Shadows from Higher Dimensions

  • George K. Francis


A hemispherical bowl has a single, circular edge. So does a disc. If you sew the disc to the hemisphere you obtain a closed, two-sided surface; it has an inside and an outside. Topologists still call this a sphere, and it could be inflated back to a geometrical sphere if you like. Now take a long, narrow strip of paper, give it a half twist and glue the ends together. The resulting surface is not only one sided, it has but a single, closed edge. What happens if this edge is sewn to the rim of a disc?


Projective Plane Triple Point Elliptical Generation Klein Bottle Picture Plane 


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Copyright information

© Springer Science+Business Media New York 2007

Authors and Affiliations

  • George K. Francis
    • 1
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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