# Bistellar simplification

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The 2-dimensional version of a bistellar move is well known in the graphics community, though it is usually called by a different name: edge flips. It can be used to find “nice” triangle meshes of some domain, which may, for example, be applied to finite element-type computations.

The idea of the bistellar move is simple. We’ll be using some terminology from here. Let’s start with a simplicial complex $K$ of dimension $d$ and a simplex $\sigma \in K$ of dimension $0\le i \le d$. (Technically, $K$ should be a closed combinatorial manifold.)

A bistellar $i$-move is $\Phi_K(\sigma) = K – \text{star}(\sigma,K) + \tau \ast \partial \sigma$, where $\tau$ is a $(d-i)$-simplex NOT in $K$ such that $\partial \tau = \text{link}(\sigma,K)$. The move is only valid if a $\tau$ satisfying these properties exists.

The animated gif below shows some bistellar moves in action. You can see why the algorithm is called a bistellar simplification — we can change the triangle mesh to have fewer triangles.

Consider an abstract graph whose nodes are simplicial complexes which are connected by an edge if there exists a single bistellar move to get from one complex to the other. This graph is known as the bistellar flip graph or Pachner graph. The bistellar_simplification algorithm is equivalent to a random walk on this infinite graph which attempts to identify the topological type of the input complex $K$ by lowering the $f$-vector of $K$.

Tracking the $f$-vector is the cheapest way to track our progress of the simplification. In fact, for a 4-dimensional sphere, the number of vertices and edges alone is enough to learn about the entire $f$-vector [Klee]. So we can actually track the progress of the simplification on a 2 dimensional graph.

Below is a video tracking the progress of our modified bistellar_simplification on our Akbulut-Kirby 4-sphere AK_I_3. Since following a single dot was difficult to see, I am displaying the $f$-vector (or, more precisely, just the vertex and edge count) of 1000 consecutive complexes at a time. For each of those 1000 complexes, I counted the number of 0-,1-,2-,3-,4-dimensional bistellar moves were made (red = 4-move grows the $f$-vector, green = 0-move makes $f$-vector smaller; green = good). The box on the bottom right displays the vertex and edge count of the complex at the specified round.

Notice that the simulated annealing strategy brings the $f$-vector down until it gets stuck for a while, then it grows the $f$-vector in an attempt to jiggle the complex out of the local min. The current implementation uses a greedy random strategy, the modifications we made to the strategy only improved the algorithm enough to recognize AK_I_3, AK_II_3, AK_III_3. The other spheres to our knowledge have not been recognized by any heuristic software we know of.

We have also tried to implement a Markov chain Monte Carlo (Metropolis) method to improve the algorithm. We have tried several different parameters for the energy. Our tests so far indicate that an MCMC strategy results in a significant slow down of the algorithm. We are now looking into testing other deep learning techniques.