How to Turn a Sphere Inside Out // 215
up as the layers are pulled apart in the two different ways, but the red layer is inside for one way and outside for the other, while the blue layer is outside for one way and inside for the other.
How red and blue layers interchange positions at the halfway stage.
The idea for a specific eversion, then, starts in the middle with an immersed projective plane. Pull it apart one way to create a sphere, with red on the outside and blue on the inside. Then deform that sphere, smoothly, until it looks like a normal round sphere, with only its red surface showing. This may not be easy, and it is not even obvious that it can be done, until you try. However, it works. Now go back to the halfway stage, and pull the projective plane apart the other way, to create a sphere with blue on the outside and red on the inside. Then deform that sphere, smoothly, until it looks like a normal round sphere, with only its blue surface showing. Fit these two deformations together by running the first one backwards. Now a sphere that is red on the outside and blue on the inside gets scrunged around, smoothly, until opposite pairs of points coincide at the midway projective plane. Pass the layers through each other, and pull them apart according to the second deformation. The result is a sphere that is blue on the outside and red on the inside.
Pull the projective plane apart two different ways . . .