Mobius Madness
Consider a Mobius strip whose width (the red line) is extended indefinitely in both directions throughout 3D space. Note that the Mobius surface will intersect itself.
Consider all the regions that this surface would divide 3D space into. How many (disconnected) regions are there? And how many of them have an infinite volume?
Enter your answer as , where is the number of disconnected regions, and is the number of regions which have infinite volume. For example, if you believe there are 4 regions, of which 3 have infinite volume, enter your answer as .
Formally, the Mobius surface as shown in the graphic above is defined as a ruled surface , with a circle as the directrix curve (green), through each point on which (big black point) passes a line (red) perpendicular to the tangent of the circle at that point. As one progresses around the directrix circle, the angle that line makes with the plane of the directrix circle varies by exactly half of the angle around the directrix circle, so that after it has gone around once, the arrow points in the opposite direction, and going around one more time brings the arrow back to the original orientation. We are then extending the red line indefinitely in both directions.
The answer is 2.2.
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1 solution
The above graphics shows an orthographic view down the z axis, showing 5 different cross-sections of this mobius surface near where z = 0 . The cross-section at z = 0 is shown in purple, and consists of the directrix circle and a straight line through it. The other cross-sections in blue and red are just above and below the purple cross-section.
At any cross-section but the purple one, the plane is divided into 3 regions. The purple cross-section is divided into 4 . However, each of the semi-circle regions of the purple cross-section is connected with the "loop" region of one color as well as the region of the other color that has a loop in it. But that region of one color that doesn't have a loop in it is connected to the region of the other color that has a loop in it. So for each color, the "loop" region is connected to the region that doesn't contain the loop. Thus, each of the semi-circle regions of the purple cross-section is connected to half of the entire 3D space, and so the answer is 2.2 , which is 2 regions, 2 that have infinite volume.
All other cross-sections, no matter how far above or below the purple cross-section exhibit the same geometry of a curve with a single loop.