Shortest Enclosures on Surface of Cube (Math Chat)
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Frank Morgan, MAA Online  
The circle provides the leastperimeter way to enclose given area in the plane. What is the leastperimeter way to enclose given area in the surface of the unit cube? Answer: Al Zimmerman correctly finds that the best way to enclose small area is not a small circle on one of the sides of the cube, but better a "circle" around one of the corners... For intermediate areas, he proposes a rectangle about one of the edges, but it turns out that you can do better than that, using circular arcs enclosing two or three of the corners...  


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