Like the cylinder , it is not a true surface, but rather a surface with boundary Henle , p. According to Madachy , the B. Trott, pers. The coefficients of the first fundamental form for this surface are. Note that although the surface closes at , this corresponds to the bottom edge connecting with the top edge, as illustrated above, so an additional must be traversed to comprise the entire arc length of the bounding edge. In addition, two strips on top of each other, each with a half-twist, give a single strip with four twists when disentangled. However, there are three surfaces that are representations of the projective plane in with self-intersections, namely the Boy surface , cross-cap , and Roman surface. Ball, W. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp.
Have You Ever Wondered...
This space exhibits interesting properties, such as having only one side and remaining in one piece when split down the middle. See also Klein bottle. Article Media. Info Print Cite. Submit Feedback. Thank you for your feedback. Home Science Mathematics.
It can be realized as a ruled surface. For example, any rectangle can be glued left-edge to right-edge with a reversal of orientation. Some, but not all, of these can be smoothly modeled as surfaces in Euclidean space. Such paper models are developable surfaces having zero Gaussian curvature , and can be described by differential-algebraic equations. A line drawn along the edge travels in a full circle to a point opposite the starting point. If continued, the line returns to the starting point, and is double the length of the original strip: this single continuous curve traverses the entire boundary. This happens because the original strip only has one edge, twice as long as the original strip. Cutting creates a second independent edge of the same length, half on each side of the scissors. Cutting this new, longer, strip down the middle creates two strips wound around each other, each with two full twists. The other is a thin strip with two full twists, a neighborhood of the edge of the original strip, with twice the length of the original strip.
You have most likely encountered one-sided objects hundreds of times in your daily life — like the universal symbol for recycling, found printed on the backs of aluminum cans and plastic bottles. This mathematical object is called a Mobius strip. Another mathematician named Listing actually described it a few months earlier, but did not publish his work until The concept of a one-sided object inspired artists like Dutch graphic designer M. For instance, try taking a pair of scissors and cutting the strip in half along the line you just drew.