Updated 7/6/01


Home
The Project
Constructs
Transforms
References

Email Steve
Email Ethan
Email Dr. Gay

The Basics


Connected Sum

The connected sum operation allows us to create a (possibly) new surface out of two surfaces we already know. To make a connected sum, we first cut a hole into each of our surfaces, forming a single closed edge on each one. Stretching these edges towards each other, we then line them up and then join the surfaces along the edges.

Connected Sum of two Spheres

The connected sum of two spheres actually yields a single sphere at the end.

Striped
Striped Sum of Two Spheres
Media Player
Quicktime
Wooden
Wooden Sum of Two Spheres
Media Player
Quicktime

Elementary Moves

As a large portion of this site is dedicated to transforming surfaces from one representation to another, we need first to understand what we are "allowed" to do. In 1985, Tatsuo Homma and Teruo Nagase classified six of these deformations, and in 1998, Dennis Roseman produced seven similar deformations (5 of which were identical to the Homma-Nagase ones). The following movies demonstrate Roseman's seven moves.


Type I Move

This first move is simply pushing one surface through a single other surface.

Type I
Media Player
Quicktime

Type II Move

In this move we push a single surface through a "double curve", the intersection of two other surfaces.

Type I
Media Player
Quicktime

Type III Move

This move pushes a surface through a "triple point", where three surfaces intersect.

Type I
Media Player
Quicktime

Type IV Move

In this move, we push the high points of a surface with a saddle through a single other surface.

Type I
Media Player
Quicktime

Type V Move

This move adds a double curve to a surface by adding a twist to the surface (in some sense, pushing part of the surface through itself)..

Type I
Media Player
Quicktime

Type VI Move

In this move, we take a surface that has a twist in it, and "untwist" the middle portion of this twist. It may be helpful to think of it as pulling the sides of the surface (but only in the middle region) until the twist tightens down to nothing.

Type I
Media Player
Quicktime

Type VII Move

This move simply extends a pinch through a single other surface.

Type I
Media Player
Quicktime