Kaleidoscope

We have a triangle of two reflecting straight lines and a reflecting circle arc.

How do we map a point into this triangle ?

The straight lines generate a dihedral group.
This group generates images of the circle arc.
They make a polygon around the center of the image.
Inside the polygon the dihedral symmetry maps any point into the triangle.

How do we map a point into this polygon ?

Use inversions at the circles.

At which circle ?

Use "ray tracing": Draw a straight line from the center to the point and take the circle that intersects first with this line.

Repeat until the point is inside the polygon.

Choose the rotational symmetries:

center  
left
right

view:

sum of angles = o

Results:

Periodic image for a sum of exactly 180o.

Decoration of elliptic space for more than 180o.
(Stereographic projections of repeating patterns on a sphere.)

Periodic pattern in hyperbolic space for less than 180o.
(Zoom in at the border of the Poincaré disc representation.)

Rapid convergence of the iterated mapping:

The distance between points and the boundary of the Poincaré disc increases nearly exponentially with the number of circle inversions.