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Projective Geometry
Geometry is a discipline which has long been subject to mathematical fashions of the ages. In classical Greece, Euclid’s elements (Euclid pictured above) with their logical axiomatic base established the subject as the pinnacle on the “great mountain of Truth” that all other disciplines could but hope to scale. However the status of the subject fell greatly from such heights and by the late 18th century it was no longer a fashionable branch to study. The revival of interest in geometry was led by a group of French mathematicians at the start of the 1800s with their work on projective geometry. This then paved the way for the later development of non-Euclidean geometry and led to deep philosophical questions as to geometry’s links with reality and indeed just what exactly geometry was.
Projective geometry is the study of geometrical properties unchanged by projection. It strips away distinctions between conics, angles, distance and parallelism to create a geometry more fundamental than Euclidean geometry. For example the diagram below shows how an ellipse has been projected onto a circle. The ellipse and the circle are therefore projectively equivalent which means that projective results in the circle are also true in ellipses (and other conics).
Projective geometry can be understood in terms of rays of light emanating from a point. In the diagram above, the triangle IJK drawn on the glass screen would be projected to triangle LNO on the ground. This projection does not preserve either angles or side lengths – so the triangle on the ground will have different sized angles and sides to that on the screen. This may seem a little strange – after all we tend to think in terms of angles and sides in geometry, however in projective geometry distinctions about angles and lengths are stripped away (however something called the cross-ratio is still preserved).
We can see in the image above that a projection from the point E creates similar shapes when the 2 planes containing IJKL and ABCD are parallel. Therefore the Euclidean geometrical study of similar shapes can be thought of as a subset of plane positions in projective geometry.
Taking this idea further we can see that congruent shapes can be achieved if we have the centre of projection, E, “sent to infinity:” In projective geometry, parallel lines do indeed meet – at this point at infinity. Therefore with the point E sent to infinity we have a projection above yielding congruent shapes.
Projective geometry can be used with conics to associate every point (pole) with a line (polar), and vice versa. For example the point A had the associated red line, d. To find this we draw the 2 tangents from A to the conic. We then join the 2 points of intersection between B and C. This principle of duality allowed new theorems to be discovered simply by interchanging points and lines.
An example of both the symmetrical attractiveness and the mathematical potential for duality was first provided by Brianchon. In 1806 he used duality to discover the dual theorem of Pascal’s Theorem – simply by interchanging points and lines. Rarely can a mathematical discovery have been both so (mechanically) easy and yet so profoundly
beautiful.
Brianchon’s Theorem
Pascal’s Theorem
Poncelet
Poncelet was another French pioneer of projective geometry who used the idea of points and lines being “sent to infinity” to yield some remarkable results when used as a tool for mathematical proof.
Another version of Pascal’s Theorem:
Poncelet claimed he could prove Pascal’s theorem (shown above) where 6 points on a conic section joined to make a hexagon have a common line. He did this by sending the line GH to infinity. To understand this we can note that the previous point of intersection G of lines AB’ and A’B is now at infinity, which means that AB’ and A’B will now be parallel. This means that H being at infinity also creates the 2 parallel lines AC’. Poncelet now argued that because we could prove through geometrical means that B’C and BC’ were also parallel, that this was consistent with the line HI also being at infinity. Therefore by proving the specific case in a circle where line GHI has been sent to infinity he argued that we could prove using projective geometry the general case of Pascal’s theorem in any conic .
Pascal’s Theorem with intersections at infinity:
This branch of mathematics developed quickly in the early 1800s, sparking new interest in geometry and leading to a heated debate about whether geometry should retain its “pure” Euclidean roots of diagrammatic proof, or if it was best understood through algebra. The use of points and lines at infinity marked a shift away from geometry representing “reality” as understood from a Euclidean perspective, and by the late 1800s Beltrami, Poincare and others were able to incorporate the ideas of projective geometry and lines at infinity to provide their Euclidean models of non-Euclidean space. The development of projective geometry demonstrated how a small change of perspective could have profound consequences.
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.
Hyperbolic Geometry
The usual geometry taught in school is that of Euclidean geometry – in which angles in a triangle add up 180 degrees. This is based on the idea that the underlying space on which the triangle is drawn is flat. However, if the underlying space in curved then this will no longer be correct. On surfaces of constant curvature triangles will have angles greater than 180 degrees, and on surfaces of constant negative curvature triangles will have angles less than 180 degrees. Hyperbolic geometry is based on a geometry in which the underlying space is negatively curved.
