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Extra resources for Chern - A Great Geometer of the Twentieth Century
This procedure can be continued indefinitely. The golden section was an important discovery to the Pythagoreans. They used the star as their own special symbol, but they had no monopoly on the ratio. Greek architects incorporated the golden section in the proportions of the buildings that they designed. It is present in the proportions used in Greek art, and the golden section can be found throughout nature as well (see the sidebar The Golden Section). Many remarkable properties of this ratio have been uncovered during the last few millennia.
That number is represented by the Greek letter π, and in modern notation the result is written as A = πr 2, where A represents the area of the circle and the letter r represents its radius. 1), the number π is absent. This is just another indication that the Greeks investigated geometry in a way that was very different from the way that we do. Here is how the method of exhaustion is applied. 1) is false. Then it must either be true that A1 / A2 is greater than D21 / D22 or A1 / A2 is less than D21 / D22.
Axioms and postulates are like the 26 GEOMETRY rules of the game. If we change them, we change the geometry itself. They are the ultimate answer to the question, Why is this true? Any true statement in Euclidean geometry is true because in the end it is a consequence of one or more of Euclid’s axioms and postulates. Any set of axioms and postulates must meet certain criteria. First, the axioms cannot contradict one another; otherwise, we eventually uncover a statement that can be proved both true and false.