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If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!
Non Euclidean Geometry – An Introduction
It wouldn’t be an exaggeration to describe the development of non-Euclidean geometry in the 19th Century as one of the most profound mathematical achievements of the last 2000 years. Ever since Euclid (c. 330-275BC) included in his geometrical proofs an assumption (postulate) about parallel lines, mathematicians had been trying to prove that this assumption was true. In the 1800s however, mathematicians including Gauss started to wonder what would happen if this assumption was false – and along the way they discovered a whole new branch of mathematics. A mathematics where there is an absolute measure of distance, where straight lines can be curved and where angles in triangles don’t add up to 180 degrees. They discovered non-Euclidean geometry.
Euclid’s parallel postulate (5th postulate)
Euclid was a Greek mathematician – and one of the most influential men ever to live. Through his collection of books, Elements, he created the foundations of geometry as a mathematical subject. Anyone who studies geometry at secondary school will still be using results that directly stem from Euclid’s Elements – that angles in triangles add up to 180 degrees, that alternate angles are equal, the circle theorems, how to construct line and angle bisectors. Indeed you might find it slightly depressing that you were doing nothing more than re-learn mathematics well understood over 2000 years ago!
All of Euclid’s results were based on rigorous deductive mathematical proof – if A was true, and A implied B, then B was also true. However Euclid did need to make use of a small number of definitions (such as the definition of a line, point, parallel, right angle) before he could begin his first book He also needed a small number of postulates (assumptions given without proof) – such as: “(It is possible) to draw a line between 2 points” and “All right angles are equal”
Now the first 4 of these postulates are relatively uncontroversial in being assumed as true. The 5th however drew the attention of mathematicians for centuries – as they struggled in vain to prove it. It is:
If a line crossing two other lines makes the interior angles on the same side less than two right angles, then these two lines will meet on that side when extended far enough.
This might look a little complicated, but is made a little easier with the help of the sketch above. We have the line L crossing lines L1 and L2, and we have the angles A and B such that A + B is less than 180 degrees. Therefore we have the lines L1 and L2 intersecting. Lines which are not parallel will therefore intersect.
Euclid’s postulate can be restated in simpler (though not quite logically equivalent language) as:
At most one line can be drawn through any point not on a given line parallel to the given line in a plane.
In other words, if you have a given line (l) and a point (P), then there is only 1 line you can draw which is parallel to the given line and through the point (m).
Both of these versions do seem pretty self-evident, but equally there seems no reason why they should simply be assumed to be true. Surely they can actually be proved? Well, mathematicians spent the best part of 2000 years trying without success to do so.
Why is the 5th postulate so important?
Because Euclid’s proofs in Elements were deductive in nature, that means that if the 5th postulate was false, then all the subsequent “proofs” based on this assumption would have to be thrown out. Most mathematicians working on the problem did in fact believe it was true – but were keen to actually prove it.
As an example, the 5th postulate can be used to prove that the angles in a triangle add up to 180 degrees.
The sketch above shows that if A + B are less than 180 degrees the lines will intersect. Therefore because of symmetry (if one pair is more than 180 degrees, then other side will have a pair less than 180 degrees), a pair of parallel lines will have A + B = 180. This gives us:
This is the familiar diagram you learn at school – with alternate and corresponding angles. If we accept the diagram above as true, we can proceed with proving that the angles in a triangle add up to 180 degrees.
Once, we know that the two red angles are equal and the two green angles are equal, then we can use the fact that angles on a straight line add to 180 degrees to conclude that the angles in a triangle add to 180 degrees. But it needs the parallel postulate to be true!
In fact there are geometries in which the parallel postulate is not true – and so we can indeed have triangles whose angles don’t add to 180 degrees. More on this in the next post.
If you enjoyed this you might also like:
Non-Euclidean Geometry II – Attempts to Prove Euclid – The second part in the non-Euclidean Geometry series.
The Riemann Sphere – The Riemann Sphere is a way of mapping the entire complex plane onto the surface of a 3 dimensional sphere.
Circular Inversion – Reflecting in a Circle The hidden geometry of circular inversion allows us to begin to understand non-Euclidean geometry.
