Geometric Realizations of the Abstract Platonic Polyhedra

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Conference Proceeding

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The abstract Platonic polyhedra are the abstract regular 3-polytopes whose automorphism groups are the string C-groups of rank 3 arising from the finite irreducible Coxeter groups A3, B3, and H3. The objective of this work is to construct geometric realizations of these abstract polyhedra in the Euclidean space E3. The construction employs the algebraic version of the method of Wythoff construction which uses the faithful irreducible orthogonal representations of degree 3 of these Coxeter groups.