# Mathematical Ideas that Shaped the World

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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 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. Abraham Lincoln kept a copy of the Elements in his saddlebag and would read it at night by lamplight.

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

The axioms of geometry Any straight line segment can be extended indefinitely in a straight line.

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

The axioms of geometry All right angles are congruent.

The axioms of geometry 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. N L M

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?

Can we make Euclid’s axioms work on a sphere?
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 … It is always less than ‘’ and varies with every different circle drawn on the sphere.  = …  = 2

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, saying it 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. Geometry is an a-priori science: it applies to the natural world but is not derived from it = “Critique of Pure Reason”

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. Mother was illiterate and didn’t record the date of Gauss’ birth, remembering only that it was a Wednesday.

not to waste one hour's time on that problem
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 saying
I 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 …

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. Compare with, say, Greenland on a map, where the distance looks big but is actually much smaller.

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 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.

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.

Euclid’s 5th axiom of geometry is not always true
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. The Goldilocks universe!

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. Invented at about the same time as non-Euclidean geometry. =

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: How to destroy 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.