Essential Resources for IB Teachers
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There is also a lot more. I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!
Essential Resources for both IB teachers and IB students
1) Exploration Guides and Paper 3 Resources
I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission. Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator! I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.
Non Euclidean Geometry V – Pseudospheres and other amazing shapes
Non Euclidean geometry takes place on a number of weird and wonderful shapes. Remember, one of fundamental questions mathematicians investigating the parallel postulate were asking was how many degrees would a triangle have in that geometry- and it turns out that this question can be answered depending on something called Gaussian curvature.
Gaussian curvature measures the nature of the curvature of a a 3 dimensional shape. The way to calculate it is to take a point on a surface, draw a pair of lines at right angles to each other, and note the direction of their curvature. If both curve down or both curve up, then the surface has positive curvature. If one line curves up and the other down, then the surface has negative curvature. If at least one of the lines is flat then the surface has no curvature.
Positive curvature:
A sphere is an example of a shape with constant positive curvature – that means the curvature at every point is the same.
Negative curvature:
The pseudosphere is a shape which is in some respects the opposite of a sphere (hence the name pseudo-sphere). This shape has a constant negative curvature. It is formed by a surface of revolution of a called called a tractrix.
Zero curvature:
It might be surprising at first to find that the cylinder is a shape is one which is classified as having zero curvature. But one of the lines drawn on it will always be flat – hence we have zero curvature. We can think of the cylinder as analogous to the flat plane – because we could unravel the cylinder without bending or stretching it, and achieve a flat plane.
So, what is the difference between the geometries of the 3 types of shapes?
Parallel lines
Firstly, given a line m and a point p not on m, how many lines parallel to m through p can be drawn on each type of shape?
A shape with positive curvature has no such lines – and so has no parallel lines. A shape with negative curvature has many such lines – and so has many parallel lines through the same point. A shape with no curvature follows our normal Euclidean rules – and has a single parallel line through a point.
Sums of angles in a triangle and other facts
Triangles on shapes with positive curvature have angles which add to more than 180 degrees. Triangles on shapes with negative curvature have angles which add to less than 180 degrees. Triangles on shapes with no curvature are our familiar 180 degree types. Pythagoras’ theorem no longer holds, and circles no longer have pi as a ratio of their circumference and diameter outside of non-curved space.
Torus
The torus is a really interesting mathematical shape – basically a donut shape, which has the property of of having variable Gaussian curvature. Some parts of the surface has positive curvature, others zero, others negative.
The blue parts of the torus above have positive curvature, the red parts negative and the top grey band has zero curvature. If our 3 dimensional space was like the surface areas of a 4 dimensional torus, then triangles would have different angle sums depending on where we were on the torus’ surface. This is actually one of the current theories as to the shape of the universe.
Mobius Strip and Klein Bottle
These are two more bizarre shapes with strange properties. The Mobius strip only has one side – if you start anywhere on its surface and follow the curvature round you will eventually return to the same place having travelled on every part of the surface.
The Klein bottle is in someways a 3D version of the Mobius strip – and even though it exists in 3 dimensions, to make a true one you need to “fold through” the 4th dimension.
The shape of the universe
OK, so this starts to get quite esoteric – why is knowing the geometry and mathematics of all these strange shapes actually useful? Can’t we just stick to good old flat-plane Euclidean geometry? Well, on a fundamental level non-Euclidean geometry is at the heart of one of the most important questions in mankind’s history – just what is the universe?
At the heart of understanding the universe is the question of the shape of the universe. Does it have positive curvature, negative curvature, or is it flat? Is it like a torus, a sphere, a saddle or something else completely? These questions will help determine if the universe is truly infinite – or perhaps a bounded loop – in which if you travelled far enough in one direction you would return to where you had set off from. It will also help determine what will happen to universe – will it keep expanding? Slow down and stop, or crunch back in on itself? You can read more on these questions here.
This post follows on from Non-Euclidean Geometry – An Introduction – read that one first!
