TL;DRAbstract
The concept of perfection of a polytope was introduced by S. A. Robertson. Intuitively speaking, a polytope P is perfect if and only if it cannot be deformed to a polytope of dierent shape without changing the action of its symmetry group G(P) on its face-lattice F(P). By Rostami's conjecture, the perfect 4-polytopes form a particular set of Wythoan polytopes. In the present paper first this known set is briefly surveyed. In the rest of the paper two new classes of perfect 4-polytopes are constructed and discussed, hence Rostami's conjecture is disproved. It is emphasized that in contrast to an existing opinion in the literature, the classification of perfect 4-polytopes is not complete as yet.
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The concept of perfection of a polytope was introduced by S. A. Robertson. Intuitively speaking, a polytope P is perfect if and only if it cannot be deformed to a polytope of dierent shape without changing the action of its symmetry group G(P) on its face-lattice F(P). By Rostami's conjecture, the perfect 4-polytopes form a particular set of Wythoan polytopes. In the present paper first this known set is briefly surveyed. In the rest of the paper two new classes of perfect 4-polytopes are constructed and discussed, hence Rostami's conjecture is disproved. It is emphasized that in contrast to an existing opinion in the literature, the classification of perfect 4-polytopes is not complete as yet.
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