Tag Archives: Triangulation

Numerical work with sage 4 – All possible triangulations of a simplicial complex using pachner moves

So what I’ve been able to do using sagemath is triangulate a simplicial complex using Pachner moves. The complete set of all possible triangulations can be recorded, saved and plotted in 3d using jmol. The value of this is that in the path integral formulation it is necessary to be able to calculate the probability of each spin network and use the result to form a partition function.

fixed vertices all possible triangualtions via pachner moves

The next steps following on from this are to add SU(2) spins as edge labels and SU(2) interwiners as node labels. I would also like to investigate moving between simplicial complexes, spin networks and weighted digraphs as representations of discrete spacetimes.

numerical work with sage 4 fig 1

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Numerical work with sagemath 2

In my  first post about using sagemaths looked at fixed size simplicial complex, this week I have been exploring using varying size simplicial complexes – growing from a single tetrahedron to over over twenty.

sage: r = [(random(),random(),random()) for _ in range(4)]
sage: for i in range(20):
… r = r+[(random(),random(),random()) for _ in range(1)]
… PointConfiguration(r).triangulate().plot(axis=false)

I would ultimately like the growth to be constrained to satisfy  causal constraints at each stage.

simplex growth 1

Animated gif of output from tachyon viewer of sagemath code

I have also been looking at using pachner flips to generate all possible triangulations of a pointset but I haven’t converted the results to a graphical format as yet.

sage: r = [(random(),random(),random()) for _ in range(20)]
sage: points = PointConfiguration(r)
sage: triang = points.triangulate()
sage: triang.plot(axes=False)
sage: p= points.bistellar_flips()
sage: print p

Results are in text form – over 240 pages –  at the moment, so I’m working on code to process file graphically as results are generated:

pachner move output v1

                                          Sample output of pachner move sagemath code –                                                  note that sagemaths calls pachner moves bistellar flips