Once more about the 52 four-dimensional parallelotopes

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dc.creatorDeza, Michel
dc.creatorGrishukhin, Viacheslav
dc.date2003-07-11
dc.date.accessioned2026-07-07T04:59:37Z
dc.date.available2026-07-07T04:59:37Z
dc.descriptionThere are several works \cite{De} (and \cite{St}), \cite{En}, \cite{Co} and \cite{Va} enumerating four-dimensional parallelotopes. In this work we give a new enumeration showing that any four-dimensional parallelotope is either a zonotope or the Minkowski sum of a zonotope with the regular 24-cell $\{3,4,3\}$. Each zonotopal parallelotope is the Minkowski sum of segments whose generating vectors form a unimodular system. There are exactly 17 four-dimensional unimodular systems. Hence, there are 17 four-dimensional zonotopal parallelotopes. Other 35 four-dimensional parallelotopes are: the regular 24-cell $\{3,4,3\}$ and 34 sums of the regular parallelotope with non-zero zonotopal parallelotopes. For the nontrivial enumerating of the 34 sums we use a theorem discribing necessary and sufficient conditions when the Minkowski sum of a parallelotope with a segment is a parallelotope.
dc.description14 pages (submitted)
dc.identifierhttps://arxiv.org/abs/math/0307171
dc.identifierhttp://arxiv.org/abs/math/0307171
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/68060
dc.subjectMetric Geometry
dc.subjectCombinatorics
dc.titleOnce more about the 52 four-dimensional parallelotopes
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