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**Volume optimization of a cuboid**

This is an extension of the Nrich task which is currently live – where students have to find the maximum volume of a cuboid formed by cutting squares of size x from each corner of a 20 x 20 piece of paper. I’m going to use an n x 10 rectangle and see what the optimum x value is when n tends to infinity.

First we can find the volume of the cuboid:

Next we want to find when the volume is a maximum, so differentiate and set this equal to 0.

Next we use the quadratic formula to find the roots of the quadratic, and then see what happens as n tends to infinity (i.e we want to see what the optimum x values are for our cuboid when n approaches infinity). We only take the negative solution of the + – quadratic solutions because this will be the only one that fits the initial problem.

Next we try and simplify the square root by taking out a factor of 16, and then we complete the square for the term inside the square root (this will be useful next!)

Next we make a u substitution. Note that this means that as n approaches infinity, u approaches 0.

Substituting this into the expression gives us:

We then manipulate the surd further to get it in the following form:

Now, the reason for all that manipulation becomes apparent – we can use the binomial expansion for the square root of 1 + u^{2} to get the following:

Therefore we have shown that as the value of n approaches infinity, the value of x that gives the optimum volume approaches 2.5cm.

So, even though we start with a pretty simple optimization task, it quickly develops into some quite complicated mathematics. We could obviously have plotted the term in n to see what its behavior was as n approaches infinity, but it’s nicer to prove it. So, let’s check our result graphically.

As we can see from the graph, with n plotted on the x axis and x plotted on the y axis we approach x = 2.5 as n approaches infinity – as required.

**An m by n rectangle.**

So, we can then extend this by considering an n by m rectangle, where m is fixed and then n tends to infinity. As before the question is what is the value of x which gives the maximum volume as n tends to infinity?

We do the same method. First we write the equation for the volume and put it into the quadratic formula.

Next we complete the square, and make the u substitution:

Next we simplify the surd, and then use the expansion for the square root of 1 + u^{2}

This then gives the following answer:

So, we can see that for an n by m rectangle, as m is fixed and n tends to infinity, the value of x which gives the optimum volume tends to m/4. For example when we had a 10 by n rectangle (i.e m = 10) we had x = 2.5. When we have a 20 by n rectangle we would have x = 5 etc.

And we’ve finished! See what other things you can explore with this problem.

**IB Revision**

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions. This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

**Projective Geometry**

Geometry is a discipline which has long been subject to mathematical fashions of the ages. In classical Greece, Euclid’s elements (Euclid pictured above) with their logical axiomatic base established the subject as the pinnacle on the “great mountain of Truth” that all other disciplines could but hope to scale. However the status of the subject fell greatly from such heights and by the late 18th century it was no longer a fashionable branch to study. The revival of interest in geometry was led by a group of French mathematicians at the start of the 1800s with their work on projective geometry. This then paved the way for the later development of non-Euclidean geometry and led to deep philosophical questions as to geometry’s links with reality and indeed just what exactly geometry was.

Projective geometry is the study of geometrical properties unchanged by projection. It strips away distinctions between conics, angles, distance and parallelism to create a geometry more fundamental than Euclidean geometry. For example the diagram below shows how an ellipse has been projected onto a circle. The ellipse and the circle are therefore projectively equivalent which means that projective results in the circle are also true in ellipses (and other conics).

Projective geometry can be understood in terms of rays of light emanating from a point. In the diagram above, the triangle IJK drawn on the glass screen would be projected to triangle LNO on the ground. This projection does not preserve either angles or side lengths – so the triangle on the ground will have different sized angles and sides to that on the screen. This may seem a little strange – after all we tend to think in terms of angles and sides in geometry, however in projective geometry distinctions about angles and lengths are stripped away (however something called the cross-ratio is still preserved).

We can see in the image above that a projection from the point E creates similar shapes when the 2 planes containing IJKL and ABCD are parallel. Therefore the Euclidean geometrical study of similar shapes can be thought of as a subset of plane positions in projective geometry.

Taking this idea further we can see that congruent shapes can be achieved if we have the centre of projection, E, “sent to infinity:” In projective geometry, parallel lines do indeed meet – at this point at infinity. Therefore with the point E sent to infinity we have a projection above yielding congruent shapes.

Projective geometry can be used with conics to associate every point (pole) with a line (polar), and vice versa. For example the point A had the associated red line, d. To find this we draw the 2 tangents from A to the conic. We then join the 2 points of intersection between B and C. This principle of duality allowed new theorems to be discovered simply by interchanging points and lines.

