On a Fabric of Kissing Circles

TL;DR

This paper introduces a novel geometric configuration called a fabric of kissing circles, derived from circle inversion on a grid, revealing specific curvature patterns and connecting to classical configurations, with applications to solving sangaku problems.

Contribution

It constructs the fabric of kissing circles, analyzes curvature sequences within it, and applies these findings to solve classical geometric sangaku problems.

Findings

Curvatures of frame circles form a doubly infinite arithmetic sequence.

Curvatures of circles in each chain follow a quadratic bi-sequence.

The fabric connects to classical configurations like the arbelos and Pappus chain.

Abstract

Applying circle inversion on a square grid filled with circles, we obtain a configuration that we call a fabric of kissing circles. The configuration and its components, which are two orthogonal frames and two orthogonal families of chains, are in some way connected to classical geometric configurations such as the arbelos or the Pappus chain, or the Apollonian packing from the 20th century. In this paper, we build the fabric and list some of the obvious properties that result from this construction. Next, we focus on the curvature inside the individual components: we show that the curvatures of the frame circles form a doubly infinite arithmetic sequence (bi-sequence), whereas the curvatures of the circles of each chain are arranged in a quadratic bi-sequence. Because solving geometric sangaku problems was a gateway to our discovery of the fabric, we conclude this paper with two sangaku problems and their solutions using our results on curvatures.