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The Riemann Sphere
The Riemann Sphere is a fantastic glimpse of where geometry can take you when you escape from the constraints of Euclidean Geometry – the geometry of circles and lines taught at school. Riemann, the German 19th Century mathematician, devised a way of representing every point on a plane as a point on a sphere. He did this by first centering a 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 complex plane (z) can be represented as a unique point on the sphere (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 (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).
So what does this new way of representing the two dimensional (complex) plane actually allow us to see? Well, it turns on its head our conventional notions about “straight” lines. A straight line on the complex plane is projected to a circle going through North on the Riemann sphere (as illustrated above). Because North itself represents the point at infinity, this allows a line of infinite length to be represented on the sphere.
Equally, a circle drawn on the Riemann sphere not passing through North will project to a circle in the complex plane (as shown in the diagram above). So, on the Riemann sphere – which remember is isomorphic (mathematically identical) to the extended complex plane, straight lines and circles differ only in their position on the sphere surface. 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. For a two dimensional being living on a Riemann sphere, travel in what he regarded as straight lines would in fact be geodesic (a curved line joining up A and B on the sphere with minimum distance).
By the same logic, 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. This is not just science fiction – indeed Albert Einstein was one to suggest this as a possible explanation for the structure of the universe. Indeed, such a scenario would allow 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 liked this 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.
The Riemann Hypothesis Explained. What is the Riemann Hypothesis – and how solving it can win you $1 million
Are You Living in a Computer Simulation? Nick Bostrom uses logic and probability to make a case about our experience of reality.
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.
This classic clip “proves” how 25/5 = 14, and does it three different ways. Maths is a powerful method for providing proof – but we need to be careful that each step is based on correct assumptions.
One of the most well known fake proofs is as follows:
let a = b
Then a2 = ab
a2 – b2 = ab – b2
(a-b)(a+b) = b(a-b)
a+b = b (divide by a-b )
b+b = b (as a = b)
2b = b
2 = 1
Can you spot the step that causes the proof to be incorrect?
Another well known maths problem that appears to prove the impossible is the following:
This was created by magician Paul Curry – and is called Curry’s Paradox. You can work out the areas of all the 4 different coloured shapes on both triangles, and yet by simply rearranging them you created a different area.
A third “proof” shows that -1 = 1:
Let a = b = -1
a2 = b2
2a2 = 2b2
a2 = 2b2 – a2
a = √(2b2 – a2)
a = √(2(-1)2 – (-1)2)
a = √(1)
-1 = 1
And finally a proof that 1= 0. This last proof was used by Italian mathematician Guido Ubaldus as an example of a proof of God because it showed how something could appear from nothing.
0 = 0 + 0 + 0 + 0 ……
0 = (1-1) + (1-1) + (1-1) + (1-1) ……
0 = 1-1+1-1+1….
0 = 1 + (-1+1 ) + (-1+1) + ….
0 = 1
So, maths is a powerful tool for convincing people of an argument – but you always need to make sure that the maths is accurate! If you want to see the problems in the above proofs, highlight below (explanation in white text):
1) We divide by (a-b) in the 5th line. As a = b, then (a-b) = 0. We can’t divide by zero!
2) Neither of the “triangles” are in fact triangles – the hypotenuse is not actually straight. This discrepancy allows for the apparent paradox.
3) In the second to last line we square root 1, but this has 2 possible answers, 1 or -1. As a is already defined as a = -1 then there is no contradiction.
4) This is very similar to the Cesaro Summation problem which exercised mathematicians for centuries. The infinite summation of 0 + 0 + 0 + 0 … is not the same as the infinite summation 1 – 1 + 1 – 1 + 1 ….
Western music has its roots in the harmonics discovered by Pythagoras – himself a keen musician – over 2000 years ago. Pythagoras noticed that certain string ratios would produce sounds that were in harmony with each other. The simplest example is illustrated above with an electric guitar. When a string is played, and then that same string pressed half-way along its length (in the guitar’s case the 12th fret), then we get the same note – this is a whole octave.
If you were to then half the distance again you would get another octave (which explains why guitar frets get smaller and smaller near the base of the instrument – the frets mark ratios relative to the whole string).
The ratio 1: 1/2 shows the ratio of an octave. A full length string: half length string. We can multiply both sides by 2 to remove the fraction to get, 2: 1. This is the octave ratio.
All the other harmonies that are the basis of Western music can also be understood through similar ratios. The chord sequence E, A, B – which is the standard progression for blues and modern music comprises of the base note (in this case E), along with the perfect fourth (A) and the major fifth (B) of the base note.
Looking at the guitar fret we can see that the perfect fourth (A), which occurs on the fifth fret, has the ratio 1: 3/4. That is 1 whole string: 3/4 of the whole string. We can simplify this to get 4:3.
The major fifth (B) which occurs on the seventh fret has the ratio 1: 2/3 which simplifies to 3:2.
The other most likely note used in the key of E would then also be either G (the minor third) which has a ratio of 6:5, or G sharp (the major third) which has a ratio of 5:4).
It’s interesting that we find these particular whole number ratios pleasing to listen to – indeed these are the notes that often sound “right” when playing through songs. It’s also helpful to look at the circle of fifths – which shows all notes which are in the ratio 3:2 with each other. Moving around the circle again produces music which sounds nice. For an example of this, starting at C, the progression C,G,D,A,E is the one used by Jimmy Hendrix in the classic song, Hey Joe
There are lots of other areas to explore when looking at the relationship between maths and music – one of which is looking at how we can model the wave frequencies of notes using modified sine/cosine curves. The IB have included a piece of coursework on this as an example for the new exploration topics.
Another interesting exploration is looking at the strange properties of the Harmonic Sequence – which is the sequence 1, 1/2, 1/3, 1/4… This sequence like many of those found in music is said to be in harmonic progression . There are some interesting paradoxes related to the harmonic sequence – and a variety of methods of proving that the sum of this sequence (the series) actually diverges to infinity – even though you would intuitively expect it to converge. The video below provides a taster on this topic:
If you liked this post you might also like:
Synesthesia – Do Your Numbers Have Colour? What happens when 2 senses get cross-wired in the brain.
