By Pacella F., Weth T.

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**Extra resources for Symmetry of solutions to semilinear elliptic equations via Morse index**

**Example text**

DE cannot then be the unit. For if DE was the unit and is to DF in the same proportion as AC to AB, AC being greater than AB, DE, the unit, will be greater than the integer DF, which is impossible. Hence DE is not the unit, but an integer (greater than the unit). Now since AC : AB = DE : DF, it follows that also AC2 : AB2 = DE2 : DF2 . But AC2 = 2AB2 and hence DE2 = 2DF2 . Hence DE2 is an even number and therefore DE must also be an even number. For, if it was an odd number, its square would also be an odd number.

7. 8. 8. Once again we can force a contradiction in much the same manner as with the square. This time locate the line AED as a diagonal of the pentagon and suppose that the diagonal AD and the side AC of the large pentagon are commensurable. We have AD − AC = AD − AE = ED = AB and AB = BC = BF, with BF a diagonal of the inner pentagon. Since AD and AD − AE = AB = BF are commensurable, AD, the diagonal of the original pentagon must be commensurable with BF, the diagonal of the smaller, nested pentagon.

For example, if we look to the original Oxford English Dictionary’s 1908 entry for Pythagorean we see its famous editor, James Murray, allowing the capital Greek letter upsilon (Υ ) to be their representation of the two divergent paths of virtue and vice. The equally prestigious American initiative, the Century Dictionary, has the 1906 entry Hexagram attaching the regular hexagon and its associated hexagram to Pythagorean mysticism. Modern versions of these publications and others like them are consistent with their illustrious predecessors.