Polyhedral representations of discrete differential manifolds

TL;DR

This paper introduces a method to represent discrete differential manifolds using polyhedra, linking algebraic calculus with topological models and enhancing understanding of discrete geometric structures.

Contribution

It provides a novel polyhedral representation of discrete differential manifolds, connecting algebraic calculus with topological models based on finitary substitutes.

Findings

Discrete differential manifolds can be represented by polyhedra.

The representation links algebraic calculus with topological models.

Supports the adequacy of discrete calculus in geometric modeling.

Abstract

Any discrete differential manifold $M$ (finite set endowed with an algebraic differential calculus) can be represented by appropriate polyhedron ${\cal P}(M)$. This representation demonstrates the adequacy of the calculus of discrete differential manifolds and links this approach with that based on finitary substitutes of continuous spaces introduced by R.D.Sorkin.