Mathematical Ideas that Shaped the World

Non-Euclidean geometry

Plan for this class

Who was Euclid? What did he do? Find out how your teachers lied to you Can parallel lines ever meet? Why did the answer to this question

change the philosophy of centuries? Can you imagine a world in which there

is no left and right? What shape is our universe?

Your teachers lied to you!

Your teachers lied to you about at least one of the following statements. The sum of the angles in a triangle is 180

degrees. The ratio of the circumference to the diameter

of a circle is always π. Pythagoras’ Theorem Given a line L and a point P not on the line,

there is precisely one line through P in the plane determined by L and P that does not intersect L.

Euclid

Born in about 300BC, though little is known about his life.

Wrote a book called the Elements, which was the most comprehensive book on geometry for about 2000 years.

One of the first to use rigorous mathematical proofs, and to do ‘pure’ mathematics.

The Elements

A treatise of 13 books covering geometry and number theory.

The most influential book ever written? Second only to the Bible in the number of

editions published. (Over 1000!) Was a part of a university curriculum until the

20th century, when it started being taught in schools.

Partly a collection of earlier work, including Pythagoras, Eudoxus, Hippocrates and Plato.

Definitions

A straight line segment is the shortest path between two points.

Two lines are called parallel if they never meet.

The axioms of geometry The Elements starts with a set of

axioms from which all other results are derived.

1. A straight line segment can be drawn joining any two points.

The axioms of geometry

2. Any straight line segment can be extended indefinitely in a straight line.

The axioms of geometry

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.

The axioms of geometry

4. All right angles are congruent.

The axioms of geometry

5. If a straight line N intersects two straight lines L and M, and if the interior angles on one side of N add up to less than 180 degrees, then the lines L and M intersect on that side of N.

L

M

N

Axiom 5

Axiom 5 is equivalent to the statement that, given a line L and a point P not on the line, there is a unique line through P parallel to L.

This is usually called the parallel postulate.

L

P

Angles in a triangle

From Axiom 5 we can deduce that the angles in a triangle add up to 180 degrees.

Hidden infinities

People were not comfortable with Axiom 5, including Euclid himself.

There was somehow an infinity lurking in the statement: to check if two lines were parallel, you had to look infinitely far along them to see if they ever met.

Could this axiom be deduced from the earlier, simpler, axioms?

Constructing parallel lines

For example, we could construct a line parallel to a given line L by joining all points on the same side of L at a certain distance.

Problem

The problem is: how do we prove that the line we have constructed is a straight line?

Could parallel lines not exist?

Ever since Euclid wrote his Elements, people tried to prove the existence and uniqueness of parallel lines.

They all failed. But if it was impossible to prove that

Axiom 5 was true, could it therefore be possible to find a situation in which it was false?

Exhibit 1: The Earth

Can we make Euclid’s axioms work on a sphere?

Shortest distances?

Question: What is a ‘straight line’ on Earth?

Answer: The shortest distance between two points is the arc of a great circle.

What is a great circle?

A great circle’s centre must be the same as that of the sphere.

Not a straight

line

Shortest distances on a map

If we take a map of the world and draw straight lines with a ruler, these are not the shortest distances between points.

The curve depends on the distance But straight lines on a map are a good

approximation to the shortest distance if you aren’t travelling far.

The further you travel, the more ‘curved’ the path you will travel.

Axiom 5 on a sphere

Amazing fact: there are no parallel lines on a sphere.

Proof: all great circles intersect, so no two of them can be parallel.

Why does our construction fail? Studying the sphere shows why our

previous attempt to construct parallel lines went wrong.

If we take all points equidistant from a great circle, the resulting line is a small circle and is thus not ‘straight’.

Angles in a triangle

But if the parallel axiom fails, then what about angles in a triangle?

It turns out that if you draw triangles on a sphere, the angles will always add up to more than 180 degrees.

A triangle with 3 right angles! For example, draw a triangle with

angles of 270 degrees by starting at the North Pole, going down to the equator, walking a quarter of the way round the equator, then back to the North Pole.

Even the value of changes!

The ratio of the circumference of a circle to its diameter is no longer fixed at 3.14159…

It is always less than ‘’ and varies with every different circle drawn on the sphere.

= 2

= 2.8284…

Do spheres contradict Euclid then? Geometry on a sphere clearly violates

Euclid’s 5th axiom. But people were not entirely satisfied

with this counterexample, since spherical geometry also didn’t satisfy axioms 2 and 3.