Non Euclidean Geometry IV – New Universes
The 19th century saw mathematicians finally throw off the shackles of Euclid’s 5th (parallel) postulate – and go on to discover a bewildering array of geometries which no longer took this assumption about parallel lines as an axiomatic fact.
1) A curved space model
The surface of a sphere is a geometry where the parallel postulate does not hold. This is because all straight lines in this geometry will meet. We need to clarify what “straight” means in this geometry. “Straight” lines are those lines defined to be of minimum distance from a to b on the surface of the sphere. These lines therefore are defined by “great circles” which have the same radius as the sphere like those shown above.
A 2 dimensional being living on the surface of a 3D sphere would feel like he was travelling in a straight line from a to b when he was in fact travelling on the great circle containing both points. He would not notice the curvature because the curvature would be occurring in the 3rd dimension – and as a 2 dimensional being he would not be able to experience this.
2) A field model – Stereographic Projection for Riemann’s Sphere
A field model can be thought of in reverse. A curved space model is a curved surface where straight lines are parts of great circles. A field model is a flat surface where “straight lines” are curved.
This may seem rather strange, however, the German mathematician Riemann devised a way of representing every point on the sphere as a point on the plane. He did this by first centering the sphere on the origin – as shown in the diagram above. Next he took a point on the complex plane (z = x + iy ) and joined up this point to the North pole of the sphere (marked W). This created a straight line which intersected the sphere at a single point at the surface of the sphere (say at z’). Therefore every point on the sphere (z’) can be represented as a unique point on the plane (z) – in mathematical language, there is a one-to-one mapping between the two.
The only point on the sphere which does not equate to a point on the complex plane is that of the North pole itself (point w). This is because no line touching w and another point on the sphere surface can ever reach the complex plane. Therefore Riemann assigned the value of infinity to the North pole, and therefore the the sphere is a 1-1 mapping of all the points in the complex plane (and infinity).
On this field model (which is the flat complex plane), our straight lines are the stereographic projections of the great circles on the sphere. As you can see from the sketch above, these projections will give us circles of varying sizes. These are now our straight lines!
And this is where it starts to get really interesting – when we have two isometric spaces there is no way an inhabitant could actually know which one is his own reality. A 2 dimensional being could be living in either the curved space model, or the field model and not know which was his true reality.
The difference between the 2 models is that in the first instance we accept an unexplained curvature of space that causes objects to travel in “straight” lines along great circles, and that in the second instance we accept an unexplained field which forces objects travelling in “straight” lines to follow curved paths. Both of these ideas are fundamental to Einstein’s Theory of Relativity – where we must account for both the curvature of space-time and a gravitational force field.
Interestingly, our own 3 dimensional reality is isomorphic to the projection onto a 4 dimensional sphere (hypersphere) – and so our 3 dimensional universe is indistinguishable from a a curved 3D space which is the surface of a hypersphere. A hypersphere may be a bit difficult to imagine, but the video above is about as close as we can get.
Such a scenario would allow for our space to be bounded rather than infinite, and for there to be an infinite number of 3D universes floating in the 4th dimension – each bounded by the surface of their own personal hypersphere. Now that’s a bit more interesting than the Euclidean world of straight lines and circle theorems.
If you enjoyed this you might also like:
Non Euclidean Geometry V – The Shape of the Universe – the final part in the non-Euclidean Geometry series.
Imagining the 4th Dimension. How mathematics can help us explore the notion that there may be more than 3 spatial dimensions.
Geometry, Relativity and the Fourth Dimension is a fantastic (and very readable despite its daunting title!) book full of information about non-Euclidean geometry and extra dimensions.
Non Euclidean Geometry – Spherical Geometry
This article follow on from Non Euclidean Geometry – An Introduction – read that first!
Most geometers up until the 19th century had focused on trying to prove that Euclid’s 5th (parallel) postulate was true. The underlying assumption was that Euclidean geometry was true and therefore the 5th postulate must also be true.
The German mathematician Franz Taurinus made huge strides towards developing non-Euclidean geometries when in 1826 he published his work on spherical trigonometry.
Spherical trigonometry is a method of working out the sides and angles of triangles which are drawn on the surface of spheres.
