IB Maths and GCSE Maths Resources. Theory of Knowledge (ToK). Real life maths. Maths careers, Maths videos, Maths puzzles and Maths lesson resources.

The site is based around the idea of maths ToK – which tries to get students thinking about more than just syllabus content and to start exploring philosophical ideas, historical connections, cutting edge research and real life applications in mathematics.

Some of the content in the site includes:

  • An IB ToK Maths syllabus plan with a huge amount of ideas for incorporating maths ToK into lessons.
  • A large “Flipping the classroom” videos section for IB students.  These cover pretty much the entire IB HL and SL syllabus – with each topic taught in a short 10 minute or so video.  This should help students both prepare for lessons and also should be invaluable for revision.
  • A new School Code Challenge activity which allows students to practice their code breaking skills – each code hides the password needed to access the next level.
  • Over 200 ideas to help with students’ Maths Explorations – many with links to additional information to research.
  • A large number of posts on everything from imagining extra dimensions to modelling asteroid impacts.

If you would like to contact me, you can do so here. There’s loads of content – so please explore!

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How Infectious is Ebola?

Ebola is the latest virus to warrant global fears over a pandemic which infects large numbers of people.  Throughout history we have seen pandemic diseases such as the Black Death in Middle Ages Europe and the Spanish Flu at the beginning of the 20th century. More recently we have seen HIV responsible for millions of deaths.  In the last few years there have been scares over bird flu and SARS – yet neither fully developed into a major global health problem.  So, how infectious is Ebola, and how can we use mathematics to predict its spread?

The basic model is based on the SIR model.  The SIR model looks at how much of the population is susceptible to infection, how many of these go on to become infectious, and how many of these go on to recover (and in what timeframe).  However given the nature of modelling diseases with very high mortality rates like Ebola, for our Ebola model the SIR stands for Susceptible, Infectious and Dead.

SIR MODEL2

Another important parameter is R0, this is defined as how many people an infectious person will pass on their infection to in a totally susceptible population.  Some of the R0 values for different diseases are shown above.  Studies into Ebola estimate the R0 value at somewhere between 1.7 and 8.6.  Therefore whilst Ebola is infectious, it is nowhere near as infectious as a fully airbourne disease like measles.

ebola

The equations above represent a SIR (susceptible, infectious, dead) model which can be used to model the spread of Ebola.

dS/dt represents the rate of change of those who are susceptible to the illness with respect to time.  dI/dt represents the rate of change of those who are infected with respect to time.  dR/dt represents the rate of change of those who have died with respect to time.

For example, if dI/dt is high then the number of people becoming infected is rapidly increasing.  When dI/dt is zero then there is no change in the numbers of people becoming infected (number of infections remain steady).  When dI/dt is negative then the numbers of people becoming infected is decreasing.

The constants β and μ are chosen depending on the type of disease being modelled.  β represents the contact rate – which is how likely someone will get the disease when in contact with someone who is ill.  ν is the recovery rate which is how quickly people recover (and become immune.

N is the total population

μ is the per capita death rate (Calculated by μ = 1/(duration of illness) )

N – let’s take as the population of Sierra Leone (6 million)

In the case of Ebola we have the following estimated values:

μ between 1/4 and 1/10 (because it takes an infected person between 4 and 10 days to die).  Let’s take it as 1/7 ≈ 0.14

β as approximately 0.6

N – let’s take as the population of Sierra Leone (6 million)

Therefore our 3 equations for rates of change become:

dS/dt = -0.6 I S/6,000,000

dI/dt = 0.6 I S/6,000,000 – 0.14 I

dR/dt = 0.14 I

Unfortunately these equations are very difficult to solve – but luckily we can use a computer program to plot what happens.   We need to assign starting values for S, I and R – the numbers of people susceptible, infectious and dead. We have 6 million people in Sierra Leone, and currently around 8000 reported cases of Ebola.  If we assume all 6 million are susceptible, then putting this all into the program gives the following outcome:

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This graph is pretty incredible –  though it clearly shows some of our assumptions were wrong!  Given a starting point of 6 million people all susceptible to Ebola, and 8000 infectious individuals, then within 20 days the population would have crashed from 6 million to less than 1 million and within 60 days you would have nearly everyone dead.