Wau: The Most Amazing Number in the World? A great video by Vi Hart – see if you can spot the twist!
Maths IA – Maths Exploration Topics:
Scroll down this page to find over 300 examples of maths IA exploration topics and ideas for IB mathematics students doing their internal assessment (IA) coursework. Topics include Algebra and Number (proof), Geometry, Calculus, Statistics and Probability, Physics, and links with other subjects. Suitable for Applications and Interpretations students (SL and HL) and also Analysis and Approaches students (SL and HL).
Essential resources for IB students:
Make sure to visit my page on Paper 3s to download 13 different Paper 3s (which would also make excellent coursework starting points) complete with full markschemes.
Also make sure to visit my page for Exploration guides to download 3 specific guides to support the whole coursework process for IB students.
Essential resources for IB teachers:
If you are a teacher then please also visit my new site: intermathematics.com. My new site has been designed specifically for teachers of mathematics at international schools. The content now includes over 3000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL and HL Applications syllabus. Some of the content includes:
- Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics. These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
- Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
- A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.
Maths IA – Maths Exploration Topics
A list with over 300 examples of maths IA exploration topics and ideas for IB mathematics students doing their internal assessment (IA) coursework. Suitable for Applications and Interpretations students (SL and HL) and also Analysis and Approaches students (SL and HL).
Algebra and number
1) Modular arithmetic – This technique is used throughout Number Theory. For example, Mod 3 means the remainder when dividing by 3.
2) Goldbach’s conjecture: “Every even number greater than 2 can be expressed as the sum of two primes.” One of the great unsolved problems in mathematics.
3) Probabilistic number theory
4) Applications of complex numbers: The stunning graphics of Mandelbrot and Julia Sets are generated by complex numbers.
5) Diophantine equations: These are polynomials which have integer solutions. Fermat’s Last Theorem is one of the most famous such equations.
6) Continued fractions: These are fractions which continue to infinity. The great Indian mathematician Ramanujan discovered some amazing examples of these.
7) Patterns in Pascal’s triangle: There are a large number of patterns to discover – including the Fibonacci sequence.
8) Finding prime numbers: The search for prime numbers and the twin prime conjecture are some of the most important problems in mathematics. There is a $1 million prize for solving the Riemann Hypothesis and $250,000 available for anyone who discovers a new, really big prime number.
9) Random numbers
10) Pythagorean triples: A great introduction into number theory – investigating the solutions of Pythagoras’ Theorem which are integers (eg. 3,4,5 triangle).
11) Mersenne primes: These are primes that can be written as 2^n -1.
12) Magic squares and cubes: Investigate magic tricks that use mathematics. Why do magic squares work?
13) Loci and complex numbers
14) Egyptian fractions: Egyptian fractions can only have a numerator of 1 – which leads to some interesting patterns. 2/3 could be written as 1/6 + 1/2. Can all fractions with a numerator of 2 be written as 2 Egyptian fractions?
15) Complex numbers and transformations
16) Euler’s identity: An equation that has been voted the most beautiful equation of all time, Euler’s identity links together 5 of the most important numbers in mathematics.
17) Chinese remainder theorem. This is a puzzle that was posed over 1500 years ago by a Chinese mathematician. It involves understanding the modulo operation.
18) Fermat’s last theorem: A problem that puzzled mathematicians for centuries – and one that has only recently been solved.
19) Natural logarithms of complex numbers
20) Twin primes problem: The question as to whether there are patterns in the primes has fascinated mathematicians for centuries. The twin prime conjecture states that there are infinitely many consecutive primes ( eg. 5 and 7 are consecutive primes). There has been a recent breakthrough in this problem.
21) Hypercomplex numbers
22) Diophantine application: Cole numbers
23) Perfect Numbers: Perfect numbers are the sum of their factors (apart from the last factor). ie 6 is a perfect number because 1 + 2 + 3 = 6.
24) Euclidean algorithm for GCF
25) Palindrome numbers: Palindrome numbers are the same backwards as forwards.
26) Fermat’s little theorem: If p is a prime number then a^p – a is a multiple of p.
27) Prime number sieves
28) Recurrence expressions for phi (golden ratio): Phi appears with remarkable consistency in nature and appears to shape our understanding of beauty and symmetry.
29) The Riemann Hypothesis – one of the greatest unsolved problems in mathematics – worth $1million to anyone who solves it (not for the faint hearted!)
30) Time travel to the future: Investigate how traveling close to the speed of light allows people to travel “forward” in time relative to someone on Earth. Why does the twin paradox work?
31) Graham’s Number – a number so big that thinking about it could literally collapse your brain into a black hole.
32) RSA code – the most important code in the world? How all our digital communications are kept safe through the properties of primes.
33) The Chinese Remainder Theorem: This is a method developed by a Chinese mathematician Sun Zi over 1500 years ago to solve a numerical puzzle. An interesting insight into the mathematical field of Number Theory.
34) Cesaro Summation: Does 1 – 1 + 1 – 1 … = 1/2?. A post which looks at the maths behind this particularly troublesome series.
35) Fermat’s Theorem on the sum of 2 squares – An example of how to use mathematical proof to solve problems in number theory.
36) Can we prove that 1 + 2 + 3 + 4 …. = -1/12 ? How strange things happen when we start to manipulate divergent series.
37) Mathematical proof and paradox – a good opportunity to explore some methods of proof and to show how logical errors occur.
38) Friendly numbers, Solitary numbers, perfect numbers. Investigate what makes a number happy or sad, or sociable! Can you find the loop of infinite sadness?
39) Zeno’s Paradox – Achilles and the Tortoise – A look at the classic paradox from ancient Greece – the philosopher “proved” a runner could never catch a tortoise – no matter how fast he ran.
40) Stellar Numbers – This is an excellent example of a pattern sequence investigation. Choose your own pattern investigation for the exploration.
41) Arithmetic number puzzle – It could be interesting to do an exploration where you solve number problems – like this one.
42) Normal Numbers – and random number generators – what is a normal number – and how are they connected to random number generators?
43) Narcissistic Numbers – what makes a number narcissistic – and how can we find them all?
44) Modelling Chaos – how we can use grahical software to understand the behavior of sequences
45) The Mordell Equation. What is the Mordell equation and how does it help us solve mathematical problems in number theory?
