Hall's conjecture was originally put forward by my undergraduate advisor, Glen Hall, in his paper Acute Triangles in the n-Ball. While trying to prove the conjecture, I generated a lot of pictures and calculations. Here are some of the files that I found helpful. The results from these figures were used in my paper Hall's Conjecture on Extremal Sets for Random Triangles.
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This Mathematica notebook contains the calculation of a lemma on Legendre polynomials that was used to understand the problem in 3 dimensions.
For this project, it was difficult for me to visualize the geometry. Geogebra was a crucial tool, because it allowed me to manipulate sketches to see how small deformations affect the geometry.
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These two Geogebra file contain a disk and the region of convex triangles given two fixed points. The first is related to the "autocorrelation term" while the second can be modified more freely.
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The "Canonical Homotopy" Geogebra figured helped me understand how small translations and small dilations of convex regions interact. This was needed for Lemma 12, which constructed a canonical homotopy.
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This Geogebra figure was useful in calculating how the "autocorrelation term" changes as one deforms a convex region. It was useful for Lemma 14.
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This Geogebra figure helped me visualize how the locus of right triangles can intersect the boundary of a convex region. This figure was helpful for Lemma 15.
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This Geogebra figure helped me picture how the volume of right triangles changes as the boundary of a convex region changes. This figure was also helpful for Lemma 15.