What Happens if Euclid Was Wrong? — Geometry on a Curved World
Everyone knows that the angles in a triangle total 180 degrees — until they don't.
For most students, geometry begins with rules that seem almost unbreakable.
Angles on a straight line add to 180 degrees.
Parallel lines never meet.
The angles inside a triangle add to 180 degrees.
Pythagoras tells us that:
a^2 + b^2 = c^2
These ideas become so familiar that it is easy to forget something rather important:
They depend on the kind of space in which we are doing the geometry.
At GCSE, and for much of A-level Mathematics, we are working with Euclidean geometry — the geometry of a flat plane.
But the surface of the Earth is not flat.
The universe may not be perfectly flat either.
And once we allow our geometry to take place on curved surfaces, some of the apparently unquestionable rules of school mathematics begin to change.
This does not mean Euclid was wrong.
It means Euclid was describing one particular kind of geometry.
And there are others.
The Triangle That Adds Up to 270 Degrees
Let us begin with something that sounds impossible.
Imagine standing at the North Pole.
You travel directly south until you reach the equator.
You then turn through 90 degrees and travel one quarter of the way around the equator.
At that point you turn through another 90 degrees and travel directly north.
Eventually you arrive back at the North Pole.
You have travelled along three sides and returned to your starting point.
In other words, you have made a triangle.
Now examine its angles.
At the first point on the equator, the angle is 90 degrees.
At the second point on the equator, the angle is also 90 degrees.
And when the two routes from the equator meet at the North Pole, they can also meet at 90 degrees.
So:
90 + 90 + 90 = 270 degrees
We have constructed a triangle whose angles add to 270 degrees.
No cheating.
No distorted ruler.
No mathematical mistake.
The only thing that has changed is the surface on which we are drawing the triangle.
We have moved from a flat plane to the curved surface of a sphere.
Try It Without Travelling to the North Pole
Fortunately, you do not need an expedition to the Arctic to investigate this.
Find a reasonably large ball.
A football will do. A globe is even better.
Use removable tape, string or a whiteboard marker if the surface allows it.
Choose a point at the top to represent the North Pole.
Now draw or mark:
a line from the North Pole to the equator;
a quarter-turn around the equator;
another line from the equator back to the North Pole.
Try estimating the three angles.
The triangle looks very strange compared with the triangles students normally draw on paper.
That is precisely the point.
Our intuition about geometry has largely developed from working on flat surfaces.
So Was Euclid Wrong?
No.
And this is perhaps the most interesting lesson in the whole subject.
Euclid's geometry is based upon a set of assumptions, or postulates.
If we accept those assumptions, Euclidean geometry follows logically from them.
One particularly important assumption concerns parallel lines.
In simplified form, Euclidean geometry tells us that through a point outside a given line, there is exactly one line parallel to the original line.
That sounds obvious.
But mathematicians spent centuries wondering whether this statement really had to be true.
What happens if we change it?
Something extraordinary happens.
We get completely different — but still logically consistent — geometries.
What Is a Straight Line on a Sphere?
This question is more difficult than it first appears.
If I draw a straight line on a piece of paper, we all know roughly what I mean.
But what counts as the equivalent of a straight line on a sphere?
On a sphere, the important paths are called great circles.
A great circle is a circle drawn around the sphere whose centre is also the centre of the sphere.
The equator is a great circle.
Lines of longitude also form great circles when continued around the entire Earth.
Most lines of latitude, however, are not great circles.
The 50 degree north line of latitude, for example, forms a smaller circle around the Earth.
Why are great circles important?
Because travelling along a great-circle route gives the shortest path between two points on a spherical surface.
That brings curved geometry directly into the real world.
Why Airline Routes Look Curved on Maps
Look at the route of a long-distance flight on a conventional flat map.
A flight from London to somewhere on the west coast of North America may appear to curve surprisingly far north.
At first sight, that can look inefficient.
Surely a straight line across the map would be shorter?
The problem is not the aircraft.
It is the map.
We are trying to represent the curved surface of the Earth on a flat sheet of paper or computer screen.
That inevitably introduces distortion.
The shortest path across the spherical Earth is approximately a section of a great circle.
When that great-circle route is transferred onto many types of flat map, it appears curved.
So the aircraft can appear to be flying along a curved route on the map while actually following something close to the shortest available route across the Earth.
This is a lovely example of mathematics changing the way we interpret something familiar.
What Happens to Parallel Lines?
Now things become even stranger.
Take two lines of longitude.
Near the equator, they are separated.
Travel north along them and they become closer together.
Eventually they meet at the North Pole.
Travel south instead and they meet at the South Pole.
So our familiar idea that parallel lines remain the same distance apart and never meet no longer works in the same way.
Indeed, on spherical geometry, great circles always eventually intersect.
There are effectively no parallel great circles.
Compare that with ordinary Euclidean geometry, where parallel lines never meet.
Already we have two very different geometrical worlds.
There Is Another Possibility: Hyperbolic Geometry
A sphere curves one way.
But mathematicians can also study spaces with a different sort of curvature.
This leads to hyperbolic geometry.
One way of imagining this is to think of a saddle-shaped surface, although the full mathematical idea is more general than simply drawing on a saddle.
In hyperbolic geometry something remarkable happens.
Through a point outside a line, more than one line can be drawn that never meets the original line.
And triangles behave differently again.
