BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
PYTHOGORAS’ THEOREM
The following is based on an article by Colin Wright (http://www.solipsys.co.uk/cgibin/sews.py) called “How High the Moon”, I think it is so good that it should be
popularised. Colin‟s work covers other applications too.
Suppose the Earth is a sphere and we have a mountain of height h at the point A.
From the top of the mountain you can see to
the point B. This is the horizon as you see it.
How far away is B?
(Assuming no obstacles or other such
annoyances.)
Pythagoras tells us that:
The radius of the Earth is roughly 6000km and h will be much smaller than this, in fact,
let‟s take h to be 5m, then h2=25 is insignificant compared to the other terms. So
The radius of the Earth is approximately 6000km and h is 0.005km, so we get
So when you are 5 metres in the air the horizon is 8km away.
Why did we choose h to be 5m?
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
Well suppose we are at point B now and we fire a projectile at the top of the 5m tall
hill (assuming no air resistance and so on). We know that the acceleration due to gravity
is approximately 10ms-2. Now if we first the projectile so that it travels from B to A in
1 second the “suvat” equation (s=ut+1/2at2) shows us that the projectile will have fallen
5 meters. To put this another way, firing the projectile from B will result in it being at
A after 1 second: so it has fallen with the curvature of the Earth, it is in orbit. This
means the orbital velocity is 8km/s.
OTHER IMPLICATIONS

This method also shows that V2=aR, i.e. the equation for velocity for circular
motion (no, awkward derivation)

This also extends to compute the distance of the Earth to the Moon.
For details see Colin‟s article.
PRACTICAL USES OF RECURRENCE RELATIONS
We are going to analyse some models of disease transmission. These will be simple
models with some significant simplifications; nevertheless we can still derive meaningful
results from them.
We assume that everyone in the population is either susceptible to a disease (S) or
infected (I). We want to monitor how S and I change over time. In certain
circumstances it is useful to consider the absolute number of people who are S or I,
for example, S=400 and I=504. In other cases it is useful to consider only the
percentage of the total population that is S or I.
We want to describe how population sizes change with time t so we let St and It denote
the number of people susceptible and infected at time t respectively. The total
population will be Nt.
In the SIS model people move from being susceptible S to being infected I when a
susceptible person comes into contact with an infected person and the infection is
passed on. For different diseases the mechanism that results in an infection varies.
The flow is summarised in the following so-called black-box model. That is, we only
specify the relationship between the components of the model, not the specifics of
these relationships.
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MEI CONFERENCE, JULY 2010
SUSCEPTIBLES
INFECTED
Before we expand our black-box model into a mathematical model, we state a number of
assumptions.
Assumption 1
We assume that Nt=N that is the population is constant. For a large number of diseases
this is a valid.
Assumption 2
All the people in the population are the same people from period to period.
We consider proportions of the population rather than the absolute values. So we
consider
st 
St
I
and it  t
Nt
Nt
So st represent the proportion of susceptibles in the population and it represents the
proportion of infected individuals in the populations.
 that represents the proportion of the population that
recovers from the disease in each time period. Thus I t is the number infected
We introduce a parameter
individuals recovering each time period.
If I t are the number of contacts which can lead to a possible infection then only
those between infected individuals and susceptible individuals will lead to a new
infection. Given that st is the proportional of the population that is susceptible we find
that
st I t
new infections occur each time period.
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
αγstIt
SUSCEPTIBLES
INFECTED
κIt
The new model is now:
I t 1  I t  I t  st I t
S t 1  S t  I t  st I t
Notice that the change in the susceptible group equals the change in the infected
group, so the population is constant.
If we divide these equations through by Nt then we get equations only involving the
proportion of the population.
it 1  it  it  st it
st 1  st  it  st it
We can use Excel to model these equations for different values of the parameters and
different initial conditions.
Simulations
Since  features only as a combined value we shall write    . Further more, since
st+it=1 we can simplify the computation implementation by writing the system as
it 1  it  it   (1  it )it
st 1  1  it 1
To keep our analysis simple we suppose that   1 ; this does not affect they type of
behaviour it is simply a rescaling of parameter values and allows us to concentrate on
observing behaviour rather than the specifics of the actual parameters.
INVESTIGATIONS
1. To start put   1.5 and use Excel to implement the model and plot a chart.
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BIG IDEAS IN CORE MATHEMATICS
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2. Slowly increase the value of  until you reach the value 4. What do you notice?
