Partitioning Regular Polygons into Circular Pieces I: Convex Partitions

We explore an instance of the question of partitioning a polygon into pieces, each of which is as ``circular'' as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of the smallest circumscribing circle to the largest inscribed disk. The problem is rich even for partitioning regular polygons into convex pieces, the focus of this paper. We show that the optimal (most circular) partition for an equilateral triangle has an infinite number of pieces, with the lower bound approachable to any accuracy desired by a particular finite partition. For pentagons and all regular k-gons, k > 5, the unpartitioned polygon is already optimal. The square presents an interesting intermediate case. Here the one-piece partition is not optimal, but nor is the trivial lower bound approachable. We narrow the optimal ratio to an aspect-ratio gap of 0.01082 with several somewhat intricate partitions.Comment: 21 pages, 25 figure

On the Square Peg Problem and some Relatives

The Square Peg Problem asks whether every continuous simple closed planar curve contains the four vertices of a square. This paper proves this for the largest so far known class of curves. Furthermore we solve an analogous Triangular Peg Problem affirmatively, state topological intuition why the Rectangular Peg Problem should hold true, and give a fruitful existence lemma of edge-regular polygons on curves. Finally, we show that the problem of finding a regular octahedron on embedded spheres in R^3 has a "topological counter-example", that is, a certain test map with boundary condition exists.Comment: 15 pages, 14 figure

Optimum Placement of Post-1PN GW Chirp Templates Made Simple at any Match Level via Tanaka-Tagoshi Coordinates

A simple recipe is given for constructing a maximally sparse regular lattice of spin-free post-1PN gravitational wave chirp templates subject to a given minimal match constraint, using Tanaka-Tagoshi coordinates.Comment: submitted to Phys. Rev.

Random packing of regular polygons and star polygons on a flat two-dimensional surface

Random packing of unoriented regular polygons and star polygons on a two-dimensional flat, continuous surface is studied numerically using random sequential adsorption algorithm. Obtained results are analyzed to determine saturated random packing ratio as well as its density autocorrelation function. Additionally, the kinetics of packing growth and available surface function are measured. In general, stars give lower packing ratios than polygons, but, when the number of vertexes is large enough, both shapes approach disks and, therefore, properties of their packing reproduce already known results for disks.Comment: 5 pages, 8 figure