Pythagorean theorem straightedge and compass. Pythagoras's theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is
Origins and architecture of the Taj Mahal organisational ones. The smaller grids were also used to establish elevational proportion throughout the complex. Koch and Barraud explain such apparently peculiar numbers as making more sense when seen as part of Mughal geometric understanding. Octagons and triangles, which feature extensively in the Taj, have particular properties in terms of the relationships of their sides. A right-angled triangle with two sides of 12 will have a hypotenuse of approximately 17 (16.97+); similarly if it has two sides of 17 its hypotenuse will be approximately 24 (24.04+). An octagon with a width of 17 will have sides of approximately 7 (7.04+),