Classification of convex polyhedra by their rotational orbit Euler characteristic

Kovič, Jurij

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Title:	Classification of convex polyhedra by their rotational orbit Euler characteristic
Authors:	ID Kovič, Jurij (Author)
Files:	RAZ_Kovic_Jurij_i2017.pdf (272,96 KB) MD5: 34C55610C9518FB2AA931FEDFA9E96B9
Language:	English
Work type:	Unknown
Typology:	1.01 - Original Scientific Article
Organization:	ZUP - University of Primorska Press
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.
Keywords:	polyhedron, rotational orbit, Euler characteristic
Year of publishing:	2017
Number of pages:	str. 23-30
Numbering:	Vol. 13, no. 1
PID:	20.500.12556/RUP-17627
UDC:	514.113.5:519.1
ISSN on article:	1855-3966
COBISS.SI-ID:	17897561
Publication date in RUP:	03.01.2022
Views:	1430
Downloads:	19
Metadata:
:	KOVIČ, Jurij, 2017, Classification of convex polyhedra by their rotational orbit Euler characteristic. Ars mathematica contemporanea [online]. 2017. Vol. 13, no. 1, p. 23–30. [Accessed 5 April 2025]. Retrieved from: https://repozitorij.upr.si/IzpisGradiva.php?lang=eng&id=17627 Copy citation

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Record is a part of a journal

Title:	Ars mathematica contemporanea
Publisher:	Društvo matematikov, fizikov in astronomov, Društvo matematikov, fizikov in astronomov, Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije
ISSN:	1855-3966
COBISS.SI-ID:	239049984

Secondary language

Language:	Slovenian
Title:	Klasifikacija konveksnih poliedrov glede na njihovo Eulerjevo karakteristiko rotacijskih orbit
Abstract:	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.
Keywords:	polieder, rotacijska orbita, Eulerjeva karakteristika

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