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from II - Polytopes of Full Rank
Published online by Cambridge University Press: 30 January 2020
This chapter is devoted to completing the classification of the regular polytopes and apeirotopes of full rank, by treating the non-classical cases. Naturally, it appeals to the listing of the classical regular polytopes and apeirotopes in the previous chapter. First treated are the cases derived from the four infinite families of regular polytopes and honeycombs which occur in all dimensions; this includes eliminating some initially plausible examples. There are two interesting non-polytopes in dimensions six and eight, the first particularly so since its facet is genuinely polytopal. The exceptional cases are to be found in dimensions two, three and four only. First considered are the remaining regular polyhedra in three dimensions, since the apeirohedra in the plane need little comment. The other examples are in four dimensions; one of these shows that the full range of possible mirror vectors can occur.
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