An engineer noted for his ability to visualize three-dimensional structure was having coffee and doughnuts. Before he dropped a sugar cube into his cup, he placed the cube on the table and thought, “If I pass a horizontal plane through the cube’s center, the cross section will, of course, be a square. If I pass it vertically through the center and four corners of the cube, the cross section will be an oblong rectangle. Now suppose I cut the cube this way with the plane.”
To his surprise, his mental image of the cross section was a regular hexagon. How was the slice made? If the cube’s side is half an inch, what is the side of the hexagon?
After dropping the cube into his coffee, the engineer turned his attention to a doughnut lying flat on a plate. “If I pass a plane horizontally through the center,” he said to himself, “the cross section will be two concentric circles. If I pass the plane vertically through the center, the section will be two circles separated by the width of the hole. But if I turn the plane so....” He whistled with astonishment. The section consisted of two perfect circles that intersected!
How was this slice made? If the doughnut is a perfect torus, three inches in outside diameter and with a hole one inch across, what are the diameters of the intersecting circles?
A cube cut in half by a plane that passes through the midpoints of six sides, as shown in the illustration below, produces a cross section that is a regular hexagon. If the cube is half an inch on each side, the side of the hexagon is √2⁄4 inch.
Amanda Montañez
(This hexagon, by the way, furnishes the solution to an interesting puzzle by Lewis Carroll that is cited by mathematician Warren Weaver in his article “The Mathematical Manuscripts of Lewis Carroll,” which was published in Proceedings of the American Philosophical Society on October 15, 1954. Can a billiard ball travel inside a cube in such a way that it touches all six sides and continues forever on the same path, all portions of the path being equal? The solution is a path joining the midpoints of the sides of the hexagon.)
To cut a torus so that the cross section consists of two intersecting circles, the plane must pass through the center and be tangent to the torus above and below, as shown in the illustration below. If the torus and hole have diameters of three inches and one inch, respectively, each circle of the section will clearly have a diameter of two inches.
Amanda Montañez
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A version of this puzzle originally appeared in the October 1960 issue of Scientific American.