An example of both the symmetrical attractiveness and the mathematical potential for duality was first provided by Brianchon. In 1806 he used duality to discover the dual theorem of Pascal’s Theorem – simply by interchanging points and lines. Rarely can a mathematical discovery have been both so (mechanically) easy and yet so profoundly

beautiful.

**Brianchon’s Theorem**

**Pascal’s Theorem**

**Poncelet**

Poncelet was another French pioneer of projective geometry who used the idea of points and lines being “sent to infinity” to yield some remarkable results when used as a tool for mathematical proof.

**Another version of Pascal’s Theorem:**

Poncelet claimed he could prove Pascal’s theorem (shown above) where 6 points on a conic section joined to make a hexagon have a common line. He did this by sending the line GH to infinity. To understand this we can note that the previous point of intersection G of lines AB’ and A’B is now at infinity, which means that AB’ and A’B will now be parallel. This means that H being at infinity also creates the 2 parallel lines AC’. Poncelet now argued that because we could prove through geometrical means that B’C and BC’ were also parallel, that this was consistent with the line HI also being at infinity. Therefore by proving the specific case in a circle where line GHI has been sent to infinity he argued that we could prove using projective geometry the general case of Pascal’s theorem in any conic .

**Pascal’s Theorem with intersections at infinity:**

This branch of mathematics developed quickly in the early 1800s, sparking new interest in geometry and leading to a heated debate about whether geometry should retain its “pure” Euclidean roots of diagrammatic proof, or if it was best understood through algebra. The use of points and lines at infinity marked a shift away from geometry representing “reality” as understood from a Euclidean perspective, and by the late 1800s Beltrami, Poincare and others were able to incorporate the ideas of projective geometry and lines at infinity to provide their Euclidean models of non-Euclidean space. The development of projective geometry demonstrated how a small change of perspective could have profound consequences.

**Modeling hours of daylight**

Desmos has a nice student activity (on teacher.desmos.com) modeling the number of hours of daylight in Florida versus Alaska – which both produce a nice sine curve when plotted on a graph. So let’s see if this relationship also holds between Phuket and Manchester.

First we can find the daylight hours from this site, making sure to convert the times given to decimals of hours.

**Phuket**

Phuket has the following distribution of hours of daylight (taking the reading from the first of each month and setting 1 as January)

**Manchester **

Manchester has much greater variation and is as follows:

Therefore when we plot them together (Phuket in green and Manchester in blue) we get the following 2 curves:

We can see that these very closely fit sine curves, indeed we can see that the following regression lines fit the curves very closely:

**Manchester:**

**Phuket:**

For Manchester I needed to set the value of b (see what happens if you don’t do this!) Because we are working with Sine graphs, the value of d will give the equation of the axis of symmetry of the graph, which will also be the average hours of daylight over the year. We can see therefore that even though there is a huge variation between the hours of daylight in the 2 places, they both get on average the same amount of daylight across the year (12.3 hours versus 12.1 hours).

**Further investigation:**

Does the relationship still hold when looking at hours of sunshine rather than daylight? How many years would we expect our model be accurate for? It’s possible to investigate the use of sine waves to model a large amount of natural phenomena such as tide heights and musical notes – so it’s also possible to investigate in this direction as well.

**IB Revision**

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions. This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

Cartoon from here

**The Gini Coefficient – Measuring Inequality **

The Gini coefficient is a value ranging from 0 to 1 which measures inequality. 0 represents perfect equality – i.e everyone in a population has exactly the same wealth. 1 represents complete inequality – i.e 1 person has all the wealth and everyone else has nothing. As you would expect, countries will always have a value somewhere between these 2 extremes. The way its calculated is best seen through the following graph (from here):

The Gini coefficient is calculated as the area of A divided by the area of A+B. As the area of A decreases then the curve which plots the distribution of wealth (we can call this the Lorenz curve) approaches the line y = x. This is the line which represents perfect equality.

**Inequality in Thailand**

The following graph will illustrate how we can plot a curve and calculate the Gini coefficient. First we need some data. I have taken the following information on income distribution from the 2002 World Bank data on Thailand where I am currently teaching:

Thailand:

The bottom 20% of the population have 6.3% of the wealth

The next 20% of the population have 9.9% of the wealth

The next 20% have 14% of the wealth

The next 20% have 20.8% of the wealth

The top 20% have 49% of the wealth

I can then write this in a cumulative frequency table (converting % to decimals):

Here the x axis represents the cumulative percentage of the population (measured from lowest to highest), and the y axis represents the cumulative wealth. This shows, for example that the the bottom 80% of the population own 51% of the wealth. This can then be plotted as a graph below (using Desmos):

From the graph we can see that Thailand has quite a lot of inequality – after all the top 20% have just under 50% of the wealth. The blue line represents how a perfectly equal society would look.

