The Action of the Modular Group on the Fundamental Domain

first

invert

translate

second

invert

translate

third

invert

translate

fourth

invert

translate

fifth

invert

translate

sixth

invert

translate

seventh

invert

translate

eighth

invert

translate

first exponent

-1

0

1

second exponent

-1

0

1

third exponent

-1

0

1

fourth exponent

-1

0

1

fifth exponent

-1

0

1

sixth exponent

-1

0

1

seventh exponent

-1

0

1

eighth exponent

-1

0

1

Choose eight transformations from the set of controls on the left. Each transformation can be either an inversion or a translation. Choose the exponent of each of these transformations (-1, 0, or 1) from the set of controls on the right. This Demonstration shows the fundamental domain and its images under the consecutive compositions of the chosen transformations.