Mathematical Ideas that
Shaped the World
Non-Euclidean geometry
Plan for this class
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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!
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Your teachers lied to you about at least one
of the following statements.
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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
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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
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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
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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.
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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.
N
L
M
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Axiom 5
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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.
L
P
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This is usually called the parallel postulate.
Angles in a triangle
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From Axiom 5 we can deduce that the angles
in a triangle add up to 180 degrees.
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Hidden infinities
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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
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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
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The problem is: how do we prove that the
line we have constructed is a straight line?
Could parallel lines not exist?
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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?
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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?
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A great circle’s centre must be the same as
that of the sphere.
Not a
straight
line
Shortest distances on a map
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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
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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
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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?
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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
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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!
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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!
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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.
 = 2.8284…
=2
Do spheres contradict Euclid then?
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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?
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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?
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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
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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)
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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
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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)
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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.
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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 .
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At the same time, there was yet another
rival to the claim of the first nonEuclidean geometry….
Lobachevsky (1792 – 1856)
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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
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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
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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
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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!
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Your teachers lied to you about at least one
of the following statements. Which one(s)?
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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!
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Answer: all of them can sometimes be false!
It depends on the space that you are
drawing lines on.
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Any space in which these statements are false
is called a non-Euclidean geometry.
Summary
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Parallel lines do different things in different
geometries:
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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
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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
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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:
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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
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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
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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.
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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?
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Our universe is currently expanding.
The shape of our universe will determine its
eventual fate.
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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
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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
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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
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For example:
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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
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A square has no
holes and 1 edge
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A sphere has no
holes and no edges
Examples
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A cylinder has 1
hole and 2 edges
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A torus has 1 hole
and no edges
2-dimensional planets
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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
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Can you imagine a shape with 1 hole and 1
edge?
It is called a Möbius strip.
The Möbius strip
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How to make one:
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Take a strip of paper and join the ends together
with a half-twist.
How to destroy one!
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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
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Repeat the experiments with a Möbius strip
which has extra twists.
Can you spot any patterns?
Non-orientability
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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
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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?
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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
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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 Möbiusshaped, we would see repeating patterns.
The Cosmic Microwave Background
Can you spot any patterns?
The evidence
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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
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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.