One of the fundamental formula for spherical trigonometry, for a sphere of radius k is:
cos(a/k) = cos(b/k).cos(c/k) + sin(b/k).sin(c/k).cosA
So, say for example we have a triangle as sketched above. We know the radius of the sphere is 1, that the angle A = 60 degrees, the length b = 1, the length c =1, we can use this formula to find out what the length a is:
cos(a) = cos(1).cos(1) + sin(1).sin(1).cos60
a = 0.99996
We can note that for the same triangle sketched on a flat surface we would be able to use the formula:
a2 = b2 + c2 – 2bc.cosA
a2= 1 + 1 – 2cos60
a = 1
Taurinus however wanted to investigate what would happen if the sphere had an imaginary radius (i). Without worrying too much about what a sphere with an imaginary radius would look like, let’s see what this does to the previous spherical trigonometric equations:
The sphere now has a radius of ik where i = √-1, so:
cos(a/ik) = cos(b/ik).cos(c/ik) + sin(b/ik).sin(c/ik).cosA
But cos(ix) = cosh(x) and sin(ix) = (-1/i)sinh(x) – where cosh(x) and sinh(x) are the hyperbolic trig functions. So we can convert the above equation into:
cosh(a/k) = cosh(b/k)cosh(c/k) – sinh(b/k).sinh(c/k).cosA
This equation will give us the relationship between angles and sides on a triangle drawn on a sphere with an imaginary radius.
Now, here’s the incredible part – this new geometry based on an imaginary sphere (which Taurinus called Log-Spherical Geometry) actually agreed with the hypothesis of the acute angle (the idea that triangles could have an angle sum less than 180 degrees).
Even more incredible, if you take the limit as k approaches infinity of this new equation, you are left with:
a2 = b2 + c2 – 2bc.cosA
What does this mean? Well, if we have a sphere of infinite imaginary radius it stretches and flattens to be indistinguishable from a flat plane – and this is where our normal Euclidean geometry works. So, Taurinus had created a geometry for which our own Euclidean geometry is simply a special case.
So what other remarkable things happen in this new geometric world? Well we have triangles that look like this:
This triangle has angle A = 0, angle C = 90 and lines AB and AC are parallel, (they never meet). This sketch introduces a whole new concept of parallelism far removed from anything Euclid had imagined. The angle β is called the angle of parallelism – and measures the angle between a perpendicular and parallel line. Unlike in Euclidean geometry this angle does not have to be 90 degrees. Indeed the angle β will now change as we move the perpendicular along AC – as it is dependent on the length of the line a.
So, we are now into some genuinely weird and wonderful realms where normal geometry no longer makes sense. Be warned – it gets even stranger! More on that in the next post.
If you enjoyed this post you might also like:
Non Euclidean Geometry IV – New Universes – The fourth part in the non-Euclidean Geometry series.
The Riemann Sphere – The Riemann Sphere is a way of mapping the entire complex plane onto the surface of a 3 dimensional sphere.
Circular Inversion – Reflecting in a Circle The hidden geometry of circular inversion allows us to begin to understand non-Euclidean geometry.
Non-Euclidean Geometry – A New Universe
This post follows on from Non-Euclidean Geometry – An Introduction – read that one first!
The Hungarian army officer and mathematician Johan Bolyai wrote to his father in 1823 in excitement at his mathematical breakthrough with regards to the parallel postulate. “I have created a new universe from nothing.” Johan Bolyai was one of the forerunners of 19th century mathematicians who, after noting that mathematicians had spent over 2000 years trying to prove the parallel postulate, decided to see what geometry would look like if the constraint of the postulate was removed. The result was indeed, a new universe from nothing.
To recap, Euclid’s fifth postulate was as follows:
If a line crossing two other lines makes the interior angles on the same side less than two right angles, then these two lines will meet on that side when extended far enough.
It had been understood in a number of (non-equivalent) ways – that parallel lines remain equidistant from each other, that non-parallel lines intersect, that if the lines L1 and L2 in the diagram are parallel then A + B = 180 degrees, that there can only be one line through a point parallel to any given line.
