In this model the

hyperbolic plane [H.sup.2] is identified with the unit disk D = {z [member of] C : [absolute value of z] < 1} on the complex plane and hyperbolic lines are just segments of circles orthogonal to the boundary of D.

Poincare, however, described a useful model of hyperbolic geometry where the "points" in a

hyperbolic plane are taken to be points inside a fixed circle (but not the points on the circumference).

In [3, 4, 5, 10], the symmetry groups of semi-regular tilings on the

hyperbolic plane are studied and characterized.

Bryant, Griffiths exploits the natural symmetries of the problem to obtain interesting results concerning the existence of closed, free (no constraints on the arc length), elastica in the

hyperbolic plane [H.sup.2] ([4]).

Thomas presents her lecture notes for a graduate course for students with a background in basic group theory including group actions, a first course in algebraic topology, and some familiarity with Riemannian geometry, particularly the geometry of the

hyperbolic plane. She explores geometric and topological aspects first of Coxeter groups and then of buildings.

In [9], Reynold give a brief introduction to hyperbolic geometry of

hyperbolic plane [H.sup.2].

One mathematician's crocheted models of a counter-intuitive shape called a

hyperbolic plane are enabling her students and fellow mathematicians to gain new insight into startling properties.

It discusses discontinuous groups through the classical method of Poincare and the model of the

hyperbolic plane; develops automorphic functions and forms through the Poincare series, with discussion of formulas for divisors of a function and form; and details the connection between automorphic function theory and Riemann surface theory, as well as applications of Riemann-Roch theorem.

Just as a flat surface--like that of a sheet of paper--is a piece of the infinite mathematical surface known as the Euclidean plane, a saddle-shaped surface can be thought of as a small piece of the

hyperbolic plane. Picturing what the

hyperbolic plane looks like on a larger scale, however, requires some mind-bending ingenuity.

Synopsis: The winner of the Euler Book Prize (which is awarded by the Mathematical Association of America) and illustrated with more than 200 full color photographs, this newly updated and revised second edition of "Crocheting Adventures with

Hyperbolic Planes" by Daina Taimina is a non-traditional, tactile introduction to non-Euclidean geometries that also covers early development of geometry and connections between geometry, art, nature, and sciences.

In 2007, the runnerup for the prize was Robert Chenciner's definitive study Tattooed Mountain Women And Spoon Boxes Of Daghestan, and in 2010, James' Yannes's Collectible Spoons Of The 3rd Reich was a front runner before losing out to Crocheting Adventures With

Hyperbolic Planes.

Last year's winner, 'Crocheting Adventures with

Hyperbolic Planes', saw its sales soar by 1,500 percent within a month of winning the prize, which has been running since 1978.

Cut from competition was a favorite of mine, "Peek-A-Poo: What's In Your Diaper?" but "Crocheting Adventures with

Hyperbolic Planes" did nothin' for me at all.

The other five contenders on the shortlist this year are: The Changing World of Inflammatory Bowel Disease by Ellen Scherl and Marla Dubinsky; Collectible Spoons of the 3rd Reich by James A Yannes; Crocheting Adventures with

Hyperbolic Planes by Daina Taimina; Governing Lethal Behavior in Autonomous Robots by Ronald C Arkin; and What Kind of Bean is this Chihuahua?

Novel and astonishing, Taimina (mathematics, Cornell U.) crochets models of

hyperbolic planes. This beautifully illustrated volume explains how to teachers, mathematicians, and the interested reader.