MORE ABOUT THE 52 FOUR-DIMENSIONAL PARALLELOTOPES

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.