Mathieu Dutour Sikirić

Mathieu Dutour Sikirić

Periodic Triangulations of n-Dimensional Lattices

This page provides technical data and enumeration results accompanying the paper "Periodic triangulations of Zn" by Mathieu Dutour Sikirić and Alexey Garber.

5-Dimensional Lattice Triangulations

We investigated methods to generate periodic triangulations, starting from known Delaunay triangulations and applying bistellar flipping operations.

  • Triangulation Count: Iterative search and isomorphism checking yielded a set of 950 triangulations.
  • Symmetry & Type:
    • 222 are Delaunay and centrally symmetric.
    • 23 are centrally symmetric but not Delaunay.
    • 705 are neither Delaunay nor centrally symmetric.
  • Data Files:

Non-Regular Triangulations

Our research confirmed the existence of non-regular triangulations in higher dimensions.

  • Dimension 5: Identified at least one non-regular triangulation (Entry #430 in our list).
  • Proof Data: Simplices configuration (gz) – contains 3,264 simplices that cannot be represented regularly.
  • Dimension 4: Regularity remains an open question for certain identified triangulations.

4-Dimensional Enumeration

We determined the possible adjacencies between triangles in the 4D case.