46) Ramanujan’s Taxi Cab and the Sum of 2 Cubes. Explore this famous number theory puzzle.
47) Hollow cubes and hypercubes investigation. Explore number theory in higher dimensions!
48) When do 2 squares equal 2 cubes? A classic problem in number theory which can be solved through computational power.
49) Rational approximations to irrational numbers. How accurately can be approximate irrationals?
50) Square triangular numbers. When do we have a square number which is also a triangular number?
51) Complex numbers as matrices – Euler’s identity. We can use a matrix representation of complex numbers to test whether Euler’s identity still holds.
52) Have you got a Super Brain? How many different ways can we use to solve a number theory problem?
Geometry
1a) Non-Euclidean geometries: This allows us to “break” the rules of conventional geometry – for example, angles in a triangle no longer add up to 180 degrees. In some geometries triangles add up to more than 180 degrees, in others less than 180 degrees.
1b) The shape of the universe – non-Euclidean Geometry is at the heart of Einstein’s theories on General Relativity and essential to understanding the shape and behavior of the universe.
2) Hexaflexagons: These are origami style shapes that through folding can reveal extra faces.
3) Minimal surfaces and soap bubbles: Soap bubbles assume the minimum possible surface area to contain a given volume.
4) Tesseract – a 4D cube: How we can use maths to imagine higher dimensions.
5) Stacking cannon balls: An investigation into the patterns formed from stacking canon balls in different ways.
6) Mandelbrot set and fractal shapes: Explore the world of infinitely generated pictures and fractional dimensions.
7) Sierpinksi triangle: a fractal design that continues forever.
8) Squaring the circle: This is a puzzle from ancient times – which was to find out whether a square could be created that had the same area as a given circle. It is now used as a saying to represent something impossible.
9) Polyominoes: These are shapes made from squares. The challenge is to see how many different shapes can be made with a given number of squares – and how can they fit together?
10) Tangrams: Investigate how many different ways different size shapes can be fitted together.
11) Understanding the fourth dimension: How we can use mathematics to imagine (and test for) extra dimensions.
12) The Riemann Sphere – an exploration of some non-Euclidean geometry. Straight lines are not straight, parallel lines meet and angles in a triangle don’t add up to 180 degrees.
13) Graphically understanding complex roots – have you ever wondered what the complex root of a quadratic actually means graphically? Find out!
14) Circular inversion – what does it mean to reflect in a circle? A great introduction to some of the ideas behind non-euclidean geometry.
15) Julia Sets and Mandelbrot Sets – We can use complex numbers to create beautiful patterns of infinitely repeating fractals. Find out how!
16) Graphing polygons investigation. Can we find a function that plots a square? Are there functions which plot any polygons? Use computer graphing to investigate.
17) Graphing Stewie from Family Guy. How to use graphic software to make art from equations.
18) Hyperbolic geometry – how we can map the infinite hyperbolic plane onto the unit circle, and how this inspired the art of Escher.
19) Elliptical Curves– how this class of curves have importance in solving Fermat’s Last Theorem and in cryptography.
20) The Coastline Paradox – how we can measure the lengths of coastlines, and uses the idea of fractals to arrive at fractional dimensions.
21) Projective geometry – the development of geometric proofs based on points at infinity.
22) The Folium of Descartes. This is a nice way to link some maths history with studying an interesting function.
23) Measuring the Distance to the Stars. Maths is closely connected with astronomy – see how we can work out the distance to the stars.
24) A geometric proof for the arithmetic and geometric mean. Proof doesn’t always have to be algebraic. Here is a geometric proof.
25) Euler’s 9 Point Circle. This is a lovely construction using just compasses and a ruler.
26) Plotting the Mandelbrot Set – using Geogebra to graphically generate the Mandelbrot Set.
27) Volume optimization of a cuboid – how to use calculus and graphical solutions to optimize the volume of a cuboid.
28) Ford Circles– how to generate Ford circles and their links with fractions.
29) Classical Geometry Puzzle: Finding the Radius. This is a nice geometry puzzle solved using a variety of methods.
30) Can you solve Oxford University’s Interview Question?. Try to plot the locus of a sliding ladder.
31) The Shoelace Algorithm to find areas of polygons. How can we find the area of any polygon?
32) Soap Bubbles, Wormholes and Catenoids. What is the geometric shape of soap bubbles?
33) Can you solve an Oxford entrance question? This problem asks you to explore a sliding ladder.
34) The Tusi circle – how to create a circle rolling inside another circle using parametric equations.
35) Sphere packing – how to fit spheres into a package to minimize waste.
36) Sierpinski triangle – an infinitely repeating fractal pattern generated by code.
37) Generating e through probability and hypercubes. This amazing result can generate e through considering hyper-dimensional shapes.
38) Find the average distance between 2 points on a square. If any points are chosen at random in a square what is the expected distance between them?
39) Finding the average distance between 2 points on a hypercube. Can we extend our investigation above to a multi-dimensional cube?
40) Finding focus with Archimedes. The Greeks used a very different approach to understanding quadratics – and as a result had a deeper understanding of their physical properties linked to light and reflection.
41) Chaos and strange Attractors: Henon’s map. Gain a deeper understanding of chaos theory with this investigation.
Calculus/analysis and functions
1) The harmonic series: Investigate the relationship between fractions and music, or investigate whether this series converges.
2) Torus – solid of revolution: A torus is a donut shape which introduces some interesting topological ideas.
3) Projectile motion: Studying the motion of projectiles like cannon balls is an essential part of the mathematics of war. You can also model everything from Angry Birds to stunt bike jumping. A good use of your calculus skills.
4) Why e is base of natural logarithm function: A chance to investigate the amazing number e.
5) Fourier Transforms – the most important tool in mathematics? Fourier transforms have an essential part to play in modern life – and are one of the keys to understanding the world around us. This mathematical equation has been described as the most important in all of physics. Find out more! (This topic is only suitable for IB HL students).
6) Batman and Superman maths – how to use Wolfram Alpha to plot graphs of the Batman and Superman logo
7) Explore the Si(x) function – a special function in calculus that can’t be integrated into an elementary function.
