Abstract
There are several works [6] (and [13]), [8], [2] and [14] enumerating four-dimensional parallelotopes. Engel [9] was the first who distinguished 17 zonotopal parallelotopes among them. Each zonotopal parallelotope is the Minkowski sum of segments whose generating vectors form a unimodular system. We show that there are exactly 17 four-dimensional unimodular systems. Hence there are 17 four-dimensional zonotopal parallelotopes. We prove that other 35 four-dimensional parallelotopes are: the regular 24-cell $\{3,4,3\}$ and 34 sums of the 24-cell with non-zero zonotopal parallelotopes. We give a detailed description of the construction of these 35 parallelotopes.
Citation
M. Deza. V. P. Grishukhin. "MORE ABOUT THE 52 FOUR-DIMENSIONAL PARALLELOTOPES." Taiwanese J. Math. 12 (4) 901 - 916, 2008. https://doi.org/10.11650/twjm/1500404985
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