Question

What is considered the most famous impossible problem from Greek antiquity?

Answer 1

The most famous impossible problem from Greek Antiquity is doubling the cube. The problem is to construct a cube whose volume is double that of a given one. It is often denoted to as the Delian problem due to a myth that the Delians had look up Plato on the subject. In another form, the story proclaims that the Athenians in 430 B.C. consulted the oracle at Delos in the hope to break the plague devastating their country. They were advised by Apollo to double his altar that had the form of a cube. As an effect of several failed attempts to satisfy the god, the plague only got worse and at the end they turned to Plato for advice. (According to Rouse Ball and Coxeter, p 340, an Arab variant asserts that the plague had wrecked between the children of Israel but the name of Apollo had been discreetly gone astray.) According to a message from the mathematician Eratosthenes to King Ptolemy of Egypt, Euripides mentioned the Delian problem in one of his (now lost) tragedies. The other three antiquity are: angle trisection, squaring a circle, and constructing a regular heptagon. 

Answer 2

For apex, the anwer is "Doubling the cube"