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Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.
How nearly equal an odd number of triangles can be
A square cannot be cut into an odd number of triangles of equal area. It can be cut into five triangles whose areas differ by about two hundredths, seven that differ by three thousandths, nine by a ten-thousandth and a half — the spread falling by a factor of seven or more with every two triangles added, closing on equality and never reaching it. A search for the closest finds that the best five it can make have the golden ratio in their areas.
The largest polygon of diameter one
Among pentagons whose points are never more than one unit apart, the regular pentagon has the most area, and the same is true for every odd number of sides. For six sides it is false: a lopsided hexagon found in 1975 beats the regular one by four per cent. Symmetry wins half the cases and loses the other half, and the reason is parity.
Named alongside it
The objects these essays reach for when they reach for this one.
OptimisationAreaConstant widthDiameterDissectionExtremal problemGolden ratioIsodiametric inequalityMonsky theoremRegular polygonSymmetryTriangulation