Pseudosphere
The pseudosphere pictured above is an example of a surface of constant negative curvature. It was given the name pseudosphere as a usual sphere is the opposite – a surface of constant positive curvature. So the question arises, what would the geometry on a surface such as a pseudosphere look like? If we imagine 2 dimensional beings who live on the surfaces of a pseudosphere, what geometry would they believe their reality to be based on?
Poincare Disc
The Poincare disc is one way we can begin to understand what hyperbolic geometry would be like for its inhabitants. The Poincare disc is a map which translates points in the hyperbolic plane to points in the Euclidean disc. For example, we translate points from the spherical Earth onto a flat atlas – and this helps us understand our spherical reality. In the same way, we can arrive at a model of the hyperbolic plane that helps us to understand it, without actually having to picture a pseudosphere everytime we want to do some maths.
The Poincare disc maps the entire hyperbolic plane onto a finite disc. The shortest line between 2 points (which we would call a straight line in Euclidean geometry) is represented in the disc model by both circular arcs and diameters of the circle. These circular arcs are orthogonal (at 90 degrees) to the boundary circle edge. So our model shows that if we wanted to travel between points E and F on a pseudosphere, that the shortest route would be the curved path of the arc shown. Indeed, being able to calculate the shortest distance between 2 points is an integral part of understanding the underlying geometry.
Distance on the Poincare Disc
The easiest case to look at is the distance from point A on the centre, to a point x on the diameter. Remember that diameters are also straight lines in the Poincare model. What we want to calculate is what this distance from A to x actually represents in the hyperbolic plane. The formula to calculate this is:
Hyperbolic distance = 2tanh-1x
Where tanh-1x is the inverse of the hyperbolic tangent function. This can be calculated with a calculator, or we can use the alternative definition for tanh-1x
2tanh-1x = ln(1+x) – ln(1-x)
So, if our point F is 0.5 units away from A, then this relates to a hyperbolic distance of 2tanh-10.5 = 1.0986.
As we get further to the boundary of the unit circle this represents ever larger distances in the hyperbolic plane. For example if our point F is 0.9 units away from A, then this relates to a hyperbolic distance of 2tanh-10.9 = 2.9444
And if our point is 0.999999 units away from A we get a hyperbolic distance of 2tanh-10.999999 = 14.5087
In fact, as we get closer and closer to the edge of the disc (say x = 1) then the hyperbolic distance gets closer and closer to infinity. In this way the Poincare Disc, even though it has a radius of 1, is able to represent the entire (infinite) hyperbolic plane. We can see this behavior of 2tanh-1x by considering the graph of tanh-1x:
y = tanh-1x has 2 vertical asymptotes at x =1 and x = -1, so as we approach these values, the graph tends to infinity.
So, remarkably the Poincare disc is able to represent an infinite amount of information within a finite space. This disc model was hugely helpful for mathematicians as they started to investigate hyperbolic geometry.
Mathematical art
Escher was a Dutch artist who worked in the 20th century. He used mathematical concepts within his pieces. The above piece is one of his most famous, which is art based on the Poincare disc. It consists of tessellations of the same shape – but because distance is not the same as in Euclidean geometry, those fish which are closer to the boundary edge appear smaller. It is a fantastic illustration of the strange nature of hyperbolic geometry and a pictorial representation of infinity.
If you enjoyed this post you might also like:
Non Euclidean Geometry V – The Shape of the Universe – a look at how non Euclidean geometry is part of the physics of understanding the universe.
Circular Inversion – Reflecting in a Circle – study another transformation which is used in hyperbolic geometry.
Zeno’s Paradox – Achilles and the Tortoise
This is a very famous paradox from the Greek philosopher Zeno – who argued that a runner (Achilles) who constantly halved the distance between himself and a tortoise would never actually catch the tortoise. The video above explains the concept.
There are two slightly different versions to this paradox. The first version has the tortoise as stationary, and Achilles as constantly halving the distance, but never reaching the tortoise (technically this is called the dichotomy paradox). The second version is where Achilles always manages to run to the point where the tortoise was previously, but by the time he reaches that point the tortoise has moved a little bit further away.