To find the Gini Coefficient we first need to find the area between the 2 curves. The area underneath the blue line represents the area A +B. This is just the area of a triangle with length and perpendicular height 1, therefore this area is 0.5.

The area under the green curve can be found using the trapezium rule, 0.5(a+b)h. Doing this for the first trapezium we get 0.5(0+0.063)(0.2) = 0.0063. The second trapezium is 0.5(0.063+0.162)(0.2) and so on. Adding these areas all together we get a total trapezium area of 0.3074. Therefore we get the area between the two curves as 0.5 – 0.3074 ≈ 0.1926

The Gini coefficient is then given by 0.1926/0.5 = 0.3852.

The actual World Bank calculation for Thailand’s Gini coefficient in 2002 was 0.42 – so we have slightly underestimated the inequality in Thailand. We would get a more accurate estimate by taking more data points, or by fitting a curve through our plotted points and then integrating. Nevertheless this is a good demonstration of how the method works.

In this graph (from here) we can see a similar plot of wealth distribution – here we have quintiles on the x axis (1st quintile is the bottom 20% etc). This time we can compare Hungary – which shows a high level of equality (the bottom 80% of the population own 62.5% of the wealth) and Namibia – which shows a high level of inequality (the bottom 80% of the population own just 21.3% of the wealth).

**How unequal is the world?**

We can apply the same method to measure world inequality. One way to do this is to calculate the per capita income of all the countries in the world and then to work out the share of the total global per capita income the (say) bottom 20% of the countries have. This information is represented in the graph above (from here). It shows that there was rising inequality (i.e the richer countries were outperforming the poorer countries) in the 2 decades prior to the end of the century, but that there has been a small decline in inequality since then.

If you want to do some more research on the Gini coefficient you can use the following resources:

The intmaths site article on this topic – which goes into more detail and examples of how to calculate the Gini coefficient

The ConferenceBoard site which contains a detailed look at world inequality

The World Bank data on the Gini coefficients of different countries.

**IB Revision**

**Give your university applications a headstart on other students with Coursera. **

Applying for university as an international student is incredibly competitive – for the top universities you’ll be competing with the best students from around the world, and so giving yourself a competitive advantage to make your university application stand out is really important. One way to do this is by completing a course run by some of the world’s top Universities.

**Universities offering courses:**

Examples of the top universities offering courses include: The University of Tokyo, Caltec, University of Manchester, Imperial College London, Duke, Stanford, Yale, University of Sydney, National University of Singapore, amongst many others, alongside major companies such as IBM, Google, Intel and Goldman Sachs.

**Courses on offer**

You can sign up for free and access modules run by these universities and companies, with the possibility of obtaining a certificate at the end of the course which can then go towards your university application.

Some of the courses on offer include:

Biological science courses such as Genetics and Evolution from Duke University, Understaning the brain from the University of Chicago, Astrobiology and the search for Extraterrestrial life from Edinburgh University and Medical Neuroscience from Duke University,

Business courses such as Business foundations from University of Pennsylvania, Digital Marketing from the University of Illinois, Viral Marketing and How to Create Contagious Content with the University of Pennsylvania, the Math Behind Moneyball with the University of Houston and studying the Global Financial Crisis with Yale University.

Physical science courses such as Welcome to Game Theory with Tokyo University, Science Literacy with Erasmus University Rotterdam and The Journey of the Universe with Yale University.

Arts and humanities courses such as Creative writing from Wesleyan University, Graphics design from Californian Institute of Arts, Music Production from Berklee College of Music, Introduction to Philosophy from the University of Edinburgh.

Overall there are over 3000 courses from 170 universities and partners – so almost certainly there’ll be something worth investigating. Have a look and give your University application a boost over everyone else!

This post is inspired by the Quora thread on interesting functions to plot.

**The butterfly**

This is a slightly simpler version of the butterfly curve which is plotted using polar coordinates on Desmos as:

Polar coordinates are an alternative way of plotting functions – and are explored a little in HL Maths when looking at complex numbers. The theta value specifies an angle of rotation measured anti-clockwise from the x axis, and the r value specifies the distance from the origin. So for example the polar coordinates (90 degrees, 1) would specify a point 90 degrees ant clockwise from the x axis and a distance 1 from the origin (i.e the point (0,1) in our usual Cartesian plane).