Clearly therefore this graph is very sensitive to our initial assumed values.  Say for example Ebola was less infectious than we previous assumed – and so we had β = 0.15 with the other values the same.  Then we get the following:

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This graph is very drastically different to the last one – you have infections remaining low – though this would still be enough to see a big population drop over the 3 years of the simulation.

Modelling disease outbreaks with real accuracy is therefore an incredibly important job for mathematicians.  Understanding how diseases spread and how fast they can spread through populations is essential to developing effective medical strategies to minimise deaths.  If you want to save lives maybe you should become a mathematician rather than a doctor!

If you enjoyed this post you might also like:

Differential Equations in Real Life – some other uses of differential equations in modelling predator-prey relationships between animal populations.

Modelling Infectious Diseases –  How we can use computer modelling to look at other infectious diseases like measles.

The Poincare Conjecture and Grigori Perelman

In 2006 the Russian mathematician Grigori Perelman was awarded the mathematical equivalent of the mathematical Nobel prize (the Fields Medal).  He declined it.  In 2010 he was the first mathematician to be awarded $1 million – he turned it down.  What had Perelman done to achieve such (apparently unwanted) acclaim?  He had solved a puzzle that had frustrated mathematicians for over 100 years – the Poincare conjecture.

What is the Poincare Conjecture?

The Poincare Conjecture is that, “Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.”  At first glance that may look quite complicated – so looking at the definitions in turn:

poincare3

Simply connected means a shape without holes.  The two shapes on the left above are simply connected, the two on the right are not.  In 3 dimensions, a sphere and cube are simply connected, but a donut shape (torus) is not.

3D manifold means a 3 dimensional surface.  Imagine the surface of a sphere – that is a 2 dimensional surface.  So a 3 dimensional surface on a sphere would require a 4 dimensional sphere.  A 4 dimensional sphere is one which has a fixed radius in 4 dimensions (unlike in 3 dimensions for a sphere and 2 dimensions for a circle).

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Homeomorphic means it is mathematically equivalent in terms of the relationship between points.  Basically, if 2 shapes can be sqeezed or stretched to form another shape then they are homeomorphic.  In the above animation, the coffee mug and the donut (torus) are shown to be homeomorphic.

3-Sphere means a sphere in 4 dimensions (i.e with a 3 dimensional surface area).

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So, with those terms defined we can simplify the Poincare conjecture.  In regular 3 dimensions, conventional 3 dimensional shapes without a hole in them (cubes, cuboids etc) can all be squashed and squeezed to create a sphere.  Poincare conjectured that the same would be true in higher dimensions – i.e 4 dimensional cubes (a tesseract, as shown above) could be squashed and squeezed to make a 4 dimensional sphere.

poincare2

Grigori Perelman however was not interested in either the acclaim or the money on offer for solving one of the world’s most difficult mathematics problems.  In explaining why he turned down $1 million he said that the prize, “was completely irrelevant for me. Everybody understood that if the proof is correct, then no other recognition is needed.”

If you liked this post you might also like:

Imagining the 4th Dimension. How mathematics can help us explore the notion that there may be more than 3 spatial dimensions.

Non Euclidean Geometry V – The Shape of the Universe – Using mathematics to understand one of the most important questions of all.

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Batman and Superman Maths

Wolfram Alpha is an incredibly powerful mathematical tool – which has been developed to allow both complex calculations and data analysis. It is able to generate images like that shown above, of the Batman logo. What’s really impressive however is that you can see the underlying graph input that would generate this image:

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At first glance this look indecipherable – but we can actually understand this a little better by breaking these inequalities down and looking at the individually.

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The first inequality defines the area inside an ellipse.  All ellipses have a general formula:

  \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1.