Dichotomy Paradox
The first version we can think of as follows:
Say the tortoise is 2 metres away from Achilles. Initially Achilles halves this distance by travelling 1 metre. He halves this distance again by travelling a further 1/2 metre. Halving again he is now 1/4 metres away. This process is infinite, and so Zeno argued that in a finite length of time you would never actually reach the tortoise. Mathematically we can express this idea as an infinite summation of the distances travelled each time:
1 + 1/2 + 1/4 + 1/8 …
Now, this is actually a geometric series – which has first term a = 1 and common ratio r = 1/2. Therefore we can use the infinite summation formula for a geometric series (which was derived about 2000 years after Zeno!):
sum = a/(1-r)
sum = 1/(1-0.5)
sum = 2
This shows that the summation does in fact converge – and so Achilles would actually reach the tortoise that remained 2 metres away. There is still however something of a sleight of hand being employed here however – given an infinite length of time we have shown that Achilles would reach the tortoise, but what about reaching the tortoise in a finite length of time? Well, as the distances get ever smaller, the time required to traverse them also gets ever closer to zero, so we can say that as the distance converges to 2 metres, the time taken will also converge to a finite number.
There is an alternative method to showing that this is a convergent series:
S = 1+ 1/2 + 1/4 + 1/8 + 1/16 + …
0.5S = 1/2+ 1/4 + 1/8 + 1/16 + …
S – 0.5S = 1
0.5S = 1
S = 2
Here we notice that in doing S – 0.5S all the terms will cancel out except the first one.
Achilles and the Tortoise
The second version also makes use of geometric series. If we say that the tortoise has been given a 10 m head start, and that whilst the tortoise runs at 1 m/s, Achilles runs at 10 m/s, we can try to calculate when Achilles would catch the tortoise. So in the first instance, Achilles runs to where the tortoise was (10 metres away). But because the tortoise runs at 1/10th the speed of Achilles, he is now a further 1m away. So, in the second instance, Achilles now runs to where the tortoise now is (a further 1 metre). But the tortoise has now moved 0.1 metres further away. And so on to infinity.
This is represented by a geometric series:
10 + 1 + 0.1 + 0.01 …
Which has first time a = 10 and common ratio r = 0.1. So using the same formula as before:
sum = a/(1-r)
sum = 10/(1-0.1)
sum = 11.11m
So, again we can show that because this geometric series converges to a finite value (11.11), then after a finite time Achilles will indeed catch the tortoise (11.11m away from where Achilles started from).
We often think of mathematics and philosophy as completely distinct subjects – one based on empirical measurement, the other on thought processes – but back in the day of the Greeks there was no such distinction. The resolution of Zeno’s paradox by use of calculus and limits to infinity some 2000 years after it was first posed is a nice reminder of the power of mathematics in solving problems across a wide range of disciplines.
The Chess Board Problem
The chess board problem is nothing to do with Zeno (it was first recorded about 1000 years ago) but is nevertheless another interesting example of the power of geometric series. It is explained in the video above. If I put 1 grain of rice on the first square of a chess board, 2 grains of rice on the second square, 4 grains on the third square, how much rice in total will be on the chess board by the time I finish the 64th square?
The mathematical series will be:
1+ 2 + 4 + 8 + 16 +……
So a = 1 and r = 2
Sum = a(1-r64)/(1-r)
Sum = (1-264)/(1-2)
Sum = 264 -1
Sum = 18, 446,744, 073, 709, 551, 615
This is such a large number that, if stretched from end to end the rice would reach all the way to the star Alpha Centura and back 2 times.
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.
Unbelievable: 1+2+3+4…. = -1/12 ?
The above video by the excellent team at Numberphile has caused a bit of an internet stir – by providing a proof that 1+2+3+4+5+… = -1/12
It’s well worth watching as an example of what proof means – if something is proved which we “know” is wrong, then should we accept it as true? The particular proof as offered in the video is certainly open to question – even if the end result: 1+2+3+4+5+… = -1/12 can actually be proved under certain definitions, using the Riemann Zeta function.
