Adding the Triple Point:
The transition from level 3 to level 4 is one of the hardest to visualize. What happens is that the top part of the lower loop is pulled across the top of the upper loop, as shown in the diagram below:
As the red segment passes through the point of intersection formed by the upper loop, we have three sheets of the surface passing through a single point, thus forming the triple point that is required in any immersion of the projective plane [More].

This sequence in the polyhedral case is shown below:

[Right] Level 5: The critical level
[Left] Level 3: The initial self-intersection
[Left] Comparison to Kuiper's level sets
[Up] Other pictures

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