Applet: A Möbius strip is not orientable
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A Möbius strip is shown with a normal vector. If you drag the red point on the slider, the normal vector moves along the Möbius strip. Once you drag the slider half way, the normal vector has moved all the way around the surface. However, now the normal vector is pointing in the opposite direction. Since the normal vector didn't switch sides of the surface, you can see that Möbius strip actually has only one side. For this reason, the Möbius strip is not orientable.
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The applet was made using three.js and requires Javascript as well as a browser that supports WebGL. For most three.js applets, you can drag with the mouse to rotate the view, drag with the right button to pan, and zoom in/out with the mouse wheel. Many applets contain points that you can drag to change values of variables.