Euclid's puzzling parallel postulate - Jeff Dekofsky

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Euclid, known as the "Father of Geometry," developed several of modern geometry's most enduring theorems--but what can we make of his mysterious fifth postulate, the parallel postulate? Jeff Dekofsky shows us how mathematical minds have put the postulate to the test and led to larger questions of how we understand mathematical principles.

Lesson by Jeff Dekofsky, animation by The Leading Sheep Studios.

As any current or past geometry student knows,

the father of geometry was Euclid,

a Greek mathematician who lived in Alexandria, Egypt

around 300 B.C.E.

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 13 volumes filled 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 weren't able 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 ask this question,

people like Ibn al-Haytham,

Omar Khayyam,

Nasir al-Din al-Tusi,

Giovanni Saccheri,

Janos 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.

Well, 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 positively and negatively curved surfaces

display very different characteristics.

At first these alternative geometries seemed a bit strange

but were soon found to be equally adept

at describing the world around us.

Navigating our planet requires elliptical geometry

while the much of the art of M.C. Escher

displays hyperbolic geometry.

Albert Einstein used non-Euclidean geometry as well

to describe the way that space time

becomes work in the presence of matter

as part of his General Theory of Relativity.

The big mystery here is whether or not Euclid

had any inkling of the existence of these different geometries

when he wrote his mysterious postulate.

We may never know the answer to this question,

but it seem hard to believe

that he had no idea whatsoever of their nature,

being the great intellect that he was

and understanding the field as thoroughly as he did.

Maybe he did know

and intentionally wrote the Parallel Postulate in such a way

as to leave curious minds after him

to flush out the details.

If so, he's probably quite pleased.

These discoveries could never have been made

without gifted, progressive thinkers

who are able to suspend their preconceived notions

and think outside of what they have been taught.

We, too, must be willing at times

to put aside our preconceived notions and physical experiences

and look at the larger picture,

or we risk not seeing the rest of the story.