On a flat Euclidean plane:
Triangle angle total = 180 degrees
On a sphere:
Triangle angle total > 180 degrees
In hyperbolic geometry:
Triangle angle total < 180 degrees
So there is nothing universally sacred about 180 degrees.
It belongs to one particular geometry.
A Surprisingly Deep Connection Between Area and Angles
For A-level students wanting to take the idea slightly further, spherical triangles contain another beautiful result.
Suppose the angles of a spherical triangle are A, B and C.
Calculate how much their total exceeds 180 degrees.
This difference is called the spherical excess.
For example, our North Pole triangle has:
A + B + C = 270 degrees
so its excess is:
270 - 180 = 90 degrees
On a sphere, this excess is related directly to the area of the triangle.
In a more advanced treatment, if the excess E is measured in radians:
Area = R^2 x E
where R is the radius of the sphere.
That means the angles of a spherical triangle tell us something about its physical size.
That is quite different from ordinary plane geometry.
On a flat sheet of paper, I can draw a tiny triangle and an enormous similar triangle with exactly the same three angles.
On a sphere, curvature changes the relationship.
An Experiment With Three Different-Sized Triangles
This makes a useful investigation.
Take a globe or large ball and construct several spherical triangles.
Make one fairly small.
Make another much larger.
Try to measure their angles as accurately as possible.
You should find that very small triangles behave rather like ordinary Euclidean triangles.
Their angles may add to something very close to 180 degrees.
But make the triangle large enough and the curvature becomes important.
The departure from 180 degrees becomes increasingly noticeable.
This gives us another fascinating idea.
Euclidean Geometry Can Be a Very Good Approximation
The surface of the Earth is curved.
Yet if I draw a triangle on my desk, I do not need spherical geometry to calculate its angles.
Why?
Because my triangle is tiny compared with the Earth.
Across a sufficiently small region, the curved surface looks almost flat.
It is rather like standing in a large field.
The ground beneath you appears flat even though you know that the Earth as a whole is approximately spherical.
This idea — that something curved can look flat when examined over a sufficiently small region — appears in many areas of mathematics and physics.
From School Geometry to Einstein
There is an even bigger reason why non-Euclidean geometry matters.
It eventually became essential to modern physics.
Einstein's general theory of relativity describes gravity not simply as a mysterious force pulling objects towards one another, but in terms of the geometry of spacetime.
Mass and energy affect the geometry of spacetime.
Objects then move through that geometry.
The mathematics required to describe this is far beyond GCSE and A-level Mathematics, but the underlying idea is accessible:
Geometry does not merely have to describe shapes drawn on paper. It can describe the structure of the universe itself.
A subject that can begin with rulers, compasses and triangles can eventually lead towards black holes, gravitational lensing and the expansion of the universe.
Why This Matters to Mathematics Students
There is a broader lesson here that I particularly like students to encounter.
At school it is very easy to get the impression that mathematics consists of a collection of rules.
Learn the rule.
Apply the rule.
Get the answer.
But mathematics is much more interesting than that.
We also ask:
Why is the rule true?
What assumptions does it depend upon?
What happens if we change those assumptions?
Does the new system remain logically consistent?
Those questions move us from simply using mathematics towards actually thinking mathematically.
The statement:
"The angles of a triangle add to 180 degrees"
is therefore incomplete.
A better statement would be:
"The angles of a triangle drawn in a Euclidean plane add to 180 degrees."
That extra qualification changes everything.
A Challenge for GCSE Students
Suppose someone tells you:
"Parallel lines never meet."
Ask:
Where?
On an ordinary flat plane, yes.
On the surface of a sphere, our equivalent "straight lines" — great circles — do meet.
Now consider the Earth.
Which of the following are great circles?
the equator;
the Greenwich meridian;
the Tropic of Cancer;
the Arctic Circle.
The equator is a great circle.
A complete meridian, together with the opposite meridian, forms a great circle.
The Tropic of Cancer and Arctic Circle do not.
That distinction matters when calculating shortest routes over the Earth.
A Challenge for A-Level Students
Try researching the three major geometrical possibilities:
Euclidean geometry
Flat curvature.
Triangle angles total 180 degrees.
Spherical geometry
Positive curvature.
Triangle angles total more than 180 degrees.
Hyperbolic geometry
Negative curvature.
Triangle angles total less than 180 degrees.
Then ask a much more difficult question:
How could you determine the geometry of the space you were living in without being able to look at it from outside?
One possibility would be to construct extremely large triangles and measure their angles very accurately.
If they consistently total 180 degrees, space may be approximately flat.
If they total more, that suggests positive curvature.
If they total less, that suggests negative curvature.
Suddenly measuring the angles of a triangle has become an experiment about the nature of space itself.
The Most Important Lesson Is Not About Triangles
For me, the most valuable part of this topic is not remembering the words "spherical geometry" or "hyperbolic geometry".
It is discovering something about mathematics itself.
Students spend years being taught mathematical statements that appear absolute.
Then they encounter a subject like non-Euclidean geometry and discover that mathematics often starts with assumptions.
Change those assumptions carefully, and a completely different mathematical world may emerge.
A triangle does not always have to contain 180 degrees.
Parallel lines do not always have to behave as expected.
A "straight line" depends partly upon the space through which we are travelling.
And the geometry learned at school turns out to be one member of a much larger family of possible geometries.
Euclid was not wrong.
He was describing a flat world.
The remarkable discovery was that mathematics did not have to stop there.
Everyone knows that the angles in a triangle total 180 degrees — until they ask what sort of world the triangle is drawn on.
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