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November 23, 2003
More Cubic Bisection
The other day I asked:
I know you've been breathlessly waiting for the answers, so here you go. First, no, a cube cannot be sectioned to create a regular pentagon. The closest you can do is the figure shown above. This is a "full house" pentagon; three of the sides are the same length, and the other two sides are the same length as each other, but longer than the other three. {Note: it is not necessary that one of the pentagon's vertices be coincident with a vertex of the cube.} Second, the regular hexagon is not the section with the greatest area. I didn't mean for this to be a trick question, but I guess it was. The section with the greatest area is this one:
Here's the regular hexagon again:
There are some other candidates as well. In the two figures above, consider rotating the section about the dashed line as an axis. That yields the following section (a diamond, not a square):
And continuing the rotation, this section, a square with the minimum area of any section which passes through the center of the cube:
Another interesting section is this one, the largest triangular section:
Finally, here's today's bonus question:
Posted by ole at
06:20 PM


