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Topology is useful for telling the difference between, say, a doughnut and a coffee cup. Or not. (Depending how many holes the doughnut has. Or how many handles on the coffee cup.)

Edited by Strange

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Early in the morning it can be difficult to tell the difference between your coffee cup and your doughnut- but it is very important to avoid pouring hot coffee over yourself!  So topology is important.

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The Moebius Flask, which can have a liquid poured in, but cannot contain it as it pours out. The three-dimensional surface having volume but no inside or outside defined surface.

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During the year I taught high school math in very rural Missouri, I calmed down several boisterous classes by predicting the results of cutting two strips of paper apart, which I explained would totally befuddle them. They listened, observed, disbelieved, then owed they would spring this on friends and family! The Moebius Strip.

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A strip of thin material, rolled into a cylindrical form, must present height, diameter, and volume. The strip if twisted first, such that it's inner and outer edges are reversed, when rolled into a cylinder, become a moebius solid having no inside or outside, nor presenting a viable volume.   imp

 

EDIT: Even more befuddling is the process of taking a moebius strip prepared of paper of reasonable size, about a foot in length, which is then pierced anywhere at it's center between edges, then cut with scissors completely around it's length to produce two (?) pieces. Worse, the strip pierced 1/3 of it's width away from one edge, then cut lengthwise continuously until separated into two (?)  pieces becomes.....?? Two new volumes, one a cylinder, the other a new moebius strip having TWO twists, AND they are of differing diameters!  Try it!   imp

Edited by tinkerer

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