 Talk on Crescent configurations in 2D. So the advisor is Avi Pallison. And so the presenters are Fiona Young, Sean Haight, Sam Kotler, and Kami Hughes, James Hughes. James. Hello. All right, so we're doing Crescent configurations in two dimensions. Our problem is based on Erdos's distinct distances problem, which comes from a paper he published in 1946, where he asked what the minimal number of distinct distances between endpoints in the plane is. And he conjectured that this was on the order of n over root log n. And this conjecture was proven in 2011, more or less. And so our problem is similar. But to describe it, we need to first define a Crescent configuration. A Crescent configuration is a set of endpoints in the plane with n minus 1 distinct distances labeled d1, d2, up to dn minus 1, such that d1 occurs once, d2 occurs twice, and so on. We also have to require that the Crescent configuration satisfy the condition that no three points can lie on any line and no four points can lie on any circle. Because otherwise, we would be able to construct arbitrarily large Crescent configurations, such as this line here, where if you equally space the points, you can have d3 occurring three times between each point, d2 occurring twice, and d1 occurring between the endpoints. And you can create an arbitrarily large Crescent configuration doing that if you relax the condition. But Erdisch conjectured that the number of Crescent configurations as we've defined them goes to zero as n grows very large. And so far, only Crescent configurations up to eight points have been found. And so we're looking at Crescent configurations on a smaller number of points to get an idea of how to build them more systematically. OK. So there's just two example Crescent configurations on five points. If we look at the configurations, we can see that the purple distance occurs four times, the green distance occurs three times, the blue distance occurs twice, and the red distance occurs exactly once, and no three points lie on a line and no four points lie on a circle. Our main reference for the project was work by some students under A.V. Paulson in which they attempted to classify Crescent configurations on four and five points. The general strategy was as follows. The first thing to notice is that instead of viewing them as points in the plane, you can view them as labeled graphs. And so in order to generate all possible Crescent configurations, they first wanted to generate all the possible adjacency matrices that could correspond to a Crescent configuration and then group the adjacency matrices by isomorphism and then consider one representative from each isomorphism class and then determine which of these adjacency matrices actually correspond to a configuration of points in the plane and which of these points actually lie in general position. There was a subtle error in their work where they threw isomorphism classes where graphs could contain an isosceles trapezoid. The logic there being that isosceles trapezoids are always inscribable on circles. And so if they had an isosceles trapezoid, they couldn't be in general position. However, it's possible to have two graphs in the same isomorphism class, one with a trapezoid and the other without a trapezoid. And so by throwing out the entire isomorphism class, you're missing out on configurations like that on the right. So our work was basically to kind of remedy that error and look for Crescent configurations that they would have thrown out. The basic process was rerun their code but this time take out the line that throws out the trapezoid configurations. So we started with around 12,600 adjacency matrices that got grouped into 98 isomorphism classes. 32 of which we determined could have been thrown out in their process, 17 of which had geometric realizations and 10 that could be realized in general position. Our fourth group member, Sam, worked on optimizing a lot of the code so that it could possibly be run on six points and classify Crescent configurations there. Cool, sorry. So as John just said, we were trying to kind of fix this error where you could have a trapezoid, an isosceles trapezoid isomorphic to that second image that was shown earlier, which is actually, there's a subgraph of a parallelogram inscribed in there. And so at the very beginning when we were trying to find code and algorithm for all of this, we wanted to first try to visualize an example of the situation. And so to do that, we created the Crescent configuration on four points. Oh, I should probably move. Yeah, a Crescent configuration on four points at the bottom. So if you look at the four points at the bottom, that's a parallelogram. And we were trying to kind of just like add a fifth point and see by solving a system of nonlinear equations where that fifth point can possibly go and have the fifth point, have the five point configuration still be a Crescent configuration. And so later we were thinking maybe we could come up with a different approach where as long as every Crescent configuration on five points has a subgraph, which is a Crescent configuration on four points, we can just build up instead of whittling them down from the original isomorphism classes and whatnot. Unfortunately, so well, first of all, in this slide, if you look at the top, there's some examples of Crescent configurations on four points. And at the bottom are the 27 that were in the paper of Crescent configurations on five points. And so I've circled two of them if you look at the top one and the bottom one that are circled in red, you can see that the one on bottom is on five points, but there's actually a subgraph hidden in there which looks like the one on four points. But unfortunately, this wasn't the case. If you simply look at number two, which I've blown up for you on the side, it turns out that no matter which one of the five points you remove, it will not look like any of the Crescent configurations on four points. So some further work could be done in this area possibly if somebody wanted to try to build up instead of narrowing down. So one possibility is that a lot of the Crescent configurations on five points do have a subgraph minus a small exception and then we can try to find properties of that exception to try to build up from there. Yep, that's the end.