In our inequality, the a simply stands for an arbitrary constant (because the Batman logo has no scale).  To keep things simple we can set a = 1.  This gives an equation:

wolfram4

which generates the ellipse:

wolfram5

When we now make this the inequality:

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Then this simply has the effect of shading in the area contained within the ellipse.  So, comparing this to the original Batman shape we can see that the ellipse we have drawn forms the wings of the logo.

Next, let’s look at the next inequality:

wolfram7Which, if we again choose a = 1 for simplicity, we will get

batman

when the part of this graph which is greater or equal to 4.

I’m not going to go through each part – as that would take too long!  Let’s look at one more inequality though:

wolfram9

This will generate the part of the graph that looks like:

wolfram10

This will form part of the Batman logo cape.

Superman Logo

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Now if you thought that was hard, have a look at the inequalities needed for the Superman logo above:

wolfram12

Now this really is almost indecipherable!  I can at least explain what the min(a,b) means.  For example, say we had:

y = min(cosx,sinx)

This would simply mean that for any x value, I would find out what cosx was equal to, find out what sinx was equal to, and then plot the smallest value as my y value.  For example, when x = 0, I would have cos(0) = 1 and sin(0) = 0.  So I choose my y value as 0 when x = 0.  Plotting this graph would give:

  wolfram13wolfram14

Which is an interesting periodic function that shares some of the features of the regular trig graphs.  Anyway, the real Superman inequality is much harder than this – and demonstrates just how powerful Wolfram Alpha is.

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

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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 = 1/2 + 1/4 + 1/8 + 1/16 + …

0.5S = 1/4 + 1/8 + 1/16 + …

S – 0.5S = 1/2

0.5S = 1/2

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

Screen Shot 2014-08-18 at 10.19.42 AM

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.  (Interestingly this number, 264 -1 is also a special type of prime number called a Mersenne Prime.  These are prime numbers of the form 2-1).

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Fourier Transform

The Fourier Transform and the associated Fourier series is one of the most important mathematical tools in physics. Physicist Lord Kelvin remarked in 1867:

“Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics.”

The Fourier Transform deals with time based waves – and these are one of the fundamental building blocks of the natural world. Sound, light, gravity, radio signals, Earthquakes and digital compression are just some of the phenomena that can be understood through waves. It’s not an exaggeration therefore to see the study of waves as one of the most important applications of mathematics in our modern life.

Here are some real life applications in a wide range of fields:

JPEG picture and MP3 sound compression – to allow data to reduced in size.

Analysing DNA sequences – to allow identification of specific regions of genetic code

Apps like Shazam which can recognise a song from a sample of music

Processing mobile phone network data and WIFI data

Signal processing - in everything from acoustic guitar amps or electrical currents through capacitors

Radio telescopes - used to construct images of the night sky

Building’s natural frequencies - architects can design buildings to better withstand earthquakes.

Medical imaging such as MRI scans

There are many more applications – this Guardian article is a good introduction to some others.

So, what is the Fourier Transform? It takes a graph like the graph f(t) = cos(at) below:

 

fourier1

and transforms it into:

fourier2

From the above cosine graph we can see that it is periodic time based function. Time is plotted on the x axis, and this graph will tell us the value of f(t) at any given time. The graph below with 2 spikes represents this same information in a different way. It shows the frequency (plotted on the x axis) of the cosine graph. Now the frequency of a function measures how many times it repeats per second. So for a graph f(t) = cos(at) it can be calculated as the inverse of the period. The period of cos(at) is 2pi/a so it has a frequency of a/2pi.

Therefore the frequency graph for cos(ax) will have spikes at a/2pi and -a/2pi.

But how does this new representation help us? Well most real life waves are much more complicated than simple sine or cosine waves – like this trumpet sound wave below:

fourier3

But the remarkable thing is that every continuous wave can be modelled as the sum of sine and cosine waves. So we can break-down the very complicated wave above into (say) cos(x) + sin(2x) + 2cos(4x) . This new representation would be much easier to work with mathematically.

