20.500.12556/RUP-17627 Classification of convex polyhedra by their rotational orbit Euler characteristic Klasifikacija konveksnih poliedrov glede na njihovo Eulerjevo karakteristiko rotacijskih orbit 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}$▫ may be arbitrarily large or small. Naj bo ▫$\mathcal P$▫ polieder, katerega površje sestoji iz ploskih poligonskih lic na neki kompaktni ploskvi ▫$S(\mathcal P)$▫ (ne nujno homeomorfni sferi ▫$S^{2}$)▫. Naj bodo ▫$vo_{R}(\mathcal P), eo_{R}(\mathcal P)$▫, ▫$ fo_{R}(\mathcal P)$▫ ševila rotacijskih orbit vozlišč, povezav in lic, določena z grupo ▫$G = G_{R}(P)$▫ vseh takšnih rotacij evklidskega prostora ▫$E^{3}$▫, ki ohranjajo polieder ▫$\mathcal P$▫. Definiramo ''Eulerjevo karakteristiko rotacijskih orbit'' poliedra ▫$\mathcal P$▫ kot število ▫$Eo_{R}(\mathcal P) = vo_{R}(\mathcal P) - eo_{R}(\mathcal P) + fo_{R}(\mathcal P)$▫. S pomočjo Burnsidove leme dobimo spodnjo in zgornjo mejo za ▫$Eo_{R}(\mathcal P)$▫, ki ju izrazimo kot funkcijo reda ploskve ▫$S(P)$▫. Dokažemo, da je ▫$Eo_{R} \in \lbrace 2,1,0,-1\rbrace $▫ za vsak konveksen polieder ▫$\mathcal P$▫. V nekonveksnem primeru je ▫$Eo_{R}$▫ lahko poljubno velik ali majhen. polyhedron rotational orbit Euler characteristic polieder rotacijska orbita Eulerjeva karakteristika true true false Angleški jezik Slovenski jezik Neznano 2022-01-03 01:11:12 2022-01-03 01:11:12 2024-03-01 14:39:22 0000-00-00 00:00:00 2017 0 0 str. 23-30 no. 1 Vol. 13 2017 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 514.113.5:519.1 1855-3966 17897561 RAZ_Kovic_Jurij_i2017.pdf RAZ_Kovic_Jurij_i2017.pdf 1 34C55610C9518FB2AA931FEDFA9E96B9 81499255fb666325233bb6852a818dada9434b9263f4fdc25f418d682cf343c5 7834e445-c3d6-11ed-b5df-005056bf369b https://repozitorij.upr.si/Dokument.php?lang=slv&id=25945 Založba Univerze na Primorskem 0 0 0