Abstract
Images resulting from multiple inversion and reflection in intersecting circles and straight lines are presented. Three
circles and lines making a triangle give the well-known tilings of spherical, Euclidean or hyperbolic spaces. Four
circles and lines can form a quadrilateral or a triangle with a circle around its center. Quadrilaterals give tilings
of hyperbolic space or fractal tilings with a limit set that resembles generalized Koch snowflakes. A triangle with
a circle results in a Poincaré disc representation of tiled hyperbolic space with a fractal covering made of small
Poincaré disc representations of tiled hyperbolic space. An example is the Apollonian gasket. Other such tilings can
simultaneously be decorations of hyperbolic, elliptic and Euclidean space. I am discussing an example, which is a
self-similar decoration of both a sphere with icosahedral symmetry and a tiled hyperbolic space. You can create your
own images and explore their geometries using public browser apps.