OUTCOMES
1)
As we can see the simulation shows convergence to the predicted steady-state. We now
examine how the model behaves as we change  .
2) When   2 we see convergence to a single common steady-state where the
population is an exact 50/50 split between infected and susceptible individuals.
1
0.9
0.8
0.7
0.6
I_t
0.5
S_t
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
30
35
When   2.5 we see convergence again, but this time there appears to be small
oscillations before convergence settles in.
1
0.9
0.8
0.7
0.6
I_t
0.5
S_t
0.4
0.3
0.2
0.1
0
0
5
10
15
20
5
25
30
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
In the next series of charts we change  to the following values respectively
  3.0,3.21,3.3,3.5,4 .
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
I_t
0.5
I_t
0.5
S_t
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
S_t
0
0
5
10
15
20
25
30
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
35
0
5
10
15
20
25
30
35
0.6
I_t
0.5
S_t
I_t
0.5
0.4
0.4
0.3
0.3
0.2
0.2
S_t
0.1
0.1
0
0
0
5
10
15
20
25
30
0
35
5
10
15
20
25
30
35
1.2
1
0.8
0.6
I_t
S_t
0.4
0.2
0
0
5
10
15
20
25
30
35
-0.2
This is an example of a period-doubling bifurcation and eventually leads to chaos.
FURTHER INVESTIGATIONS
What values of the parameters give a steady-value? What values give oscillations with
period 2, 4, 8, 16? It is possible to find oscillations with period 3, but the range of
values is small.
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
The “x-axis” here represents the value of  and you can see how as  is increased we
get a period 2 state, then 4 then 8 and so on. There is a small region where you can
observe a period 3 oscillation.
OTHER QUESTIONS
1. The model gives steady states, where the size of the population in each
compartment does not change. (A steady-state is not where individuals cease to
be infected, it simply means the numbers in each compartment remain constant
because the flows between them are equal.) How do you find the steady states?
Do your values agree with the simulations? What is happening to the steadystates when the simulations become periodic?
2. Can you determine conditions on the parameters to control an epidemic?
For more information on this type of equation see:
http://mathworld.wolfram.com/LogisticMap.html
PROJECT IDEAS
1. This type of compartment model can be adapted to other situations. The SIR model adds a
third compartment. This compartment represents the “removed” (or recovered) class. How
would the model look and behave in this case? (This is the so-called S-I-R model)
2. The S-E-I-R model introduces an exposed class, what happens in this case?
3. How do you introduce birth-death rates into the model?
All these situations are classic examples of mathematics modelling biological situations
and have far reaching implications which we don‟t have time to investigate.
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
BROKEN PHOTOCOPIER (RECUSION AGAIN)
The recursive photocopier does not work like a normal photocopier; it only copies
certain „blocks‟ from the original document. It has several settings:
Setting 1
Any rectangle like this:
Example:
will be copied to an identical rectangle.
would become
as everything would remain unchanged.This setting is a little dull so we will look at the
more „exciting‟ settings.
Setting 2
Any rectangle like this:
will be copied to
Complete this set of photocopies. Each time, the copy is fed back into the photocopier.
What could you start with that would be unchanged by repeated copying?
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
Setting 3
Any rectangle like this:
will be copied to
Complete this set of photocopies. Each time, the copy is fed back into the photocopier.
What could you start with that would be unchanged by repeated copying?
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
LANGTON’S ANT (RECUSION AGAIN)
We‟re going to start with a very simply universe. It consists of an infinite grid of white
squares. This universe has one occupant, an Ant. The ant is facing up on the page:
The ant obeys some very simple rules in its life (different people have the ant facing in
different directions, and have left where we have right, it doesn‟t matter):
Move forward.
Look at the colour of the square you are standing on. If it black turn right by 90
degrees (clockwise) and if it white turn left by 90 degrees (anticlockwise).
 Change the colour of the square you are standing on (if the square is black
change it to white and if it is white change it to black).
 Repeat until dead (go to 1).
Can you work out what the ant will do?


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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
It would take a significant amount of time to simulate 10,000 moves of this ant. We can
get computers to help us simulate this universe:
See for example:
http://www.annanardella.it/ant.html
http://www.math.umd.edu/~wphooper/ant/
http://www.tiac.net/~sw/LangtonsAnt/LangtonsAnt.html
Running the ant simulation seems to produce totally random behaviour.
