http://www.geom.umn.edu/apps/lafite/about.html (World Wide Web Directory, 06/1995)

About Lafite (Exploring Hyperbolic Symmetry Groups)

Lafite

Exploring Hyperbolic Symmetry Groups

by Adam Deaton

lafite is a program designed to help one understand the structure of hyperbolic symmetry groups. It was written in the spring of 1993 and runs on SGI workstations. The World Wide Web front-end supports only a fraction of the features of lafite . Read on for a little background, or click here to begin using lafite.

What can it do?

In priciple, lafite will allow you to work with any hyperbolic plane symmetry group. Starting from an intuitive notation devised by J.H. Conway, lafite calculates a fundamental region and the generators of the group. In cases where the shape of the fundamental region is not uniquely determined lafite lets you explore the family of possible shapes.

lafite supports three models of the hyperbolic plane and contains a basic drawing program, to allow you to make patterns which can be replicated by the action of the group you chose.

Understanding the Notation

The main reference for Conway's notation is [1] . What we want is a way to describe the action of a group of isometries of the hyperbolic plane. The original insight, due to Thurston [2] is to describe the structure of the quotient of the manifold by the group. Such an object is no longer just a manifold: it has additonal structure, in the form of ``singular points'' where the group action had fixed points. Such an object is called an orbifold

Conway's orbifold notation describes both the topology of the quotient manifold, and also the structure of the singular points. The topology is indicated with the symbols o: indicating a handle; x, a cross-cap; and * a perforation (the removal of an open disk). Digits 2, 3, 4,..., and i (for infinity) indicate the singularities: before a * symbol, a digit n indicates a ``cone point'' where the original group of isometries acted by a rotation of 2Pi/n. After a * symbol, a digit indicates a ``corner point'' where the original group acted by a reflection: the orbifold has a ``corner'' here, with an angle of Pi/n.

Some examples

These examples only show part of the plane for clarity.

*237

A ``triangle group,'' the fundamental region is a triangle with angles Pi/2, Pi/3, and Pi/7. (Of course, these angles sum to less than Pi) The group acts by reflection in each of the sides. Here shown with an arrow motif.

237

Looks like the *237 group, except the fundamental region is two copies of the (Pi/2, Pi/3, Pi/7) triangle, and the group acts by rotation at each of the corners. Here one half of the fundamental region is colored blue, and the other half red.

This shows an orbifold with non-trival topology. This is a torus with an order 2 cone point. The two colored arrows show the hyperbolic isometries that correspond to the two translations of the familiar flat Euclidean torus. Can you spot the order two rotation?

iii

This orbifold has three infinite order cone points.

Make some pictures.

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