  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.