The way to find out what these constituent sine and cosine waves are that make up a complicated wave is to use the Fourier Transform. By transforming a function into one which shows the frequency peaks we can work out what the sine and cosine parts are for that function.

fourier4

For example, this transformed graph above would show which frequency sine and cosine functions to use to model our original function. Each peak represents a sine or cosine function of a specific frequency. Add them all together and we have our function.

The maths behind this does get a little complicated. I’ll try and talk through the method using the function f(t) = cos(at).

\\1.\ f(t) = cosat\\

So, the function we want to break down into its constituent cosine and sine waves is cos(at). Now, obviously this function can be represented just with cos(at) – but this is a good demonstration of how to use the maths for the Fourier Transform. We already know that this function has a frequency of a/2pi – so let’s see if we can find this frequency using the Transform.

\\2.\ F(\xi) = \int_{-\infty}^{\infty} f(t)(e^{-2\pi i\xi t})dt\\

This is the formula for the Fourier Transform. We “simply” replace the f(t) with the function we want to transform – then integrate.

\\3.\ f(t)= 0.5({e}^{iat}+ {e}^{-iat})\\

To make this easier we use the exponential formula for cosine. When we have f(t) = cos(at) we can rewrite this as the function above in terms of exponential terms.

\\4.\ F(\xi) = 0.5\int_{-\infty}^{\infty} (e^{iat}+e^{-iat})(e^{-2\pi i\xi t})dt\\

We substitute this version of f(t) into the formula.

\\5.\ F(\xi) = \frac{1}{2} \int_{-\infty}^{\infty} e^{it(a-2\pi \xi) }dt + \frac{1}{2} \int_{-\infty}^{\infty}e^{it(-a-2\pi \xi)}dt\\

Next we multiply out the exponential terms in the bracket (remember the laws of indices), and then split the integral into 2 parts. The reason we have grouped the powers in this way is because of the following step.

\\6.\ \delta (a-2\pi \xi) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{it(a-2\pi \xi)}\\

This is the delta function – which as you can see is very closely related to the integrals we have. Multiplying both sides by pi will get the integral in the correct form. The delta function is a function which is zero for all values apart from when the domain is zero.

\\7.\ F(\xi) =\pi [ \delta (a-2\pi \xi ) + \delta (a+2\pi \xi ) ]\\

So, the integral can be simplified as this above.

\\8.\ a-2\pi \xi = 0 \ or \ a+2\pi \xi = 0\\

So, our function F will be zero for all values except when the delta function is zero. This gives use the above equations.

\\9.\ \xi = \pm\frac{a}{2\pi }\\

Therefore solving these equations we get an answer for the frequency of the graph.

\\10.\ frequency\ of\ cosat = \frac{a}{2\pi }

This frequency agrees with the frequency we already expected to find for cos(at).

A slightly more complicated example would be to follow the same process but this time with the function f(t) = cos(at) + cos(bt). If the Fourier transform works correctly it should recognise that this function is composed of one cosine function with frequency a/2pi and another cosine function of b/2pi. If we follow through exactly the same method as above (we can in effect split the function into cos(at) and cos(bt) and do both separately), we should get:

\\7.\ F(\xi) =\pi [ \delta (a-2\pi \xi ) + \delta (a+2\pi \xi ) + \delta (b-2\pi \xi ) + \delta (b+2\pi \xi ) ]\\

This therefore is zero for all values except for when we have frequencies of a/2pi and b/2pi. So the Fourier Transform has correctly identified the constituent parts of our function.

If you want to read more about Fourier Transforms, then the Better Explained article is an excellent start.

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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:

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A sphere is an example of a shape with constant positive curvature – that means the curvature at every point is the same.

Negative curvature:

 

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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:

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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?

euclid23

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

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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.

euclid24

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

euclid25

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.

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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?

euclid27

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.

 

euclideanThis 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

euclid18

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

Joh-RiemannSphere01

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).

euclid19

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.

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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.

euclid14

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:

euclid15

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.

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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.

euclid3

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:

euclid7

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:

euclid8

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.

euclid12

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

euclid13

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.

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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. 

euclid3

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.

euclid2

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.

euclid3

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:

euclid4

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.

euclid5

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.

 

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