2. **Fermat’s Spiral**

This is plotted by the polar equation:

The next 3 were all created by my students.

3. **Chaotic spiral (by Laura Y9)**

I like how this graph grows ever more tangled as it coils in on itself. This was created by the polar equation:

4. **The flower (by Felix Y9)**

Some nice rotational symmetries on this one. Plotted by:

5. **The heart (by Tiffany Y9)**

Simple but effective! This was plotted using the usual x,y coordinates:

You can also explore how to draw the Superman and Batman logos using Wolfram Alpha here.

This is a quick example of how using Tracker software can generate a nice physics-related exploration. I took a spring, and attached it to a stand with a weight hanging from the end. I then took a video of the movement of the spring, and then uploaded this to Tracker.

**Height against time**

The first graph I generated was for the height of the spring against time. I started the graph when the spring was released from the low point. To be more accurate here you can calibrate the y axis scale with the actual distance. I left it with the default settings.

You can see we have a very good fit for a sine/cosine curve. This gives the approximate equation:

y = -65cos10.5(t-3.4) – 195

(remembering that the y axis scale is x 100).

This oscillating behavior is what we would expect from a spring system – in this case we have a period of around 0.6 seconds.

**Momentum against velocity**

For this graph I first set the mass as 0.3kg – which was the weight used – and plotted the y direction momentum against the y direction velocity. It then produces the above linear relationship, which has a gradient of around 0.3. Therefore we have the equation:

p = 0.3v

If we look at the theoretical equation linking momentum:

p = mv

(Where m = mass). We can see that we have almost perfectly replicated this theoretical equation.

**Height against velocity**

I generated this graph with the mass set to the default 1kg. It plots the y direction against the y component velocity. You can see from the this graph that the velocity is 0 when the spring is at the top and bottom of its cycle. We can then also see that it reaches its maximum velocity when halfway through its cycle. If we were to model this we could use an ellipse (remembering that both scales are x100 and using x for vy):

If we then wanted to develop this as an investigation, we could look at how changing the weight or the spring extension affected the results and look for some general conclusions for this. So there we go – a nice example of how tracker can quickly generate some nice personalised investigations!

**Finger Ratio Predicts Maths Ability?**

Some of the studies on the 2D: 4D finger ratios (as measured in the picture above) are interesting when considering what factors possibly affect mathematical ability. A 2007 study by Mark Brosnan from the University of Bath found that:

*“Boys with the longest ring fingers relative to their index fingers tend to excel in math. The boys with the lowest ratios also were the ones whose abilities were most skewed in the direction of math rather than literacy.*

*With the girls, there was no correlation between finger ratio and numeracy, but those with higher ratios–presumably indicating low testosterone levels–had better scores on verbal abilities. The link, according to the researchers, is that testosterone levels in the womb influence both finger length and brain development.*

*In men, the ring (fourth) finger is usually longer than the index (second); their so-called 2D:4D ratio is lower than 1. In females, the two fingers are more likely to be the same length. Because of this sex difference, some scientists believe that a low ratio could be a marker for higher prenatal testosterone levels, although it’s not clear how the hormone might influence finger development.”*

In the study, Brosnan photocopied the hands of 74 boys and girls aged 6 and 7. He worked out the 2D:4D finger ratio by dividing the length of the index finger (2D) with the length of the ring finger (4D). They then compared the finger ratios with standardised UK maths and English tests. The differences found were small, but significant.

Another study of 136 men and 137 women, looked at the link between finger ratio and aggression. The results are plotted in the graph above – which clearly show this data follows a normal distribution. The men are represented with the blue line, the women the green line and the overall cohort in red. You can see that the male distribution is shifted to the left as they have a lower mean ratio. (Males: mean 0.947, standard deviation 0.029, Females: mean 0.965, standard deviation 0.026).

The 95% confidence interval for average length is 0.889-1.005 for males and 0.913-1.017 for females. That means that 95% of the male and female populations will fall into these categories.

The correlation between digit ratio and everything from personality, sexuality, sporting ability and management has been studied. If a low 2D:4D ratio is indeed due to testosterone exposure in the womb (which is not confirmed), then that raises the question as to why testosterone exposure should affect mathematical ability. And if it is not connected to testosterone, then what is responsible for the correlation between digit ratios and mathematical talent?

