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

2. 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

3. 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

4. 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.

5. 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

6. 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?

7. 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

8. 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