On a few occasions you get some symmetrical results (after 386 steps)
With these simulations people noticed that after approximately 10,000 steps the ant
did something very strange, it makes a highway, which consists of 104 repeated steps:
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
No-one in the world has yet been able to explain why the ant‟s rules lead to this
highway. In fact, you don‟t have to start with an all white grid; it is suspected that any
starting grid of white and black squares will eventually lead to the highway.
The most anyone knows is that the ant is eventually very far from its starting point!
CONWAY’S GAME OF LIFE (RECUSION ONCE MORE)
How to play
The game is played on an infinite, two-dimensional grid of squares. Each square (cell) on
the grid is either occupied (live) or unoccupied (dead). An initial pattern of live and
dead cells is set up.
Rules for births and deaths
A live cell with fewer than 2 neighbours dies (from loneliness).
A live cell with more than 3 neighbours dies (from overcrowding).
Any dead cell with exactly 3 live neighbours comes to life.
Births and deaths occur at the same instant before moving on to the next round.
initial
stage 1
stage 2
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stage 3
stage 4
BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
Langton‟s Ant and Conway‟s Game of Life are both great examples of the problems
mathematicians meet when they try to understand recursive processes (algorithms).
Conway‟s Game of Life is equivalent to a universal Turing machine.
Berlekamp, E. R.; Conway, John Horton; Guy, R.K. (2001 2004), Winning Ways for your Mathematical
Plays (2nd ed.), A K Peters Ltd,
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
BINOMIAL THEOREM
A quick puzzle: How many ways are there from A to B going only “right” and “down”?
A
B
SOLUTION
A
1
1
1
1
1
2
3
4
5
1
3
6
10
15
1
4
10
20
35
1
5
15
35
70
14
=B
BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
A new puzzle:
How many odd numbers are there on the 65th (or indeed any) row of Pascal‟s triangle?
Colouring gives:
More colouring gives:
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
Eventually we get:
Practical applications of the theory of fractals include film productions:
See, for example, Terragen: http://www.planetside.co.uk/
Other interesting properties of Pascal’s triangle:
http://mathworld.wolfram.com/PascalsTriangle.html
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
DIFFERENTIAL EQUATIONS AND CATASTROPHE
In Core 4 students need to formulate and solve differential equations. A typical example could
be population growth:
We consider a single species which at time t has population N(t). The rate of
change of the population dN/dt is determined by several factors, in this initial model
we shall assume:
To simplify the situation further we shall assume there is no migration and that
births and deaths are proportional to the population size. Our initial model then
takes the form
The solution is N(t)=N0e(b-d)t. The constants b and d are the birth and death rates respectively.
The term N0 represents the initial population.
The world‟s population has shown roughly exponential growth:
There are certain processes that limit a population's growth; competition for resources and
geographical considerations. In 1836 Verhulst proposed the following adjustment
Here r and K are constants. The constant K is called the carrying capacity.
Good students should be able to show the solution to this equation is (using partial fractions):
Questions to ask:
1. What happens in the long term?
2. What happens in the long term if N0<K?
3. What happens in the long term if N0>K?
Given the solution we can determine the long term behaviour of N(t):
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
Looking at
If N0<K, then dN/dt>0, so N(t) increases to K. Conversely if N0>K, then dN/dt<0, so N(t)
decreases to K.
Question to ask:
How important was the explicit solution in this analysis? Not important at all! The answers to
parts 2 and 3 do not use the solution at all! What about question 1?
A steady-state of a differential equation is a solution to dN/dt=0. So
So N=0 or N=K.
We can derive significant information from a differential equation, without solving it!
For a more dramatic illustration consider the following model for population growth with
predation is given by (Spruce-Budworm model):
QUESTIONS TO ASK?
Can you find N(t) in terms of t (and the constants r and K)?
This model, like almost every other practical model used today, has no closed form solution. (Or
at least not one that is of any practical use.) We can still derive considerable information
without solving the equation.
STEADY-STATES
We need to solve:
Either we try to solve a cubic equation or we examine:
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MEI CONFERENCE, JULY 2010
One solution is N=0 and the others are given by
So the solutions are the intersections of a straight line (the left-hand side) and a curve (righthand side).
So we have either one, two or three solutions: so one, two or three steady-states. (Geogebra is
very useful for illustrating these features.)
FURTHER INVESTIGATION
Fix q = 6.9. When r = 0.4 there is one steady-state, which we call u1. We now increase the value
of r. When r is 0.55 we have three steady states u1 < u2 < u3.