I think this would make a really interesting Internal Assessment investigation at either Studies or Standard Level. Also it works well as a class investigation at KS3 and IGCSE into correlation and scatter diagrams. Does the relationship still hold for when you look at algebraic skills rather than numeracy? Or is algebraic talent distinct from numeracy talent?

A detailed academic discussion of the scientific literature on this topic is available here.

If you enjoyed this post you might also like:

Simulations -Traffic Jams and Asteroid Impacts

NASA, Aliens and Binary Codes from the Stars

**IB Revision**

**Amanda Knox and Bad Maths in Courts**

This post is inspired by the recent BBC News article, “Amanda Knox and Bad Maths in Courts.” The article highlights the importance of good mathematical understanding when handling probabilities – and how mistakes by judges and juries can sometimes lead to miscarriages of justice.

**A scenario to give to students:**

*A murder scene is found with two types of blood – that of the victim and that of the murderer. As luck would have it, the unidentified blood has an incredibly rare blood disorder, only found in 1 in every million men. The capital and surrounding areas have a population of 20 million – and the police are sure the murderer is from the capital. The police have already started cataloging all citizens’ blood types for their new super crime-database. They already have nearly 1 million male samples in there – and bingo – one man, Mr XY, is a match. He is promptly marched off to trial, there is no other evidence, but the jury are told that the odds are 1 in a million that he is innocent. He is duly convicted. The question is, how likely is it that he did not commit this crime? *

**Answer:**

*We can be around 90% confident that he did not commit this crime. Assuming that there are approximately 10 million men in the capital, then were everyone cataloged on the database we would have on average 10 positive matches. Given that there is no other evidence, it is therefore likely that he is only a 1 in 10 chance of being guilty. Even though P(Fail Test/Innocent) = 1/1,000,000, P(Innocent/Fail test) = 9/10.
*

**Amanda Knox**

Eighteen months ago, Amanda Knox and Raffaele Sollecito, who were previously convicted of the murder of British exchange student Meredith Kercher, were acquitted. The judge at the time ruled out re-testing a tiny DNA sample found at the scene, stating that, “The sum of the two results, both unreliable… cannot give a reliable result.”

This logic however, whilst intuitive is not mathematically correct. As explained by mathematician Coralie Colmez in the BBC News article, by repeating relatively unreliable tests we can make them more reliable – the larger the pooled sample size, the more confident we can be in the result.

**Sally Clark**

One of the most (in)famous examples of bad maths in the court room is that of Sally Clark – who was convicted of the murder of her two sons in 1999. It has been described as, “one of the great miscarriages of justice in modern British legal history.” Both of Sally Clark’s children died from cot-death whilst still babies. Soon afterwards she was arrested for murder. The case was based on a seemingly incontrovertible statistic – that the chance of 2 children from the same family dying from cot-death was 1 in 73 million. Experts testified to this, the jury were suitably convinced and she was convicted.

The crux of the prosecutor’s case was that it was so statistically unlikely that this had happened by chance, that she must have killed her children. However, this was bad maths – which led to an innocent woman being jailed for four years before her eventual acquittal.

**Independent Events**

The 1 in 73 million figure was arrived at by simply looking at the probability of a single cot-death (1 in 8500 ) and then squaring it – because it had happened twice. However, this method only works if both events are independent – and in this case they clearly weren’t. Any biological or social factors which contribute to the death of a child due to cot-death will also mean that another sibling is also at elevated risk.

**Prosecutor’s Fallacy**

Additionally this figure was presented in a way which is known as the “prosecutor’s fallacy” – the 1 in 73 million figure (even if correct) didn’t represent the probability of Sally Clark’s innocence, because it should have been compared against the probability of guilt for a double homicide. In other words, the probability of a false positive is not the same as the probability of innocence. In mathematical language, P(Fail Test/Innocent) is not equal to P(Innocent/Fail test).

Subsequent analysis of the Sally Clark case by a mathematics professor concluded that rather than having a 1 in 73 million chance of being innocent, actually it was about 4-10 times more likely this was due to natural causes rather than murder. Quite a big turnaround – and evidence of why understanding statistics is so important in the courts.

This topic has also been highlighted recently by the excellent ToK website, Lancaster School ToK.

If you enjoyed this topic you might also like:

Benford’s Law – Using Maths to Catch Fraudsters

The Mathematics of Cons – Pyramid Selling

**IB Revision**