8) The Remarkable Dirac Delta Function. This is a function which is used in Quantum mechanics – it describes a peak of zero width but with area 1.
9) Optimization of area – an investigation. This is an nice example of how you can investigation optimization of the area of different polygons.
10) Envelope of projectile motion. This investigates a generalized version of projectile motion – discover what shape is created.
11) Projectile Motion Investigation II. This takes the usual projectile motion ideas and generalises them to investigate equations of ellipses formed.
12) Projectile Motion III: Varying gravity. What would projectile motion look like on different planets?
13) The Tusi couple – A circle rolling inside a circle. This is a lovely result which uses parametric functions to create a beautiful example of mathematical art.
14) Galileo’s Inclined Planes. How did Galileo achieve his breakthrough understanding of gravity? Follow in the footsteps of a genius!
Statistics and modelling 1 [topics could be studied in-depth]
1) Traffic flow: How maths can model traffic on the roads.
2) Logistic function and constrained growth
3) Benford’s Law – using statistics to catch criminals by making use of a surprising distribution.
4) Bad maths in court – how a misuse of statistics in the courtroom can lead to devastating miscarriages of justice.
5) The mathematics of cons – how con artists use pyramid schemes to get rich quick.
6) Impact Earth – what would happen if an asteroid or meteorite hit the Earth?
7) Black Swan events – how usefully can mathematics predict small probability high impact events?
8) Modelling happiness – how understanding utility value can make you happier.
9) Does finger length predict mathematical ability? Investigate the surprising correlation between finger ratios and all sorts of abilities and traits.
10) Modelling epidemics/spread of a virus
11) The Monty Hall problem – this video will show why statistics often lead you to unintuitive results.
12) Monte Carlo simulations
13) Lotteries
14) Bayes’ theorem: How understanding probability is essential to our legal system.
15) Birthday paradox: The birthday paradox shows how intuitive ideas on probability can often be wrong. How many people need to be in a room for it to be at least 50% likely that two people will share the same birthday? Find out!
16) Are we living in a computer simulation? Look at the Bayesian logic behind the argument that we are living in a computer simulation.
17) Does sacking a football manager affect results? A chance to look at some statistics with surprising results.
18) Which times tables do students find most difficult? A good example of how to conduct a statistical investigation in mathematics.
19) Introduction to Modelling. This is a fantastic 70 page booklet explaining different modelling methods from Moody’s Mega Maths Challenge.
20) Modelling infectious diseases – how we can use mathematics to predict how diseases like measles will spread through a population
21) Using Chi Squared to crack codes – Chi squared can be used to crack Vigenere codes which for hundreds of years were thought to be unbreakable. Unleash your inner spy!
22) Modelling Zombies – How do zombies spread? What is your best way of surviving the zombie apocalypse? Surprisingly maths can help!
23) Modelling music with sine waves – how we can understand different notes by sine waves of different frequencies. Listen to the sounds that different sine waves make.
24) Are you psychic? Use the binomial distribution to test your ESP abilities.
25) Reaction times – are you above or below average? Model your data using a normal distribution.
26) Modelling volcanoes – look at how the Poisson distribution can predict volcanic eruptions, and perhaps explore some more advanced statistical tests.
27) Could Trump win the next election? How the normal distribution is used to predict elections.
28) How to avoid a Troll – an example of a problem solving based investigation
29) The Gini Coefficient – How to model economic inequality
30) Maths of Global Warming – Modeling Climate Change – Using Desmos to model the change in atmospheric Carbon Dioxide.
31) Modelling radioactive decay – the mathematics behind radioactivity decay, used extensively in science.
32) Circular Motion: Modelling a Ferris wheel. Use Tracker software to create a Sine wave.
33) Spotting Asset Bubbles. How to use modeling to predict booms and busts.
34) The Rise of Bitcoin. Is Bitcoin going to keep rising or crash?
35) Fun with Functions!. Some nice examples of using polar coordinates to create interesting designs.
36) Predicting the UK election using linear regression. The use of regression in polling predictions.
37) Modelling a Nuclear War. What would happen to the climate in the event of a nuclear war?
38) Modelling a football season. We can use a Poisson model and some Excel expertise to predict the outcome of sports matches – a technique used by gambling firms.
39)Modeling hours of daylight – using Desmos to plot the changing hours of daylight in different countries.
40) Modelling the spread of Coronavirus (COVID-19). Using the SIR model to understand epidemics.
41) Finding the volume of a rugby ball (or American football). Use modeling and volume of revolutions.
42) The Martingale system paradox. Explore a curious betting system still used in currency trading today.
Statistics and modelling 2 [more simplistic topics: correlation, normal, Chi squared]
1) Is there a correlation between hours of sleep and exam grades?Studies have shown that a good night’s sleep raises academic attainment.
2) Is there a correlation between height and weight? (pdf). The NHS use a chart to decide what someone should weigh depending on their height. Does this mean that height is a good indicator of weight?
3) Is there a correlation between arm span and foot height? This is also a potential opportunity to discuss the Golden Ratio in nature.
4) Is there a correlation between smoking and lung capacity?
5) Is there a correlation between GDP and life expectancy? Run the Gapminder graph to show the changing relationship between GDP and life expectancy over the past few decades.
7) Is there a correlation between numbers of yellow cards a game and league position?
Use the Guardian Stats data to find out if teams which commit the most fouls also do the best in the league.
8) Is there a correlation between Olympic 100m sprint times and Olympic 15000m times?
Use the Olympic database to find out if the 1500m times have got faster in the same way the 100m times have got quicker over the past few decades.
9) Is there a correlation between time taken getting to school and the distance a student lives from school?
10) Does eating breakfast affect your grades?
11) Is there a correlation between stock prices of different companies? Use Google Finance to collect data on company share prices.
12) Is there a correlation between blood alcohol laws and traffic accidents?
13) Is there a correlation between height and basketball ability? Look at some stats for NBA players to find out.
14) Is there a correlation between stress and blood pressure?
15) Is there a correlation between Premier League wages and league positions?
16) Are a sample of student heights normally distributed? We know that adult population heights are normally distributed – what about student heights?
17) Are a sample of flower heights normally distributed?
