Euclid's mysterious parallel postulate - Jeff Dekofsky

As any current or past geometry student knows, the father of geometry was Euclid, a Greek mathematician who lived in Alexandria, Egypt around 300 BCE.

Euclid is known as the author of a singularly influential work known as Elements.

You think your math book is long? Euclid's Elements is thirteen volumes full of just geometry.

In Elements, Euclid structured and supplemented the work of many mathematicians that came before him, such as Pythagoras, Eudoxus, Hippocrates, and others.

Euclid laid it all out as a logical system of proof built up from a set of definitions, common notions, and his five famous postulates.

Four of these postulates are very simple and straightforward.

Two points determine a line, for example.

The fifth one, however, is the seed that grows our story.

This fifth mysterious postulate is known simply as the parallel postulate.

You see, unlike the first four, the fifth postulate is worded in a very convoluted way.

Euclid's version states that if a line falls on two other lines so that the measure of the two interior angles on the same side of the transversal add up to less than two right angles, then the lines eventually intersect on that side and therefore are not parallel.

Wow, that is a mouthful.

Here's the simpler, more familiar version.

In a plane, through any point not on a given line, only one new line can be drawn that's parallel to the original one.

Many mathematicians over the centuries tried to prove the parallel postulate from the other four, but were unable to do so.

In the process, they began looking at what would happen logically if the fifth postulate were actually not true.

Some of the greatest minds in the history of mathematics asked this question.

People like Ibn al-Haytham, Omar Khayyam, Nasir al-Din al-Tusi, Giovanni Saccheri, János Bolyai, Carl Gauss, and Nikolai Lobachevsky.

They all experimented with negating the parallel postulate, only to discover that this gave rise to entire alternative geometries.

These geometries became collectively known as non-Euclidean geometries.

While we'll leave the details of these different geometries for another lesson, the main difference depends on the curvature of the surface upon which the lines are constructed.

Turns out that Euclid did not tell us the entire story in Elements.

He merely described one possible way to look at the universe.

It all depends on the context of what you're looking at.

Flat surfaces behave one way, while both positively and negatively curved surfaces display very different characteristics.