Points and lines configurations for perpendicular bisectors of convex cyclic polygons

We characterize the topological configurations of points and lines that may arise when placing n points on a circle and drawing the n perpendicular bisectors of the sides of the corresponding convex cyclic n-gon. We also provide exact and asymptotic formulas describing a random realizable configuration, obtained either by sampling the points uniformly at random on the circle or by sampling a realizable configuration uniformly at random.

Cube moves for $s$-embeddings and $alpha$-realizations

For every $alpha$ in $mathbb{R}^*$, we introduce the class of $alpha$-embeddings as tilings of a portion of the plane by quadrilaterals such that the side-lengths of each quadrilateral $ABCD$ satisfy $AB^alpha+CD^alpha=AD^alpha+BC^alpha$. When $alpha$ is $1$ (resp. $2$) we recover the so-called $s$-embeddings (resp. harmonic embeddings). We study existence and uniqueness properties of a local transformation of $alpha$-embeddings (and of more general $alpha$-realizations, where the quadrilaterals may overlap) called the cube move, which consists in flipping three quadrilaterals that meet at a vertex, while staying within the class of $alpha$-embeddings. The special case $alpha=1$ (resp. $alpha=2$) is related to the star-triangle transformation for the Ising model (resp. for resistor networks). In passing, we give a new and simpler formula for the change in coupling constants for the Ising star-triangle transformation.