(That is, straight lines cannot be extended indefinitely, and circles cannot be drawn with any radius.)

What if axiom 5 were not true? Is it possible to construct a kind of

geometry that does satisfy Euclid’s axioms 1-4 and only contradicts axiom 5?

For a long time, people didn’t even think to try.

And when they did try, they were unable to overcome the force of their intuition.

What if axiom 5 were not true? The Italian mathematician Saccheri was

unable to find a contradiction when he assumed the parallel postulate to be false.

Yet he rejected his own logic, sayingit is repugnant to the nature of straight

lines

Kant’s philosophy

In 1781 the philosopher Immanuel Kant wrote his Critique of Pure Reason.

In it, Euclidean geometry was held up as a shining example of a priori knowledge.

That is, it does not come from experience of the natural world.

The players in our story

Some people were willing to change the status quo, or at least to think about it…

Carl Friedrich Gauss Farkas Bolyai János Bolyai Nikolai Ivanovich Lobachevsky

Carl Friedrich Gauss (1777 – 1855) Born in Braunschweig,

Germany, to poor working class parents.

A child prodigy, completing his magnum opus by the age of 21.

Often made discoveries years before his contemporaries but didn’t publish because he was too much of a perfectionist.

Father and son

Gauss discussed the theory of parallels with his friend, Farkas Bolyai, a Hungarian mathematician, who tried in vain to prove Axiom 5.

Farkas in turn taught his son János about the theory of parallels, but warned him

not to waste one hour's time on that problem

He went on…

An imploring letter

"I know this way to the very end. I have traversed this bottomless night, which

extinguished all light and joy in my life… It can deprive you of your leisure, your health, your peace of mind, and your

entire happiness… I turned back when I saw that no man can reach the bottom of

this night. I turned back unconsoled, pitying myself and all mankind. Learn from

my example…"

An imploring letter

“For God's sake, please give it up. Fear it no less than the sensual

passion, because it, too, may take up all your time and deprive you of

your health, peace of mind and happiness in life.”

An imploring letter

His son ignored him.

János Bolyai (1802 – 1860)

Born in Kolozsvár (Cluj), Transylvania.

Could speak 9 languages and play the violin.

Mastered calculus by the age of 13 and became obsessed with the parallel postulate.

János Bolyai

In 1823 he wrote to his father sayingI have discovered things so wonderful that I was astounded ... out of nothing I

have created a strange new world. His work was published in an appendix

to a book written by his father.

A reply from Gauss

Bolyai was excited to tell the great mathematician Gauss about his discoveries.

Imagine his dismay, then, at receiving the following reply:

To praise it would amount to praising myself. For the entire content of the work ...

coincides almost exactly with my own meditations which have occupied my mind

for the past thirty or thirty-five years .

At the same time, there was yet another rival to the claim of the

first non-Euclidean geometry….

Lobachevsky (1792 – 1856) Born in Nizhny

Novgorod, Russia. Was said to have had

18 children. Was the first person to

officially publish work on non-Euclidean geometry.

Some people have accused him of stealing ideas from Gauss, but there is no evidence for this.

Hyperbolic geometry

In hyperbolic geometry, there are many lines parallel to a given line and going through a given point.

In fact, ‘parallel’ lines diverge from one another.

Angles in triangles add up to less than 180 degrees.

is bigger than 3.14159…

The Poincaré disk model

Distances in a hyperbolic circle get larger the closer you are to the edge.

Imagine a field which gets more muddy at the boundary.

A straight line segment is one which meets the boundary at right angles.

Hyperbolic geometry in real life Although hyperbolic geometry was

completely invented by pure mathematicians, we now find it crops up surprisingly often in the real world…

Plants

Mushrooms

Coral reefs

The hyperbolic crochet coral reef

Brains

Art (especially Escher!)

Your teachers lied to you!

Your teachers lied to you about at least one of the following statements. Which one(s)? The sum of the angles in a triangle is 180

degrees. The ratio of the circumference to the diameter

of a circle is always π. Pythagoras’ Theorem Given a line L and a point P not on the line,

there is precisely one line through P in the plane determined by L and P that does not intersect L.

Your teachers lied to you!

Answer: all of them can sometimes be false!

It depends on the space that you are drawing lines on.

Any space in which these statements are false is called a non-Euclidean geometry.

