Data on periodic triangulations of n-dimensional lattice

In this web page we give some technical information on our paper Periodic triangulations of Zⁿ by Mathieu Dutour Sikiric and Alexey Garber.

Periodic triangulations of 5-dimensional lattice

We consider here some way to generate periodic triangulations of n-dimensional lattice
We know from previous work 222 triangulations coming from Delaunay triangulations
The idea is to take a triangulation and do bistellar flipping when possible
By iterating and checking for isomorphism we get 950 triangulations
This could be the complete list of triangulations in dimension 5, but we do not know
Of those 950 triangulations, 222 are Delaunay and centrally symmetric, 23 are not Delaunay but are centrally symmetric and 705 are not Delaunay and not centrally symmetric
The list of invariant is available at ListOfTriangulationsInvariant_5.pdf
The list of triangulations is available at ListOfTriangulation_5.gz. In this file the list of triangles is given

Non-regular triangulation

In dimension 5 there is a triangulation which is not regular
The open case is dimension 4. There is a triangulation for which we do not know if it is regular or not
In dimension 5 there is at least one triangulation which is not regular. It is represented by triangulation number 430 in the above list
The list of 3264 simplices that cannot be represented is available at Configuration430_proving_NonRegularity.gz

Enumeration of triangulations in dimension 4

In dimension 4 we are able to determine the possible adjacencies between triangles
The code for the enumeration in dimension 4 is enumerationtiling.tar.bz2