Grandi’s Series
The proof in the video requires that firstly we accept that the infinite summation, 1-1+1-1+1-1… = 1/2. This series is known as the Grandi’s Series – and has been the cause of arguments in the mathematical community for centuries as to what the infinite summation should actually be. One method (called Cesaro Summation) gives an answer of 1/2 – which is the answer accepted in the video.
Alternative interpretations of Grandi’s series would be to group the numbers as 1 + (-1+1) + (-1+1) +(-1+1)…. which you would expect to equal 1. Or, we could group the numbers as (1-1) + (1-1) + (1-1) … which you would expect to equal 0. Therefore it would be also mathematically valid to say that the infinite summation 1-1+1-1… has no sum.
Divergent Series are the invention of the Devil
For the proof in the video to be valid we have to therefore accept that the sum of Grandi’s series is 1/2. We also need accept that it is possible to manipulate infinite series by “shifting them along by 1” or by factorising.
However as we have already seen in the case of Grandi’s series, infinite series don’t always follow normal arithmetic rules. Indeed, the 19th century Norwegian mathematician Niels Abel, warned that that, “Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever!”
Nevertheless it is an interesting method. First they define 3 different infinite series:
S = 1 + 2 + 3 + 4 + 5 …..
S1 = 1 – 1 + 1 – 1 + 1 – 1 ….
S2 = 1 – 2 + 3 – 4 + 5…
Step 1:
The first step is to state that S1 = 1 – 1 + 1 – 1 + 1 – 1 …. = 1/2.
Step 2:
if S2 = 1-2+3-4+5…
then 2S2 = 1 -2+3-4+5…
+1-2+3-4…
Here we have “shifted along by one space” the second S2. This means that when we add the two sequences together we end up with:
2S2 = 1-1+1-1+1… = 1/2
which gives S2 = 1/4.
Step 3:
Do S – S2 = 1 + 2 + 3 + 4 + 5 …..
-(1 – 2 + 3 – 4 + 5…)
= 4 + 8 + 12 + ….
= 4(1 + 2 + 3….)
= 4(S)
Now if S – S2 = 4S we can simply rearrange this equation and substitute the value of S2 = 1/4 which we found before to give: S = -1/12
As mentioned above this is not a very rigorous proof. There is a more rigorous (and complicated) method of proving this – which is the method used by Euler, and which employs the Riemann Zeta function. You can watch this method here:
You might notice when watching this proof that at the start of the video they use the infinite summation of a geometric sequence formula – which is only valid for absolute x less than 1. Then later on they substitute x = -1 into a result derived from it. This is OK because of analytical continuation (which is a method of extending the domain of a function beyond its usual domain). This idea starts to get really complicated – but if you’re interested in the basic idea look at the post on the Riemann Sphere below. The Riemann Sphere allows infinity to be included in the domain of the complex numbers.
If you enjoyed this post you might also like:
Mathematical Proof and Paradox: How you can “prove” things like 1 = 2. Can you spot the flaws in the logic?
The Riemann Hypothesis: How the Riemann Zeta function is fundamental to understanding the prime numbers – and how solving the Riemann Hypothesis is one of the greatest puzzles in mathematics.
The Riemann Sphere – an introduction to isomorphic mappings, which is a lot more interesting than it sounds!
Essential resources for IB students:
1) Exploration Guides and Paper 3 Resources
I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.
Is maths invented or discovered? One of the most interesting questions to investigate with regards to maths Theory of Knowledge (ToK) is the relationship between maths and reality. Why does maths describe reality? Are the mathematical equations of Newton and Einstein inventions to describe reality, or did they exist prior to their discovery? If equations exist independent of discovery, then where do they exist and in what form? The below passage is a brief introduction to some of the ideas on this topic I wrote a while back. Hopefully it will inspire some further reading!
Mathematics and Reality
We live in a mathematical universe. Mathematics describes the reality we see, the reality that we can’t, and the reality that we suppose. Mathematical models describe everything from the orbital path of Jupiter’s moons, to the flight of a football through the air, from the spiral pattern of a shell to the evolution of honey bee hives, from the chaotic nature of weather, to the expansion of the universe.
But why should maths describe reality? Why should there be an equation linking energy and mass, or one predicting the decay of a radioactive atom or one even linking three sides of a triangle? We take the amazing predictive powers of mathematics for granted, and yet these questions lead onto one the most fundamental questions of all – is mathematics a human invention, created to understand the universe, or do we simply discover the equations of mathematics, which are themselves woven into the fabric of reality?