Summary

Parallel lines do different things in different geometries: In flat space, there are unique parallel lines In a spherical geometry, there are no parallel

lines In a hyperbolic geometry, there are infinitely

many lines parallel to a given line going through a particular point.

Euclid’s 5th axiom of geometry is not always true

Reaction of the philosophers

Mathematics, and in particular Euclid, had always been examples of perfect truth.

The concept of mathematical proof meant that we could know things absolutely.

The fact that Euclid had got one of his basic axioms ‘wrong’ meant that a large part of philosophy needed to be re-written.

Was there such a thing as absolute truth?

Another dimension

All the geometry we’ve looked at so far has been in 2 dimensions – what happens in 3D?

There are also 3 kinds of geometry: Flat (180 degree triangles) Spherical (> 180 degree triangles) Hyperbolic (< 180 degree triangles)

What shape is our universe? What would make space curvy?

The theory of relativity

In 1917 Albert Einstein published his theory of general relativity.

Einstein’s theory would not have been possible without non-Euclidean geometry.

Relativity tells us that spacetime gets ‘curved’ by matter, and this is what causes the effect of gravity.

Light gets bent and time slows down in the gravitational influence of large objects.

Matter changes geometry The more matter there is in the

universe, the more curved it will be. The symbol is a measure of the

density of matter in the universe. If =1 then space is flat. (No curvature) If >1 then space is spherical. (Positive

curvature) If <1 then space is hyperbolic. (Negative

curvature)

Why does it matter?

Our universe is currently expanding. The shape of our universe will

determine its eventual fate. If space is spherical, it will eventually stop

expanding, and contract again in a ‘big crunch’.

If space is hyperbolic, the expansion will continue to get faster, resulting in the ‘big freeze’ or ‘big rip’.

If space is flat, the expansion will gradually slow down to a fixed rate.

Current estimates

Our current best guess is that is about 1, so that space is flat.

But finding relies on being able to tell how much dark matter there is.

New question: if space is flat, what does it look like?

Topology

To answer the question about what space looks like, we will need some topology.

But this area of maths is not concerned with measuring distances on objects.

Topology is about the properties of a shape that get preserved when a shape is wiggled and stretched like a sheet of rubber. =

Wiggling and stretching

For example: How many holes does the object have? Does the object have an edge? How many ‘sides’ does the object have? How does the object sit inside another

object? Can you get from one point to any other

point?

Examples

A square has no holes and 1 edge

A sphere has no holes and no edges

Examples

A cylinder has 1 hole and 2 edges

A torus has 1 hole and no edges

2-dimensional planets

Imagine you are an ant living on some kind of surface.

How would you decide what the surface was?

Answer: you have to travel around it. All surfaces look the same locally.

Think how long it took people to discover that we live on a sphere!

Weirder topologies

Can you imagine a shape with 1 hole and 1 edge?

It is called a Möbius strip.

The Möbius strip

How to make one: Take a strip of paper and join the ends

together with a half-twist. How to destroy one!

Cut the strip in half along the long edge. What do you think will happen?

What if you cut it in half again? Cut the strip in thirds. Should the answer be

different from before?

Homework

Repeat the experiments with a Möbius strip which has extra twists.

Can you spot any patterns?

Non-orientability

The Möbius strip is the simplest example of a shape which is non-orientable.

This means that the concepts of left and right make no sense to a creature living on the surface.

An alien living on the surface could travel around it and come home to find that everything had been reversed.

Non-orientability

The Klein bottle Here is another example of a non-

orientable surface, called the Klein bottle.

Unlike the Möbius strip, it has no edge. It is a surface which has no inside or

outside.

What shape is our universe?

Just like the ant on the surface, we cannot tell the shape of our universe without travelling around it.

Or…by seeing what happens to light that travels around it.

Does the universe have holes in it? Does it have an edge? Could it be non-orientable?

The microwave background

Scientists study radiation from the very earliest universe: the Cosmic Microwave Background Radiation.

In the beginning the universe was much smaller, so this light has had time to go all around it.

If the universe were toroidal or Möbius-shaped, we would see repeating patterns.

The Cosmic Microwave Background

Can you spot any patterns?

The evidence

There is currently no conclusive evidence for anything other than a flat universe.

Some scientists believe that the universe is a strange kind of 3D dodecahedron.

Lessons to take home

Don’t believe everything your teachers tell you!

That ‘straight lines’ can have different meanings depending on where you draw them.

That crazy ideas in pure maths often turn out to be useful.

That our universe could be much more complicated than we ever imagined.