The paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind:\ first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in $\mathbb{Z}[\tau]$ (with $\tau$ the golden ratio), to construct new regular $4$-polytopes of hyperbolic types $\{3,5,3\}$ and $\{5,3,5\}$ with automorphism groups given by finite orthogonal groups.

Keywords:

abstract polytopes, regular and chiral, Coxeter groups, modular reduction abstract polytopes, regular and chiral, Coxeter groups, modular reduction

MSC Classifications:

51M20, 20F55 show english descriptions Polyhedra and polytopes; regular figures, division of spaces [See also 51F15]
Reflection and Coxeter groups [See also 22E40, 51F15] 51M20 - Polyhedra and polytopes; regular figures, division of spaces [See also 51F15]
20F55 - Reflection and Coxeter groups [See also 22E40, 51F15]

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