New THE MATHEMATICA JOURNAL article:
by PAUL R. MCCREARY, TERI JO MURPHY, CHRISTAN CARTER
ABSTRACT: The action of Möbius transformations with real coefficients preserves the hyperbolic metric in the upper half-plane model of the hyperbolic plane. The modular group is an interesting group of hyperbolic isometries generated by two Möbius transformations, namely, an order-two element 
and an element of infinite order 
. Viewing the action of the group elements on a model of the hyperbolic plane provides insight into the structure of hyperbolic 2-space. Animations provide dynamic illustrations of this action.
