Duplication, Trisection and Quadrature

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Presentation transcript:

Duplication, Trisection and Quadrature The three famous problems of greek antiquity

Some History of Greek Mathematics 1200 B.C. Dorians moved southward into Greek peninsula Dorians – major division of Greeks with 3 tribes Chief Tribe – Spartans Invaded region - inhabitants moved to Asia minor and Ionian islands Ionian School was formed 546 B.C. Persia conquered Asia Minor and Ionian Islands Greeks living there fled to South Italy Schools of Philosophy and Math Developed Crotona – Pythagoras Elea – Xenophanes, Zeno and Parmenides

More War and Invasions 499 B.C. Athens helped Ionians revolt against Persians - SUCCESS 492 B.C. Persian King tried to get Ionian cities back – FAILURE Now Athens became the Greek Leaders 480 B.C. Xerxes from Persia tried again – and FAILED (Battle of Thermopylae) 50 years of peace and intellectualism

Peace came to and End 431 B.C. Peloponnesian War between Athens and Sparta Long, Drawn out Athens was successful .. at first Plague wiped out ¼ of population 404 B.C. Athens accepted defeat. 371 B.C Sparta was overthrown by rebellion This was the era of Socrates, Plato and Eudoxus.

Plato 427 B.C. Born in Athens 387 B.C. Returned to Athens Studied philosophy under Socrates Wandered around African Coast and Italy 387 B.C. Returned to Athens Founded Academy – pursuit of philosophy and scientific inquiry 347 B.C. Death at age of 80, was still teaching Many other schools rose around Greece from students of Plato, and mathematics evolved rapidly

Problem 1 – Duplication of Cube Given a cube of volume V, can one construct (using only compass and straightedge) a cube of volume 2V? Using ILLEGAL marked straightedges, one can easily construct cube root of 2. Why does the picture above work?

Trisection of an Angle Given an angle, we know we can bisect it using only compass and straightedge. Can we trisect it using only compass and straightedge? Wantzel proved this algebraically impossible in 1836

Quadrature of a Circle Given a circle of area, A, can we construct (using only compass and straightedge) a square of area A. Lindemann – Weierstrass proved this impossible in 1882 by showing pi is transcendental. Hippocrates Quadrature of the Lune gave false hope. The shaded area equal to the area of triangle ABC