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The Dodecahedron as a Voronoi Cell - and its (minor) importance for the Kepler conjecture

DOI zum Zitieren dieses Dokuments: https://doi.org/10.15495/EPub_UBT_00004313
URN to cite this document: urn:nbn:de:bvb:703-epub-4313-1

Title data

Leppmeier, Max:
The Dodecahedron as a Voronoi Cell - and its (minor) importance for the Kepler conjecture.
Bayreuth , 2019 . - 7 S.
DOI: https://doi.org/10.15495/EPub_UBT_00004313

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Abstract

The regular dodecahedron has a 2% smaller volume than the rhombic dodecahedron which is the Voronoi cell of a fcc packing. From this point of view it seems possible that the dodecahedral aspect which is the core of the so-called dodecahedral conjecture, will play a major part for an elementary proof of the Kepler conjecture. In this paper we will show that the icosahedral configuration caused by dodecahedron leads to tetrahedra with significantly larger volume than the fcc fundamental parallelotope tessellation tetrahedra. Therefore on the basis of a tetrahedral based point of view for sphere packing densities we will demonstrate the minor importance of the dodecahedron as a Voronoi cell for the Kepler conjecture.

Abstract in another language

Im Vergleich zum rhombischen Dodekaeder, der Voronoi-Zelle einer fcc-Packung, hat das reguläre Dodekaeder ein um 2% kleineres Volumen. Aus dieser Perspektive erscheint es möglich, dass der dodekaedrische Aspekt, der den Kern der sogenannten dodekaedrischen Vermutung bildet, auch eine entscheidende Rolle für eine elementare Lösung der Kepler-Vermutung spielt. Hier zeigen wir, dass eine mit dem Dodekaeder als Voronoi-Zelle zusammenhängende, ikosaedrische Konfiguration zu Tetraedern mit signifikant größerem Volumen im Vergleich zu den Zerlegungstetraedern eines fcc-Fundamentalparallelotops führt. Somit legt die tetraedrische Perspektive für Kugelpackungsdichten eine untergeordnete Bedeutung des Dodekaeders als Voronoi-Zelle für das Kepler-Hilbert-Problem nahe.

Further data

Item Type: Preprint, postprint
Keywords: Kepler conjecture; dodecahedron; Voronoi cell; icosahedron; Delone tetrahedron; dodecahedral conjecture
DDC Subjects: 500 Science
500 Science > 510 Mathematics
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics and Didactics
Faculties
Language: English
Originates at UBT: Yes
URN: urn:nbn:de:bvb:703-epub-4313-1
Date Deposited: 12 Apr 2019 05:37
Last Modified: 12 Apr 2019 10:51
URI: https://epub.uni-bayreuth.de/id/eprint/4313

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