Collectively these assumptions lead to the basis of numerous geometric proofs – such as the fact that angles in a triangle add up to 180 degrees and that angles in a quadrilateral add up to 360 degrees.
Gerolamo Saccheri
A geometry not based on the parallel postulate could therefore contain 3 possibilities, as outlined by the Italian mathematician Gerolamo Saccheri in 1733:
1) A quadrilateral with (say) 2 right angles A,B and two other angles C,D also both right angles. This is the hypothesis of the right angle – the “normal” geometry of Euclid.
2) A quadrilateral with (say) 2 right angles A,B and two other angles C,D both obtuse. This is the hypothesis of the obtuse angle – a geometry in which the angles in quadrilaterals add up to more than 360 degrees.
3) A quadrilateral with (say) 2 right angles A,B and two other angles C,D also both acute. This is the hypothesis of the acute angle – a geometry in which the angles in quadrilaterals add up to less than 360 degrees.
Don’t be misled by the sketch above – the top line of the quadrilateral is still “straight” in this new geometry – even if it can’t be represented in flat 2 dimensions.
Adrien Legendre
Mathematicians now set about trying to prove that both the cases (2) and (3) were false – thus proving that the Euclidean system was the only valid geometry. The French mathematician Adrien Legendre, who made significant contributions to Number Theory tried to prove that the hypothesis of the obtuse angle was impossible. His argument went as follows:
1) Take a straight line and divide it into n equal segments. In the diagram these are the 4 lines A1A2, A2A3, A3A4, A4A5
2) Complete the diagram as shown above so that the lengths B1B2, B2B3, B3B4, B4B5 are all equal. From the sketch we will have lines A1B1 and A2B2 (and subsequent lines) equal.
3) Now we see what will happen if angle β is greater than α. We compare the two triangles A1B1A2 and A2B2A3. These have 2 sides the same. Therefore if β is greater than α then the length A1A2 must be larger than B1B2.
4) Now we note that the distance A1B1 + B1B2 + B2B3 + … BnBn+1 + Bn+1An+1 is greater than A1A2 + A2A3 + …AnAn+1. In other words, the distance starting at A1 then travelling around the shape missing out the bottom line (the yellow line) is longer than the bottom line (green line).
5) Therefore we can write this as
A1B1 + nB1B2 + An+1Bn+1 > nA1A2
(Here we have simplified the expression by noting that as all the distances B1B2, B2B3 etc are equal)
6) Therefore this gives
2A1B1 > n(A1A2 -B1B2)
(Here we simplify by noting that A1B1 = An+1Bn+1 and then rearranging)
7) But this then gives a contradiction – because we can make the RHS as large as we like by simply subdividing the line into more pieces (thus increasing n), but the LHS remains bounded (as it is a fixed value). Therefore as n tends to infinity, this inequality must be broken.
8) This means that β is not greater than α, so we can write β ≤ α. This will therefore mean that the angles in the triangle A1B1A2 will be ≤ 180. To see this
We can work out the angles in A1B1A2 by noting that c = (180-α)/2 . Therefore
angles in A1B1A2 = (180-α)/2 + (180-α)/2 + β
angles in A1B1A2 = 180 + β – α
But we know that β ≤ α. Therefore β – α ≤ 0
So angles in A1B1A2 = 180 + β – α ≤ 180
Adrien Legendre therefore concluded that the hypothesis of the obtuse angle was impossible. In fact, it isn’t – and the flaw wasn’t in the logic of his proof but in the underlying assumptions contained within it. This will be revealed in the next post!
If you enjoyed this you might also like:
Non Euclidean Geometry III – Breakthrough Into New Worlds – The third part of the series on non-Euclidean Geometry.
The Riemann Sphere – The Riemann Sphere is a way of mapping the entire complex plane onto the surface of a 3 dimensional sphere.
Circular Inversion – Reflecting in a Circle The hidden geometry of circular inversion allows us to begin to understand non-Euclidean geometry.