Since u1 is stable the system remains at the state u1. If we continue to increase
r to 0.6 then we find the steady states u1 and u2 have vanished leaving only the
larger state u3 which is stable.
So the population jumps from the low-level u1 state to the high level u3 state.
We now run this process in reverse. We start with r = 0.6 and q = 6.9. The system will be at the
stable u3 value. We decrease the value of r and yet again we see the emergence of three
steady-states. However, since the system is at the state u3 and this state is stable, it will
remain there. We continue to decrease r and it isn't until the only steady-state is u1 that the
population jumps back to the low-level state: this phenomenon is called a hysteresis effect.
It is important to note that even though the parameters are changed in a continuous manner the
system “jumps” discontinuously from one state to another.
This is an example of a (mathematical) catastrophe which is modelled with the surface:
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
In the case illustrated above we start at point A move along the surface to reach B, where
there are now two solutions. As we move from B to C there are three solutions, but since we
started on the bottom sheet of the fold we remain there (since this point is stable). When we
reach C the number of solutions drops to one. In particular, our current state ceases to exist,
so we jump to the upper sheet. So a small change in the parameters causes a sudden and
dramatic change in behaviour. In this case we have a population explosion!
In reverse, we start at D: the large population state. From D we move to C, there are now two
solutions, but we remain on the upper sheet of the fold since the system is stable. As we move
from C to B we see there are three solutions, but we remain on the upper sheet since the
system is stable. The key observation is that we have reduced the parameter below the level
that caused the population explosion without causing a population crash. It is not until we reach
B that the population will crash to its lower level again.
The hysteresis effect is very important in a number of different situations like stock markets,
crowd behaviour, riots, markets, and even the study of diseases. The key point is that once the
situation has jumped from one state to another, it requires great effort (and often cost) to
under this effect. In certain circumstance, it is actually impossible to undo the effect!
Catastrophe theory: http://en.wikipedia.org/wiki/Catastrophe_theory
Zeeman catastrophe machine (a classic demonstration in mechanics):
http://www.math.sunysb.edu/~tony/whatsnew/column/catastrophe-0600/cusp4.html
http://lagrange.physics.drexel.edu/flash/zcm/
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BIG IDEAS IN CORE MATHEMATICS
MEI CONFERENCE, JULY 2010
APPENDIX (DIFFERENCE EQUATIONS IN EXCEL)
Consider the system
it 1   (1  it )it
st 1  1  it 1
Procedure (For Excel 2003)
Follow the following instructions to set-up your spreadsheet:
1.
2.
3.
4.
5.
6.
In cell B1 enter the text “Kappa”. In cell D1 enter the text “Beta” and in cell F1 enter
the text “kappa/beta”.
Cells C1 will contain the numerical value of kappa, cell E1 the numerical value of beta
used in your simulations.
In cell G1 enter the formula “=C1/E1”.
In cell A3 enter the text “t”, in cell B3 enter the text “I_t” and in cell C3 enter the
text “S_t”.
Enter the numerical value 0 into cell A4.
Cell B4 contains the initial numerical value for the infectives and cell C4 contains the
initial number of susceptibles. Since we have a constant population size we have st=1-it.
So we can enter the formula “=1-B4” into cell C4.
If you choose the following values
  1;   4, it  0.1 then your spreadsheet with the formula
showing will be:
We now need to work out the values of it and st for a number of values of t. We do this as
follows:
7. In cell A5 enter the formula “=A4+1” this increases the time step by one.
8. In cell B5 enter the formula “=$E$1*(1-B4)*B4” this if the formula for it+1 written for
Excel. (The $ sign ensure that when we copy the formula we continue to refer to the
value of beta given in cell E1. Technically one $ sign is not required.)
9. In cell C5 enter the formula “=1-B5”
At this point we should have the following formulae in Excel
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BIG IDEAS IN CORE MATHEMATICS
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We are now in a position to simulate the number of infected and susceptibles for any required
time period.
10. Select cells A5 to C5. Then click with the mouse on the small square on the bottom right
of the black square your selection has created.
11. Pull the black square down to copy the formulae and generate a time series of
simulations.
12. You now have a set of data that you can plot using Excel‟s graph function. Select (in this
case) cells as shown below and click on the chart wizard.
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13. Select XY (Scatter) and select the Chart sub-type that scatters the data points
connected by lines without markers. Select “Finish”, this will plot the values of it and st
against time t.
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