Table of Figures:

The following entries will display the picture using an extern GIF viewer. The arrow links (indicated by "[More]") will take you to the page where the figure is used.

Movies: [More]

  1. Levels sets of the polyhedral surface (60K)
  2. Building up the polyhedral surface (200K)
  3. The polyhedral surface rotating (340K)

Interactive Pictures: [More]

  1. The complete polyhedral surface
  2. The central projective plane

Geomview objects: [More]

  1. The complete tight surface
  2. The central projective plane
  3. The complete surface with transparent convex envelope

Views of the polyhedral object: [More]

  1. The complete surface from above
  2. Front right corner, from above
  3. Front right corner, from the side
  4. Right-hand side
  5. Back right corner
  6. Front left corner, from above
Level sets: [More]
  1. Levels for the polyhedral object: [More]
    1. The initial circle
    2. Pulling one side across the other
    3. Adding the triple point
    4. The triple point
    5. Completely across the loop, moving toward a saddle
    6. Just before the saddle
    7. The critical level
    8. Just after the saddle
    9. Pulling back toward the edge
    10. The final circle

  2. Levels drawn by Kuiper:
    1. All levels
    2. The initial circle
    3. Pulling one side across the other
    4. Adding the triple point
    5. Just before the saddle
    6. The critical level
    7. Just after the saddle
    8. Pulling back toward the edge
    9. The final circle

Templates for building your own model: [More]

(not yet ready)
Pictures from Kuiper's Paper [K2]:
  1. The level sets for an immersed projective plane [More] [More]
  2. Ruled surfaces to connect the levels [More]
  3. A tight two-handled torus [More]
  4. A tight Klein bottle with a handle [More]


  1. Tight polyhedral torus, two-handled torus, and Klein bottle with a handle [More]
  2. How the triple point is formed within the level sets [More] [More]
  3. Polyhedral analogues of cusps and folds [More]
  4. A Klein bottle [More]
  5. The M+ and M- regions for a Klein bottle [More]
  6. A Möbius band [More]
  7. An exotic polyhedral monkey saddle [More]
  8. Adding a polyhedral handle tightly [More] [More]
  9. Some polyhedral surfaces that are not tight [More]
  10. The neighborhood of a non-bounding embedded curve on the projective plane [More]
  11. A tight torus and its decomposition into the M+ and M- regions [More]
  12. The triangulation of the projective plane that forms the tight model, together with its double curve [More] [More]
  13. An immersed, but not embedded, topset [More]

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8/13/94 -- The Geometry Center