Non Euclidean Geometry – An Introduction
It wouldn’t be an exaggeration to describe the development of non-Euclidean geometry in the 19th Century as one of the most profound mathematical achievements of the last 2000 years. Ever since Euclid (c. 330-275BC) included in his geometrical proofs an assumption (postulate) about parallel lines, mathematicians had been trying to prove that this assumption was true. In the 1800s however, mathematicians including Gauss started to wonder what would happen if this assumption was false – and along the way they discovered a whole new branch of mathematics. A mathematics where there is an absolute measure of distance, where straight lines can be curved and where angles in triangles don’t add up to 180 degrees. They discovered non-Euclidean geometry.
Euclid’s parallel postulate (5th postulate)
Euclid was a Greek mathematician – and one of the most influential men ever to live. Through his collection of books, Elements, he created the foundations of geometry as a mathematical subject. Anyone who studies geometry at secondary school will still be using results that directly stem from Euclid’s Elements – that angles in triangles add up to 180 degrees, that alternate angles are equal, the circle theorems, how to construct line and angle bisectors. Indeed you might find it slightly depressing that you were doing nothing more than re-learn mathematics well understood over 2000 years ago!
All of Euclid’s results were based on rigorous deductive mathematical proof – if A was true, and A implied B, then B was also true. However Euclid did need to make use of a small number of definitions (such as the definition of a line, point, parallel, right angle) before he could begin his first book He also needed a small number of postulates (assumptions given without proof) – such as: “(It is possible) to draw a line between 2 points” and “All right angles are equal”
Now the first 4 of these postulates are relatively uncontroversial in being assumed as true. The 5th however drew the attention of mathematicians for centuries – as they struggled in vain to prove it. It is:
If a line crossing two other lines makes the interior angles on the same side less than two right angles, then these two lines will meet on that side when extended far enough.
This might look a little complicated, but is made a little easier with the help of the sketch above. We have the line L crossing lines L1 and L2, and we have the angles A and B such that A + B is less than 180 degrees. Therefore we have the lines L1 and L2 intersecting. Lines which are not parallel will therefore intersect.
Euclid’s postulate can be restated in simpler (though not quite logically equivalent language) as:
At most one line can be drawn through any point not on a given line parallel to the given line in a plane.
In other words, if you have a given line (l) and a point (P), then there is only 1 line you can draw which is parallel to the given line and through the point (m).
Both of these versions do seem pretty self-evident, but equally there seems no reason why they should simply be assumed to be true. Surely they can actually be proved? Well, mathematicians spent the best part of 2000 years trying without success to do so.
Why is the 5th postulate so important?
Because Euclid’s proofs in Elements were deductive in nature, that means that if the 5th postulate was false, then all the subsequent “proofs” based on this assumption would have to be thrown out. Most mathematicians working on the problem did in fact believe it was true – but were keen to actually prove it.
As an example, the 5th postulate can be used to prove that the angles in a triangle add up to 180 degrees.
The sketch above shows that if A + B are less than 180 degrees the lines will intersect. Therefore because of symmetry (if one pair is more than 180 degrees, then other side will have a pair less than 180 degrees), a pair of parallel lines will have A + B = 180. This gives us:
This is the familiar diagram you learn at school – with alternate and corresponding angles. If we accept the diagram above as true, we can proceed with proving that the angles in a triangle add up to 180 degrees.
Once, we know that the two red angles are equal and the two green angles are equal, then we can use the fact that angles on a straight line add to 180 degrees to conclude that the angles in a triangle add to 180 degrees. But it needs the parallel postulate to be true!
In fact there are geometries in which the parallel postulate is not true – and so we can indeed have triangles whose angles don’t add to 180 degrees. More on this in the next post.
If you enjoyed this you might also like:
Non-Euclidean Geometry II – Attempts to Prove Euclid – The second part in the non-Euclidean Geometry series.
The Riemann Sphere – The Riemann Sphere is a way of mapping the entire complex plane onto the surface of a 3 dimensional sphere.
Circular Inversion – Reflecting in a Circle The hidden geometry of circular inversion allows us to begin to understand non-Euclidean geometry.
IB Maths Resources from Intermathematics 