Boy's surface

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Animated cutaway and annular section views of Boy's surface[edit]

In the animation above, a small disk is cut out from Boy's surface and expands until the boudary nearly doubles upon itself, leaving a Möbius band with one an a half twists (the boundary is a trefoil knot). The animation then reverses itself and repeats in a loop.

In the next view (to the right) the same surface is presented with animated successive annular sections flowing along Boy's surface.

Looping animated cutaway view of Boy's surface.
Looping animated annular section view.

Views of Boy's surface from different directions[edit]

Sections of the Boy's surface[edit]

The Boy's surface can be cut into six sections. Let them be called A, B, C, D, E, and F. Then sections A, C, and E are mutually congruent, and sections B, D, and F are mutually congruent.

These six sections arrange themselves into a circle, or rather a hexagon: each section corresponding to one side of the hexagon. The sections are arranged in this order: A, B, C, D, E, F -- counterclockwise around the hexagon. Each section has three sides which have been shown as orange, green, and blue. Each section can be converted through a homotopy into a triangle. The colors of the edges show how the sections are supposed to fit together. Only sides of the same color are allowed to coincide.

Section A′s green edge matches section D′s, B′s green edge with E, C′s green edge with F. Thus, opposite sides of the hexagon match through the green sides.

Notice that if A is rotated counterclockwise by 120°, it looks the same as C, and if it is rotated further another 120° then it looks the same as E. A similar case holds for B, D, and F.

Pathways on a Boy's surface[edit]

Let a "topological ant" start out walking from the bottom of the Boy's surface (shown in Figure 12), on the outside. Let this ant walk along the green path into a cave entrance. This cave entrance is located under an archway which is like one of the tentacles of an octopus.