on article: | 1855-3966 |
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| COBISS.SI-ID: | 17897561 |
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| Publication date in RUP: | 03.01.2022 |
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| Views: | 913 |
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| Downloads: | 18 |
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