Comparison to Kuiper's Level Sets:

In Kuiper's original paper [K2], he drew the following sequence of levels to describe an immersion of the real projective plane in three-space:
The figure has three critical points (the minimum possible), indicated by the arrows. Kuiper used this picture to show how to build a tight model of a smooth projective plane with two handles [More], but was not able to produce a tight version with only one handle.

The core of the tight polyhedral model presented here [More] follows Kuiper's original drawings quite closely. Starting with this initial projective plane, the top and bottom are cut off and a tube added around the outside of the object to complete the tight immersion of the projective plane with one handle.

The following pages describe the relationship between Kuiper's drawings and the polyhedral level sets in more detail:

[Right] Level 1: The initial circle.
[Right] Level 2: Pulling one side across.
[Right] Level 3: The initial self-intersection.
[Right] Level 4: Adding the triple point.
[Right] Level 5: The critical level.
[Right] Level 6: After the critical level.
[Right] Level 7: No more intersection.
[Right] Level 8: The final circle.

[Left] Other pictures
[Up] The polyhedral solution

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