The Altar of Apollo: The Great Geometric Challenge

According to legend, the Greek city of Delos was stricken with a horrible plague sent by the god Apollo. Desperate the citizens went to the oracle of Delphi for advice. Calmly the oracle told them that in order to appease Apollo and end the plague was to simply double the volume of his altar.

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A mosaic of Apollo, the Greek god of health and music, he was also said to move the sun across the sky in his chariot.

The old altar was a cube. The citizens quickly crafted a cube with twice the dimensions but the plague didn’t subside. It turns out the citizens made the cube eight times larger than the original size, not two. Puzzled they gathered their best mathematicians and philosophers to tackle the problem, unfortunately they were defeated.

The ancient Greeks did geometry by constructing shapes and objects using a ruler and a compass. For example an equilateral triangle can be easily created with a straight edged ruler and a compass, increasing the volume of that triangle by four can also be easily done. Many great geometry problems were solved with only these two tools however finding a way to double the volume of a cube wasn’t possible.

Construction of an equilateral triangle and an equilateral triangle four times the area using only a straight edge and a compass
Construction of an equilateral triangle and an equilateral triangle four times the area using only a straight edge and a compass

In order to double volume of a cube the dimensions of the cube would have to be multiplied by \sqrt[3]{2}. Thus the Greeks would have to be able to construct the value with a straight edge and compass. Natural and rational numbers were easy to construct but irrational numbers took a lot more work and thought. Unfortunately for the citizens of Delos, it is impossible to construct the cube root of 2 with only a compass and a straight edge.

Plato, one of the founders of western philosophy believed that mathematics was an essential part in philosophical education
Plato, one of the founders of western philosophy believed that mathematics was an essential part in philosophical education

The Delos Problem is in fact only one of three geometric challenges that are impossible to solve using only those two tools. The other two are: construct a square with the same area as a circle (“squaring the circle”) and dividing and angle into three equal angles (“Trisecting the Angle”). The great Philosopher Plato was known to have challenged his students to solve these problems, none of them could even provide him a solution.

These problems where finally resolved in the 19th century was mathematicians with more sophistical tools and resource showed that it these problems were impossible to be solved using only a straight edge and a compass.

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