The Second Law of Motion which links force, mass and acceleration, drawn up by Sir Isaac Newton in 1687, works just as well on the surface of Mars as it does on Earth. Einstein’s equations explaining the warping of space time by gravity apply in galaxies light years away from our own. Heisenberg’s uncertainty principle, which limits the information we can know simultaneously about a subatomic particle applied as well in the post Big Bang universe of 13.7 billion years ago as it does today. When such mathematical laws are discovered they do not simply describe reality from a human perspective, but a more fundamental, objective reality independent of human observation completely.
Anthropic reasoning
Anthropic reasoning could account for two of the greatest mysteries of modern science – why the universe seems so fine-tuned for life and the “unreasonable effectiveness of mathematics” in describing reality.
The predictive power of mathematics might itself be necessary for the development of any advanced civilisation. If we lived in a universe in which mathematics did not describe reality – i.e. one in which we could not use the predictive mathematical models either explicitly or implicitly then where would mankind currently be?
At the core of mathematical models are an ability to predict the consequences of actions in the natural world. A hunter gatherer on the African savannah is implicitly using a parabolic flight model when throwing a spear, if mathematical models do not describe reality, then such interactions are inherently unpredictable – and the evolutionary premium on higher cognition which has driven human progress would have been significantly diminished. Our civilisation, our progress, our technology is all founded on the mathematical models that allow us to understand and shape the world around us.
Anthropic reasoning requires that the act of conscious questioning itself is taken into account. In other words, it is certain that we would live in a universe both fine-tuned for mathematics and fine-tuned for life because if our universe was not, we would not be an advanced civilisation able to consider the question in the first place.
This reasoning does however require that we simply accept what appear to be the vanishingly small probabilities that such a universe would be created by chance. For example, Martin Rees, in his book, “Just Six Numbers” looks at six mathematical constants which were they to alter even slightly would create a universe which could not support life.
Whilst tossing a coin and getting 20 heads in a row is unbelievably unlikely, if you repeatedly do this millions of times, then such an occurrence becomes practically assured. Therefore using this mathematical logic, any vanishingly small probabilities can be resolved. The universe is the way that we observe it, precisely because it is a universe taken from the set of all universes in which we can observe it.
Mathematics as reality
An even more intriguing possibility is that maths doesn’t merely describe reality – but that maths itself is the reality. When we view a website, what we are actually viewing is the manifestation of the website source code – which provides all the rules that govern how that page looks and acts. The source code does not simply describe the page, but it is what generates the page in the first place – it is the underlying reality that underpins what we observe. Using this same reasoning could explain why our continued search for a Theory of Everything continually discovers new mathematical formulae to explain the universe – because what we are discovering is part of the universal source code, written in mathematics.
MIT physicist Max Tegemark, describes this view as “radical Platonism.” Plato contended that there exists a perfect circle – in the world of ideas – which every circle drawn on Earth is a mere imitation of. Radical Platonism takes this idea further with the argument that all mathematical structures really exist – in physical space. Therefore there is a mathematical structure isomorphic to our own universe – and that is the universe we live in.
Whilst this may seems rather far fetched, it is worth noting that in quantum mechanics it is difficult to distinguish between mathematical equations and reality. It is already clear that mathematical equations -wave functions – describe reality at the subatomic level. At this level the spatial existence of particles is described not in terms of classical co-ordinates, but in terms of a probability density function. What is still not clear after decades of debate is whether this wave function merely describes reality (e.g. the Copenhagen interpretation), or if this wave function itself is what really exists (e.g. the Many Worlds interpretation). The latter interpretation would necessitate that at its fundamental level mathematical equations are indeed reality.
It is clear that there is a remarkable relationship between mathematics and reality, indeed this relationship is one of the most fundamental mystery in science. We live in a mathematical universe. Whether that is because of nothing more than a statistical fluke, or because of the necessary condition that advanced civilisations require mathematical models or because the universe itself is a mathematical structure is still a long way from being resolved. But simply asking the question, “Why these equations and not others?” takes us on a fantastic journey to the very bounds of human imagination.
Essential resources for IB students:
1) Exploration Guides and Paper 3 Resources
I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.
IB Maths Resources from Intermathematics 