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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.

  • 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.

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