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I am interested in Euclidean gravity models such as Regge's simplicial gravity model. These models use triangulations of $\mathbb{R}^4$ with periodic boundary conditions. While there are many papers using such triagulations, I have not found an algorithm for generating such triangulations. Could someone explain how to generate a regular 4D triagulation of the unit 4 lattice ($\mathbb{Z}^4$) that is invariant under lattice translations? I have found algorithms for 2D and 3D lattices but do not understand how to generalize these.

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There is a paper by P. S. Mara : Triangulations of the cube. But easier to find the article "impact of topology in cdt quantum gravity" : https://arxiv.org/abs/1604.08786, page 5, where we explicitly write a triangulation of the hypercube (citing mara) .

A comment: In Regge calculus the deficit angle gives the curvature, thus you will not find a translation invariant smooth but discrete triangulation in 4d. This version that I cite can be used to construct Triangulations. In the appendix we even give another one, which is smaller but more complicated. Due to the 4 type of simplices, it's hard to imagine these objects.

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