An Excursion in Diagrammatic Algebra

Turning a Sphere from Red to Blue

The aim of this book is to give as detailed a description as is possible of one of the most beautiful and complicated examples in low-dimensional topology. This example is a gateway to a new idea of higher dimensional algebra in which diagrams replace algebraic expressions and relationships between diagrams represent algebraic relations. The reader may examine the changes in the illustrations in a leisurely fashion; or with scrutiny, the reader will become familiar and develop a facility for these diagrammatic computations.

The text describes the essential topological ideas through metaphors that are experienced in everyday life: shadows, the human form, the intersections between walls, and the creases in a shirt or a pair of trousers. Mathematically informed reader will benefit from the informal introduction of ideas. This volume will also appeal to scientifically literate individuals who appreciate mathematical beauty.

Contents:

• A Sphere
• Surfaces, Folds, and Cusps
• The Inside and Outside
• Dimensions
• Immersed Surfaces
• Movies
• Movie Moves
• Taxonomic Summary
• How Not to Turn the Sphere Inside-Out
• A Physical Metaphor
• Sarah's Thesis
• The Eversion
• The Double Point and Fold Surfaces

Readership: Researchers in mathematics.
Key Features:

• Most sphere eversions that are available are computer illustrations. One cannot stop the animations, look inside, and understand how one got from here to there. The illustrations that the author has given show each step in intricate detail
• This is a book rather than a film. As such, it provides insight and in-depth analyses that cannot be presented in film
• These illustrations are drawn essentially by hand with computer software. They are remarkable for their subtle nature and each illustration is a work of art unto itself