18) Are a sample of student weights normally distributed?
19) Are the IB maths test scores normally distributed? (pdf). IB test scores are designed to fit a bell curve. Investigate how the scores from different IB subjects compare.
20) Are the weights of “1kg” bags of sugar normally distributed?
21) Does gender affect hours playing sport? A UK study showed that primary school girls play much less sport than boys.
22) Investigation into the distribution of word lengths in different languages. The English language has an average word length of 5.1 words. How does that compare with other languages?
23) Do bilingual students have a greater memory recall than non-bilingual students?
Studies have shown that bilingual students have better “working memory” – does this include memory recall?
Games and game theory
1) The prisoner’s dilemma: The use of game theory in psychology and economics.
2) Sudoku
3) Gambler’s fallacy: A good chance to investigate misconceptions in probability and probabilities in gambling. Why does the house always win?
4) Bluffing in Poker: How probability and game theory can be used to explore the the best strategies for bluffing in poker.
5) Knight’s tour in chess: This chess puzzle asks how many moves a knight must make to visit all squares on a chess board.
6) Billiards and snooker
7) Zero sum games
8) How to “Solve” Noughts and Crossess (Tic Tac Toe) – using game theory. This topics provides a fascinating introduction to both combinatorial Game Theory and Group Theory.
9) Maths and football – Do managerial sackings really lead to an improvement in results? We can analyse the data to find out. Also look at the finances behind Premier league teams
10) Is there a correlation between Premier League wages and league position? Also look at how the Championship compares to the Premier League.
11) The One Time Pad – an uncrackable code? Explore the maths behind code making and breaking.
12) How to win at Rock Paper Scissors. Look at some of the maths (and psychology behind winning this game.
13) The Watson Selection Task – a puzzle which tests logical reasoning. Are maths students better than history students?
Topology and networks
1) Knots
2) Steiner problem
3) Chinese postman problem – This is a problem from graph theory – how can a postman deliver letters to every house on his streets in the shortest time possible?
4) Travelling salesman problem
5) Königsberg bridge problem: The use of networks to solve problems. This particular problem was solved by Euler.
6) Handshake problem: With n people in a room, how many handshakes are required so that everyone shakes hands with everyone else?
7) Möbius strip: An amazing shape which is a loop with only 1 side and 1 edge.
8) Klein bottle
9) Logic and sets
10) Codes and ciphers: ISBN codes and credit card codes are just some examples of how codes are essential to modern life. Maths can be used to both make these codes and break them.
11) Zeno’s paradox of Achilles and the tortoise: How can a running Achilles ever catch the tortoise if in the time taken to halve the distance, the tortoise has moved yet further away?
12) Four colour map theorem – a puzzle that requires that a map can be coloured in so that every neighbouring country is in a different colour. What is the minimum number of colours needed for any map?
13) Telephone Numbers – these are numbers with special properties which grow very large very quickly. This topic links to graph theory.
14)The Poincare Conjecture and Grigori Perelman – Learn about the reclusive Russian mathematician who turned down $1 million for solving one of the world’s most difficult maths problems.
Mathematics and Physics
1) The Monkey and the Hunter – How to Shoot a Monkey – Using Newtonian mathematics to decide where to aim when shooting a monkey in a tree.
2) How to Design a Parachute – looking at the physics behind parachute design to ensure a safe landing!
3) Galileo: Throwing cannonballs off The Leaning Tower of Pisa – Recreating Galileo’s classic experiment, and using maths to understand the surprising result.
4) Rocket Science and Lagrange Points – how clever mathematics is used to keep satellites in just the right place.
5) Fourier Transforms – the most important tool in mathematics? – An essential component of JPEG, DNA analysis, WIFI signals, MRI scans, guitar amps – find out about the maths behind these essential technologies.
6) Bullet projectile motion experiment – using Tracker software to model the motion of a bullet.
7) Quantum Mechanics – a statistical universe? Look at the inherent probabilistic nature of the universe with some quantum mechanics.
8) Log Graphs to Plot Planetary Patterns. The planets follow a surprising pattern when measuring their distances.
9) Modeling with springs and weights. Some classic physics – which generates some nice mathematical graphs.
10) Is Intergalactic space travel possible? Using the physics of travel near the speed of light to see how we could travel to other stars.
Maths and computing
1) The Van Eck Sequence – The Van Eck Sequence is a sequence that we still don’t fully understand – we can use programing to help!
2) Solving maths problems using computers – computers are really useful in solving mathematical problems. Here are some examples solved using Python.
3) Stacking cannonballs – solving maths with code – how to stack cannonballs in different configurations.
4) What’s so special about 277777788888899? – Playing around with multiplicative persistence – can you break the world record?
5) Project Euler: Coding to Solve Maths Problems. A nice starting point for students good at coding – who want to put these skills to the test mathematically.
6) Square Triangular Numbers. Can we use a mixture of pure maths and computing to solve this problem?
7) When do 2 squares equal 2 cubes? Can we use a mixture of pure maths and computing to solve this problem?
8) Hollow Cubes and Hypercubes investigation. More computing led investigations
9) Coding Hailstone Numbers. How can we use computers to gain a deeper understanding of sequences?
Further ideas:
1) Radiocarbon dating – understanding radioactive decay allows scientists and historians to accurately work out something’s age – whether it be from thousands or even millions of years ago.
2) Gravity, orbits and escape velocity – Escape velocity is the speed required to break free from a body’s gravitational pull. Essential knowledge for future astronauts.
3) Mathematical methods in economics – maths is essential in both business and economics – explore some economics based maths problems.
4) Genetics – Look at the mathematics behind genetic inheritance and natural selection.
5) Elliptical orbits – Planets and comets have elliptical orbits as they are influenced by the gravitational pull of other bodies in space. Investigate some rocket science!
6) Logarithmic scales – Decibel, Richter, etc. are examples of log scales – investigate how these scales are used and what they mean.
7) Fibonacci sequence and spirals in nature – There are lots of examples of the Fibonacci sequence in real life – from pine cones to petals to modelling populations and the stock market.
8) Change in a person’s BMI over time – There are lots of examples of BMI stats investigations online – see if you can think of an interesting twist.
9) Designing bridges – Mathematics is essential for engineers such as bridge builders – investigate how to design structures that carry weight without collapse.
10) Mathematical card tricks – investigate some maths magic.
11) Flatland by Edwin Abbott – This famous book helps understand how to imagine extra dimension. You can watch a short video on it here
12) Towers of Hanoi puzzle – This famous puzzle requires logic and patience. Can you find the pattern behind it?
13) Different number systems – Learn how to add, subtract, multiply and divide in Binary. Investigate how binary is used – link to codes and computing.
14) Methods for solving differential equations – Differential equations are amazingly powerful at modelling real life – from population growth to to pendulum motion. Investigate how to solve them.
15) Modelling epidemics/spread of a virus – what is the mathematics behind understanding how epidemics occur? Look at how infectious Ebola really is.
16) Hyperbolic functions – These are linked to the normal trigonometric functions but with notable differences. They are useful for modelling more complex shapes.
17) Medical data mining – Explore the use and misuse of statistics in medicine and science.
18)Waging war with maths: Hollow squares. How mathematical formations were used to fight wars.
19) The Barnsley Fern: Mathematical Art – how can we use iterative processes to create mathematical art?
The Gorilla in the Room and Other Great Maths Investigations
These topics are a great way to add interest to statistics and probability lessons at KS3 and KS4 level, and also a good example of investigations that IB students can conduct. They also have a nice link to ToK – how can we believe what we see or what we hear? To what extent should we trust our senses? And it shows the power of statistics and empirical testing in trying to understand what is externally real and what is our own version of reality.
For each one, have the students make a hypothesis (if possible without giving the endings away!), then collect some data as to how the students react. Then look at how the data could be collected in a larger scale experiment (or how the experiment could be modified).
The first one at the top of the page is the “Fa, Ba” test. This is a really curious experiment that shows that what we “hear” is actually often influenced by what we see.
The second one is the amazing colour changing card trick by Richard Wiseman. This is also a great way of showing how we often fail to see what is really in front of us:
The third video is even more impressive – though it doesn’t work on all students. You have to set this one up so that all students are really intently concentrating on the screen – perhaps a prize for the student who gets the answer correct? Also no talking! Students have to count basketball passes:
The last one is a good test of whether students are “right brain” or “left brain” dominant. They have to stare at a rotating woman – some students will see this going clockwise, others anticlockwise. Some will be able to switch between the 2 views. If they can’t (I initially could only see this going in an anti clockwise direction) near the end of the video it shows the woman rotating in a clockwise direction to help. Then rewinding the video to the start – and as if by magic she had changed direction.
If you liked this post you might also like:
Even Pigeons Can Do Maths A discussion about the ability of both chimps and pigeons to count
Finger Ratio Predicts Maths Ability? A post which discusses the correlation between the two.
Are You Living in a Computer Simulation?
This idea might be familiar to fans of The Matrix – and at first glance may seem somewhat unbelievable. However, Oxford University Professor Nick Bostrom makes an interesting case using both conditional probability and logic as to why it’s more likely than you might think.
The summary of Bostrom’s Computer Simulation argument is the following:
At least one of the following propositions is true: (1) the human species is very likely to go extinct before reaching a “posthuman” stage; (2) any posthuman civilization is extremely unlikely to run a significant number of simulations of their evolutionary history (or variations thereof); (3) we are almost certainly living in a computer simulation. It follows that the belief that there is a significant chance that we will one day become posthumans who run ancestor-simulations is false, unless we are currently living in a simulation.
The full paper where he makes his argument is available as a pdf here – and is well worth a read. Alternatively Bostrom makes this case in a detailed interview:
Taking the argument step by step, firstly when Bostrom talks about a “posthuman” stage he is referring to an advanced civilisation with significantly greater technological capabilities than we have at present. Such a civilisation would have the ability to run a computer simulation so accurate that it would be indistinguishable from “real life”.
This is a twist on the traditional “Brain in a Vat” thought experiment much loved by philosophers when trying to argue whether we be sure that anything exists outside our own subjective experience:
Based on the same logic, we have no way of genuinely knowing whether we are really “here” or whether we are nothing but a computer model designed to give the impression that we really exist. Interestingly, the possibility that our individual life, the world around us and indeed everything we know about the universe may be false means that we can never truly claim to have knowledge of anything.
I think that most optimists would think that civilisation has the potential to develop into a “posthuman” phase of advanced technology. Indeed, you only need to look at the phenomenal growth in computer power (see Moore’s Law) to have confidence that should we stick around long enough, we will have the computational power possible to run such simulations.
So if we optimistically accept that humans will reach a “posthuman” stage, then it’s even easier to accept the second proposition – that if an advanced civilisation is able to run such civilisations they will do. After all human nature is such that we tend to do things just because we can – and in any case running such ancestor simulations would potentially be very beneficial for real world modelling.
If we do accept both these premises, then this therefore leads to the argument that we are almost certainly living in a computer simulation. Why? Well, an advanced civilisation with the computational power to run ancestor simulations would likely run a large number of them – and if there is only one real world, then our experience of a world is likely to be one of these simulations.
As a ToK topic this is a fantastic introduction to epistemological questions about the limits of knowledge and questions of existence, and is a really good example of the power of logic and mathematics to reveal possibilities about the world outside our usual bounds of thinking.
If you enjoyed 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.
Is Maths Invented or Discovered? – A discussion about some of the basic philosophical questions that arise in mathematics.
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.
Black Swans and Civilisation Collapse
A really interesting branch of mathematics is involved in making future predictions about how civilisation will evolve in the future – and indeed looking at how robust our civilisation is to external shocks. This is one area in which mathematical models do not have a good record as it is incredibly difficult to accurately assign probabilities and form policy recommendations for events in the future.
Malthusian Catastrophe
One of the most famous uses of mathematical models in this context was by Thomas Malthus in 1798. He noted that the means of food production were a fundamental limiting factor on population growth – and that if population growth continued beyond the means of food production that there would be (what is now termed) a “Malthusian catastrophe” of a rapid population crash.
As it turns out, agrarian productivity has been able to keep pace with the rapid population growth of the past 200 years.
Looking at the graph we can see that whilst it took approximately 120 years for the population to double from 1 billion to 2 billion, it only took 55 years to double again. It would be a nice exercise to try and see what equation fits this graph – and also look at the rate of change of population (is it now slowing down?) The three lines at the end of the graph are the three different UN predictions – high end, medium and low end estimate. There’s a pretty stark difference between high end and low end estimates by 2100 – between 16 billion and 6 billion! So what does that tell us about the accuracy of such predictions?
Complex Civilisations
More recently academics like Joseph Tainter and Jared Diamond have popularised the notion of civilisations as vulnerable to collapse due to ever increasing complexity. In terms of robustness of civilisation one can look at an agrarian subsistence example. Agrarian subsistence is pretty robust against civilisation collapse – small self sufficient units may themselves be rather vulnerable to famines and droughts on an individual level, but as a society they are able to ride out most catastrophes intact.
The next level up from agrarian subsistence is a more organised collective – around a central authority which is able to (say) provide irrigation technology through a system of waterways. Immediately the complexity of society has increased, but the benefits of irrigation allow much more crops to be grown and thus the society can support a larger population. However, this complexity comes at a cost – society now is reliant on those irrigation channels – and any damage to them could be catastrophic to society as a whole.
To fast forward to today, we have now an incredibly complex society, far far removed from our agrarian past – and whilst that means we have an unimaginably better quality of life, it also means society is more vulnerable to collapse than ever before. To take the example of a Coronal Mass Ejection – in which massive solar discharges hit the Earth. The last large one to hit the Earth was in 1859 but did negligible damage as this was prior to the electrical age. Were the same event to happen today, it would cause huge damage – as we are reliant on electricity for everything from lighting to communication to refrigeration to water supplies. A week without electricity for an urban centre would mean no food, no water, no lighting, no communication and pretty much the entire breakdown of society.
That’s not to say that such an event will happen in our lifetimes – but it does raise an interesting question about intelligent life – if advanced civilisations continue to evolve and in the process grow more and more complex then is this a universal limiting factor on progress? Does ever increasing complexity leave civilisations so vulnerable to catastrophic events that their probabilities of surviving through them grow ever smaller?
Black Swan Events
One of the great challenges for mathematical modelling is therefore trying to assign probabilities for these “Black Swan” events. The term was coined by economist Nassim Taleb – and used to describe rare, low probability events which have very large consequences. If the probability of a very large scale asteroid impact is (say) estimated as 1-100,000 years – but were it to hit it is estimated to cause $35 trillion of damage (half the global GDP) then what is the rational response to such a threat? Dividing the numbers suggests that we should in such a scenario be spending $3.5billion every year on trying to address such an event – and yet which politician would justify such spending on an event that might not happen for another 100,000 years?
I suppose you would have to conclude therefore that our mathematical models are pretty poor at predicting future events, modelling population growth or dictating future and current policy. Which stands in stark contrast to their abilities in modelling the real world (minus the humans). Will this improve in the future, or are we destined to never really be able to predict the complex outcomes of a complex world?
If you enjoyed this post you might also like:
Asteroid Impact Simulation – which allows you to model the consequences of asteroid impacts on Earth.
Chaos Theory – an Unpredictable Universe? – which discusses the difficulties in mathematical modelling when small changes in initial states can have very large consequences.
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.
The Riemann Hypothesis Explained
This is quite a complex topic probably only accessible for high achieving HL IB students, but nevertheless it’s still a fascinating introduction to one of the most important (and valuable) unsolved problems in pure mathematics.
Firstly, the Riemann Hypothesis is concerned with the Riemann zeta function. This function is defined in many ways, but probably the most useful for us is this version:
In other words the Riemann zeta function consists of a sum to infinity multiplied by an external bracket. s is a complex number of the form s = σ + it. This formula is valid for Re(s) > 0 . This means that the real part of the complex number must be positive.
Now, the Riemann Hypothesis is concerned with finding the roots of the Riemann zeta function – ie. what values of complex number s cause the function to be zero. However the equation above is only valid for Re(s) > 0. To check for roots where Re(s) is less than or equal to 0 we can use an alternative representation of the Riemann zeta function:
which shows that the zeta function is zero whenever s = -2,-4,-6…. as for these values the sine term becomes zero. (s = 0 has no solution in this representation because it leaves us with a zeta function of 1 in the far right term – which produces a singularity). These values are called the trivial zeroes of the zeta function.
The other, non-trivial zeroes of the Riemann zeta function are more difficult to find – and the search for them leads to the Riemann hypothesis:
The non-trivial zeroes of the Riemann zeta function have a real part of s equal to 1/2
In other words, ζ(s) has non-trivial zeroes only when s is in the form s = 1/2 + it. This is probably easier to understand in graphical form. Below we have s plotted in the complex plane:
We can see that when s = 1 the function is not defined. This is because when s =1 in the original equation for the zeta function we get a singularity as this causes the bracket to the left of the summation to reduce to 1/0. All the non-trivial zeroes for the zeta function are known to lie in the grey boxed, “critical strip” – and the Riemann hypothesis is that they all lie on the dotted line where the real value is 1/2.
This hypothesis, made by German mathematician Bernhard Riemann in 1859 is still unsolved over 150 years later – despite some of the greatest mathematical minds of the 2oth century attempting the problem. Indeed it is considered by many mathematicians to be the most important unresolved question in pure mathematics. Mathematician David Hilbert who himself collected 23 great unsolved mathematical problems together in 1900 stated,
“If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann hypothesis been proven?”
The problem is today listed as one of the Clay Institute’s Millennium Prize Problems – anyone who can prove it will win $1 million and will quite probably go down in history as one of the greatest mathematicians of all time.
One solution for s which gives a zero of the zeta function is 0.5 + 14.134725142 i. Another one is 0.5 + 21.022039639 i. These both satisfy the Riemann Hypothesis by having a real part of 1/2. Indeed, to date, 10 trillion (10,000,000,000,000) non trivial solutions have been found – and they all have a real part of 1/2. But this is not a proof that it is true for all roots – and so the problem remains unsolved.
So, why is this such an important problem? Well, because there is a connection between the Riemann zeta function and distribution of prime numbers. The function below on the left is another way of representing the Riemann zeta function and the function on the right is an infinite product including all prime numbers:
where:
Understanding the Riemann zeta function will help mathematicians unlock some of the mysteries of the prime numbers – which are the building blocks of number theory (the study of integers). For example looking at the graph below (drawn by a Wolfram Mathlab probject) we can see the function pi(x) plotted against a function which uses both the Riemann zeta function and the distribution of its zeroes. pi(x) is blue graph and shows the number of primes less than or equal to x.
With the number of primes on the y axis, we can see that out of the first 420 numbers there are approximately 80 primes. What is remarkable about the red line is that it so accurately tracks the progress of the prime numbers.
If you are interested in reading more on this the Wikipedia page on the Riemann zeta function goes into a lot more detail. A more lighthearted introduction to the topic is given by the paper, “A Friendly Introduction to the Riemann Hypothesis”
If you enjoyed this post you might also like:
Graham’s Number – a number so large it’ll literally collapse your head into a black hole.
Twin Primes and How Prime Numbers are Distributed – some more discussion on studying prime numbers – in particular the conjecture that there are infinitely many twin primes.
The Million Dollar Maths Problems – some general introductions to the seven million dollar maths problems.
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.
Synesthesia – Do Your Numbers Have Colour?
Synesthesia is another topic which provides insights into how people perceive numbers – and how a synesthetic’s perception of the mathematical world is distinctly different to everyone else’s.
Those with synesthesia have a cross-wiring of brain activity between 2 of their senses – so for example they may hear sounds when they see images, sounds may invoke taste sensations, or numbers may be perceived as colours.
Daniel Tammet, an autistic savant with remarkable memory abilities (he can remember pi to 22 thousand places and learn a new language to fluency in one week). He also has number synesthesia which means that he “sees” numbers as each having their own distinct colour. This also allows him to multiply two numbers in his head almost instantaneously by “seeing” the two colours merge into a third one.
Dr Ramachandran (of phantom limb fame) has written a fascinating academic article looking at synesthesia – and estimates that as many as 1 in 200 people may have some form of it. A simple test of grapheme colour synesthesia (where people perceive numbers with colours) is the graphic below:
For people without synesthesia, locating the 2s from graphic on the left is a slow process, but for people with synesthesia, they can immediately see the 2s as standing out – like the graphic on the right. This test is easily able to distinguish that this type of synesthesia is real.
Those with grapheme synesthesia also report that the image below changes colour – depending on whether they look at the whole image (ie. a five) or concentrate on how it is made of smaller constituent parts (of threes):
What is truly remarkable about synesthesia is what it reveals about our brain’s innate capacity for mathematical calculations far beyond what average people can achieve. Francois Galton, the 19th Century polymath who first documented the condition (which he himself had) described how synesthetics often also experienced a tangible number line in their mind – that was not straight but curved and bent and in which some numbers were closer that others (an example is at the top of the page). This allowed him, and others like Temmet, to perform lightening fast mental calculations of unimaginable complexity. In the above video Daniel is able to divide 13 by 97 in a matter of seconds to over 30 decimal places.
Numberphile have also made a short video in which they interview a lady with synesthesia:
Could one day we all unlock this potential? And what does this condition tell us about whether numbers exist in any tangible sense? Do they exist in a more real sense for a grapheme synesthic than someone else?
If you enjoyed this topic you may also like:
Even Pigeons Can Do Maths – a discussion about the ability of both chimps and pigeons to count
Does finger ratio predict maths ability? – a post which discusses the correlation between the two.
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.
Imagining the 4th Dimension
Imagining extra dimensions is a fantastic ToK topic – it is something which seems counter-intuitively false, something which we have no empirical evidence to support, and yet it is something which seems to fit the latest mathematical models on string theory (which requires 11 dimensions). Mathematical models have consistently been shown to be accurate in describing reality, but when they predict a reality that is outside our realm of experience then what should we believe? Our senses? Our intuition? Or the mathematical models?
Carl Sagan produced a great introduction to the idea of extra dimensions based on the Flatland novel. This imagines reality as experienced by two dimensional beings.
Mobius strips are a good gateway into the weird world of topology – as they are 2D shapes with only 1 side. There are some nice activities to do with Mobius strips – first take a pen and demonstrate that you can cover all of the strip without lifting the pen. Next, cut along the middle of the strip and see the resulting shape. Next start again with a new strip, but this time start cutting from nearer the edge (around 1/3 in). In both cases have students predict what they think will happen.
Next we can move onto the Hypercube (or Tesseract). We can see an Autograph demonstration of what the fourth dimensional cube looks like here.
The page allows you to model 1, then 2, then 3 dimensional traces – each time representing a higher dimensional cube.
It’s also possible to create a 3 dimensional representation of a Tesseract using cocktail sticks – you simply need to make 2 cubes, and then connect one vertex in each cube to the other as in the diagram below:
For a more involved discussion (it gets quite involved!) on imagining extra dimensions, this 10 minute cartoon takes us through how to imagine 10 dimensions.
It might also be worth touching on why mathematicians believe there might be 11 dimensions. Michio Kaku has a short video (with transcript) here and Brian Greene also has a number of good videos on the subject.
All of which brings us onto empirical testing – if a mathematical theory can not be empirically tested then does it differ from a belief? Well, interestingly this theory can be tested – by looking for potential violations to the gravitational inverse square law.
The current theory expects that the extra dimensions are themselves incredibly small – and as such we would only notice their effects on an incredibly small scale. The inverse square law which governs gravitational attraction between 2 objects would be violated on the microscopic level if there were extra dimensions – as the gravitational force would “leak out” into these other dimensions. Currently physicists are carrying out these tests – and as yet no violation of the inverse square law has been found, but such a discovery would be one of the greatest scientific discoveries in history.
Other topics with counter-intuitive arguments about reality based on mathematical models are Nick Bostrom’s Computer Simulation Hypothesis, the Hologram Universe Hypothesis and Everett’s Many Worlds quantum mechanics interpretation. I will blog more on these soon!
If you enjoyed this topic you may also like:
Wolf Goat Cabbage Space – a problem solved by 3d geometry.
Graham’s Number – a number literally big enough to collapse your head into a black hole.
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 