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Maths and Evolutionary Biology

  Maths and Evolutionary Biology Mathematics is often utilised across many fields – lets look at an example from biology, evolutionary biology and paleontology, in trying to understand the development of homo-sapiens.  We can start with a large data set which gives us the data for mammal body mass and brain size in grams (downloaded from here).  I then tidied up this to remove the rows with NA…

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A Cat and Mouse Game

A Cat and Mouse Game The Numberphile video above talks through an investigation in which a mouse is swimming in a pond, with a hungry cat prowling around the edge.  The cat can’t swim, but can run at a speed of 4m/s.  The mouse can swim at a speed of 1m/s and can run faster than the cat on the land.  The question is, can the mouse ever escape and run to safety? Modelling the cat The cat, C…

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Aliquot sequence: An unsolved problem

Aliquot sequence: An unsolved problem At school students get used to the idea that we know all the answers in mathematics – but the aliquot sequence is a simple example of an unsolved problem in mathematics.  The code above (if run for long enough on a super-computer!) might be enough to disprove a conjecture about this sequence.  Let’s look at this sequence and the associated conjecture. Aliquot…

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Time dependent gravity exploration

Time dependent gravity exploration In our universe we have a gravitational constant – i.e gravity is not dependent on time.  If gravity changed with respect to time then the gravitational force exerted by the Sun on Earth would lessen (or increase) over time with all other factors remaining the same. Interestingly time-dependent gravity was first explored by Dirac and some physicists have tried…

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Cowculus - the farmer and the cow

Cowculus – the farmer and the cow The Numberphile video above is an excellent starting point for an investigation – so I thought I’d use this to extend the problem to a more general situation.  The simple case is as follows: A farmer is at point F and a cow at point C. There is a river represented by the line passing through 𝐴𝐵. The distance 𝐴𝐹 is 2km, the distance 𝐵𝐶 is 6km and the distance 𝐴𝐵…

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Getting a 7 in IB Maths Coursework

Getting a 7 in IB Maths Coursework Are you a current IB student or IB teacher? Do you want to learn the tips and tricks to produce excellent Mathematics coursework? Gain the inside track on what makes a good coursework piece from an IB Maths Examiner as you learn all the skills necessary to produce something outstanding.  This course is written for current IB Mathematics students.  It is also…

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Lissajous Curves: Roller Coasters

Roller Coaster design This post continues from the previous post on Lissajous Curves.  Make sure to read that one first! We can design a rollercoaster track by using the following Lissajous Curve: This gives the following graph: Ground level is given by the line y = −50. Distances are in metres and t is measured in seconds. Customers ride an elevator to the starting point (0,50) where the ride…

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AI Masters Olympiad Geometry

AI Masters Olympiad Geometry The team behind Google’s Deep Mind have just released details of a new AI system:  AlphaGeometry This has been specifically trained to solve classical geometry problems – and already is now at the level of a Gold Medalist at the International Olympiad (considering only geometry problems).  This is an incredible achievement – as in order to solve classical geometry…

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Lissajous Curves

Lissajous Curves Lissajous Curves were explored by French Physicist Jules Lissajous in the 1850s.  The picture above (Wikimedia Commons) shows him investigating Lissajous curves through a telescope. Lissajous curves include those which can be written in the form: This parametric form allows us to represent complicated curves which are difficult to write in terms of x and y only.  A simple…

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Using matrices to make fractals

Using matrices to make fractals We start with a triangle ABC, with coordinates 𝐴(0,0) , 𝐵(1,0) , 𝐶( 0,1) as shown above.  We can this triangle F_0 and we then write this as the following matrix: We then have the following algorithm to generate the next triangle F_1. In effect this means that the triangle F_1 is made by combining the coordinates from F_1a, F_1b and F_1c.  So we now need to know…

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Chi Square: Language Recognition II

Chi Square: Language Recognition II I thought I would build on the last post by making a simple spreadsheet that can then easily show which language is being used.  I chose the groupings of letters such that as long as there are at least 1000 letters in the text it will satisfy the Chi square condition of no expected values less than 1 and no more than 20% of expected values less than 5.  Data I…

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Google Page Rank: Trillion dollar maths

Google Page Rank: Billion dollar maths In the early 1990s search engines used to be text based – and would rank pages based on how many times a key word appeared. But this did not discriminate between useful pages and less useful pages. Larry Page and Segei Brin used some maths to come up with an alternative method of ranking pages and made a trillion dollar company! Let’s see the basis of…

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Ladybirds vs Aphids

Ladybirds vs Aphids At t=0 we have a ladybird on the edge of a leaf at point A(0,10) in cm, and an aphid at point B(0,10).  The ladybird is in pursuit of the aphid.  In each time interval of 1 second the ladybird travels 1cm by heading towards the aphid following the shortest straight-line path.  However during this time interval the aphid moves 1cm along the branch in the positive x…

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The Holy Grail of Maths: Langlands. (specialization vs generalization).

This year’s TOK question for Mathematics is the following: “How can we reconcile the opposing demands for specialization and generalization in the production of knowledge? Discuss with reference to mathematics and one other area of knowledge” This is a nice chance to discuss the Langlands program which was recently covered in a really excellent Numberphile video.  image by Bjorn Oberg for Quanta…

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Winning at Snakes and Ladders

Winning at Snakes and Ladders The fantastic Marcus de Sautoy has just made a video on how to use Markov chains to work out how long it will take to win at Snakes and Ladders.  This uses a different method to those I’ve explored before (Playing Games with Markov Chains) so it’s well worth looking at in more detail.   The game This simplified version of Snakes and Ladders will have 9 squares, one…

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Chi Square: Language Detection + Code Breaking

Chi Square: Language Detection + Code Breaking We can use the power of maths to allow computers to accurately recognise which language someone is writing in – even without needing to have understanding of any language at all.  How?  With the Chi Square goodness of fit test.  Every language in the world has its own unique distribution of letter frequencies – in English E is the most commonly used…

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Roll or bust? A strategy for dice games

Roll or bust? A strategy for dice games Let’s explore some strategies for getting the best outcome for some dice games. Game 1: 1 dice, bust on 1. We roll 1 dice.  However we can roll as many times as we like and add the score each time.  We can choose to stop when we want.  However If we roll a 1 at anytime then we are bust, lose all out points and the game is over. What is the best strategy for…