## Type 1.

Annuluses formed by an infinite family of concentric circles
in geometric progression with common ratio r (0 < r < 1).

Equations of the circles:
x² + y² = (1 + r + r² + r³ + ... + r^k)², k=0,1,2,3,....

Regions determined by inequalities formed by five infinite families
of concentric annuluses.
r = 1 / golden ratio = 2 / (1+√5) ~ 0.618
h = 2  (h is the displacement parameter.)

r = 0.75, h = 2.5

r = 0.8, h = -3

r = 1 / golden ratio, h = 1.5

r = 0.75, h = 2.5

## Type 2.

Annuluses formed by an infinite family of concentric circles
in geometric progression with common ratio r (0 < r < 1).

Equations of the circles:
x² + y² = (r^k)², k=0,1,2,3,....

r = 0.5, h = 0.25

r = 0.8, h = 0.5

r = 0.999, h = 1

r = 0.8, h = 0.5

## Type 3.

Annuluses formed by two infinite families of concentric circles.
r1 = the common ratio of the outer family  (0 < r1 < 1)
r2 = the common ratio of the inner family  (0 < r2 < 1)

r1 = r2 = 0.5, h = 1.5

r1 = 0.65, r2 = 0.8, h = 2.5

r1 = 1 / golden ratio, r2 = 0.75, h = 2

r1 = r2 = 0.5, h = 1.5

( Mathematical software used: Graph )

Related posts: