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The limit does not exist!

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What’s more fetch than toaster strüdel, foot cream in a jar and a fertility vase from the Ndebele tribe? Arguably maths. More specifically, the mathematics of Mean Girls. If you haven’t seen Mean Girls, it’s a not-too-bad movie about teenage girls, teenage boys, teenage drama, and (a little bit of) maths.

Stop trying to make fetch happen. It’s not going to happen.

It’s not exactly October 3rd*, but its been 13 years since Mean Girls came out, and

  • $13 \times 2=26$
  • $26+2\times2017=4060$
  • $4060-3010=1050$, and
  • $1050\log(2)+\frac{\pi}{1423211345}$ is… just a number.

What we’ll be looking at in this article is the last 10 minutes of the movie, where Cady and the Mathletes from North Shore compete in a maths state championship. After 87 minutes of play it’s a tie, and so the two teams enter the sudden death round, where the teams choose a member of the opposition (happening to both be female) to battle for the trophy.

T. Pak as in ‘Trang Pak the grotsky little byotch’? No, just her brother.

In one corner, Cady Heron, AKA ‘Africa girl’, AKA the ‘used-to-be maths geek but not so much anymore after she met the Plastics but is kind of a geek again because her attempted sabotage of Regina George backfired and descended into chaos’. She is also the protagonist of the movie.

In the other, is Caroline Krafft, the female member of Marymount Prep, who “seriously needed to pluck her eyebrows, whose skirt looks like it was picked out by a blind Sunday school teacher, and had some 99-cent lipgloss on her snaggletooth”. This was when I realised, making fun of Caroline Krafft wasn’t going make me any better at writing this article (or maths for that matter).

The two were asked to solve the following;

Caroline Krafft hastily states the incorrect answer, $-1$. If Cady manages to state the correct one then the North Shore mathletes ultimately win.

In the midst of all this Cady then has a miraculous flashback to the week that Aaron got his hair cut, and as she takes a moment to see straight pass Aaron’s face and onto the board behind, then has an epiphany (the kind we would like to have in our exams sometimes) and quickly realises the function diverges.

Aaron Samuels is now a spin class instructor, so he won’t be expected to know what a factorial is anymore. Thank God.

So we don’t exactly know how Cady did this in under 10 seconds, but we can try to figure out the question for ourselves.

So the question reads$$ \lim_{x\to 0} \frac{\ln(1-x)-\sin x} {1-\cos^2 x}.$$

We’ll consider L’Hôpital’s Rule, seeing as its the most straightforward way of going about it.

If we substitute $x$ with 0 we will find$$\frac{\ln(1)-\sin(0)} {1-\cos^2(0)} = \frac{0}{0}.$$
This satisfies the criteria for being able to apply L’Hôpital’s Rule, which is as follows,$$ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}.$$

Let’s apply it to our function, starting by considering the limit from the right;

\begin{align}
\lim_{x\to 0+} \frac{\frac{-1}{1-x}-\cos x} {\sin 2x} &= \lim_{x\to 0+} \frac{-x \cos x + \cos x + 1}{ (x-1) \sin 2x} \\
&= \lim_{x\to 0+} [-x \cos x + \cos x + 1] \lim_{x\to 0+}\left[\frac{1}{x-1}\right] \lim_{x\to 0+}\left[\frac{1}{\sin 2x}\right].
\end{align}
Now,\begin{align}
\lim_{x\to 0+} \left[-x\cos x +\cos x +1 \right] &=2, \\
\lim_{x\to 0+} \left[\frac{1}{\sin 2x } \right] &= \infty, \\
\lim_{x\to 0+} \left[\frac{1}{x-1} \right] &= -1,
\end{align}
so by properties of infinity we get
$$\lim_{x\to 0+} [-x \cos x + \cos x + 1] \lim_{x\to 0+}\left[\frac{1}{x-1}\right] \lim_{x\to 0+}\left[\frac{1}{\sin 2x}\right] = 2 \cdot \infty \cdot -1 = -\infty.$$

Repeating this method for the limit from the left, we will obtain the following,
$$ \lim_{x\to 0-} \frac{-x \cos x + \cos x + 1}{ (x-1) \sin 2x} = +\infty.$$

Just checking.

And so since the limit from the right and the left differ, the whole thing diverges. Yay.

Another approach would be to look at the Taylor series approximations for the separate terms at $x$ near 0. We won’t do that though since it’s not as mathsy, but it might have been the quicker way to go, on the assumption that Cady really knows her Taylor series expansions. But

“Cause the next time you see her she’ll be like, Ohh Kevin G!”

*The date where Aaron Samuels asked Cady what the date was.

CC-BY-SA: Mathletes, Matty McRib. Mean Girls, Flickr. Screenshots copyright of Paramount Pictures.

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Review: Mathematical socks

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This Christmas, I received mathematical socks. A great gift, you might think. But is it good maths or fake maths? Can you wear them and be taken mathematically seriously? Thus I have undertaken this important review.

Unboxing

The socks come beautifully packaged and folded, tied together with a fancy red label, which gives a nifty standing suggestion.

Beautifully packed mathematical socks

Beautifully packed mathematical socks


Unboxing grade: A

Mathematical content

There are five distinct mathematical items on the socks. I have graded them individually. The younger reader may wish to refer to this helpful guide to converting to new grades.

1. Proof of Pythagoras

Sock v Elements

Pythagoras’ Theorem proved on a sock (left) and in the Elements (right)

Continue reading

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Chalkdust Review of the Year 2016

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Another busy year for Chalkdust is gone(well…almost). That is 50 online articles, 2 new issues, 1 advent calendar a few quizzes and loads more. In the remaining few hours of 2016 we look back to some of the amazing articles written by us and our friends. From everyone on the Chalkdust team enjoy this post and look forward to even better blogs next year.

Attributions:

[Pictures: 1 – adapted from Flickr.com – Moscow New Year 2016 by Valeri Fortuna, CC-BY 2.0;  other pictures by Chalkdust]

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The Indisputable Existence of Santa Claus – A Review

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Welcome to the 22nd day of the 2016 Chalkdust Advent Calendar. Today we have for you, a book review!

Christmas, a time of celebration, joy and meeting family you never knew you had. Regardless of how joyous Christmas can be it also is undoubtedly stressful for some. What present could I get person $x$? Why does my Christmas tree look so ugly even though I spent 2 weeks decorating it? The simple answer is because you probably drenched it in tinsel and epileptic seizure inducing lights. The mathematical answer?

The Indisputable Existence of Santa Claus” is a recent release by Dr Hannah Fry and Dr Thomas Oléron Evans, Dr Hannah Fry who has previously written “The Mathematics of Love”. This book arranges itself as a step-by-step guide on how to prepare for the (mathematically) perfect Christmas, covering every detail from how to wrap presents according to their surface area to volume ratio, to using Markov chains as a means of perfecting the Queen’s speech. 

We shall start with the big question of whether Father Christmas himself actually exists. The first chapter gives a ‘seemingly’ valid proof of Santa’s existence, from another ‘seemingly’ valid proof of how 1+1=0. You must be thinking, what? Obviously there must be a flaw in the progression of the proof somewhere, which there is, but you can discover it for yourself by reading the book. 

Moving on, let’s have a look at the inconspicuous game of Secret Santa. For those unfamiliar with the concept, the classic approach to this game is simply, you write your name on a small piece of paper, fold it up and throw it into a hat. After a bit of shaking, someone picks out a piece and takes on the enormous responsibility of finding their victim, *coughs* I mean colleague, a present usually around the price range of £5. So why is this a rather inadequate way of organising this game? 

Well first, you risk the chance of picking out your own name. You might think that’s easy enough a problem to solve, just put it back in the hat…But what if you were the last person? Or even the second last person? Everybody knows your name is back in the hat, and they have a greater chance of picking your name, and in some case, 2 people will have each other’s name, Secret Santa is ruined. Goodnight. The book proposes another method of making sure that no one has their own name, and no one else knows who has their name. A clever yet simple solution involving derangements, alas, Christmas is saved, now lets hope your secret Santa isn’t a Scrooge

My favourite chapter in this book has to be the one on the Queen’s Speech. The beginning of the chapter is an analysis of the Queen’s vocabulary score (based solely on the number of unique words in the first 35K words of her speech. Surprisingly, poor old Lizzy scored lower than her counterparts Jay-Z and Shakespeare. Well, it seems as though maths might be able to give her a hand with that, with something special called a Markov chain, that determines the next word to place in a sentence given the word before. I won’t go into too much detail as to give it away; it just so happens that earlier this week I was reading about how Markov chains are used in determining the probability of flipping a coin and getting a certain outcome which is equivalent to another outcome, so this chapter peaked my interest even further. 

What is particularly good about this book is how accessible it is; at the end of every chapter there are endnotes that explain some of the maths mentioned and includes some extra reading material. The only questionable thing in this book might be the chapter about cooking turkey. They used a chicken instead. Enough said.

Overall, a great read, not too technical but with just enough maths to get you thinking. So even if your grumpy aunt Hilda despises anything to do with maths, there’s now a very slight chance she might enjoy it. 

Also the amount of cracker pulling rules I’ve never heard of is remarkable. I am now going to stop pulling both ends of my own cracker, despite my competitive nature.

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The Chalkdust Christmas card

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Recently, some of you may have received a Chalkdust Christmas card. If not, it’s not because we hate you, it’s just that we couldn’t find your address… Unless we hate you, in which case it is because we hate you.

The card initially looks very boring: it is just a grid of squares with “Merry Christmas” written below it. Definitely NOT HOT… But there’s more. There’s a puzzle inside that leads you to add some colour to the squares to reveal a Christmassy picture.

Without giving any more away, here is the puzzle. If you’d like to give it to someone as a Christmas card (or just want to actually be able to colour it in), you can print and fold this lovely pdf.

Christmas Card 2016

The grid (click to enlarge)

Instructions

  1. Solve the puzzles below.
  2. Convert the answers to base 3.
  3. Write the answers in the boxes on the front cover.
  4. Colour squares containing a 1 green. Colour squares containing a 2 red. Leave squares containing a 0 unshaded.

Puzzles

  1. The square number larger than 1 whose square root is equal to the sum of its digits.
  2. The smallest square number whose factors add up to a different square number.
  3. The largest number that cannot be written in the form $23n+17m$, where $n$ and $m$ are positive integers (or 0).
  4. Write down a three-digit number whose digits are decreasing. Write down the reverse of this number and find the difference. Add this difference to its reverse. What is the result?
  5. The number of numbers between 0 and 10,000,000 that do not contain the digits 0, 1, 2, 3, 4, 5 or 6.
  6. The lowest common multiple of 57 and 249.
  7. The sum of all the odd numbers between 0 and 66.
  8. One less than four times the 40th triangle number.
  9. The number of factors of the number 2756×312.
  10. In a book with 13,204 pages, what do the page numbers of the middle two pages add up to?
  11. The number of off-diagonal elements in a 27×27 matrix.
  12. The largest number, $k$, such that $27k/(27+k)$ is an integer.

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Snowflake, the symbol of winter: different sizes, infinite shapes

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Winter is coming. Or, to be more precise, the winter season begins on 21 December. Some people hate it, while others (like me) love it. Every time we hear the word winter, we think about that cold time of the year when we wear our scarves, jackets and coats, and we gather with our loved ones and eat (lots) of delicious food. But if you were asked to describe the word ‘winter’ with a simple symbol, what would that symbol be?  Most of us would probably think of a white, beautiful snowflake. And we are not the only ones thinking of that: if you Google the word ‘winter’ and go to the image section, you will find lots of pictures of them.

snowflakepaper

Paper snowflakes made by me

snowbentley

First pictures of snoflakes taken in 1931 by Wilson Bentley

snowflake

Recent pictures of snoflakes (made in a laboratory) taken by Prof. Kenneth G. Libbrecht

The snowflake is the most iconic symbol representing cold weather, and is also a traditional image used during the Christmas period. It is well known that most snowflakes have six-sides (hexagonal pattern) and many branches around them; but apart from that snowflakes come in a large variety of shapes and sizes, leading to the common phrase “no two snowflakes are alike”. But how is a snowflake formed? In this blog, we will describe the process of snowflake formation, and explain why they exist in a variety of shapes and sizes. To do that, we just need to understand some basic concepts: humidity and supersaturation.

Construct your own snowflakes using just paper and scissors. Follow 
the instructions here.

Humidity and average conditions

Many of us, before going to work, college or university, will check the weather forecast in order to decide if we need to wear our coats or bring an umbrella with us. If you visit the BBC weather page, you can find plenty of information about the forecast of a particular day; and you will probably be familiar with the minimum and maximum temperature, visibility and chance of rain. But there is another piece of information that is always reported which is relevant for this blog: the relative humidity of a city, which is normally expressed as a percentage. What does that value mean? In simple words, humidity is a measurement of the amount of water vapour in the air, and this system can be seen as a mixture, with the air as the solvent and the water as the solute. Or, you can think about it in a simpler way: as a wet sponge, where the air is the sponge that will be holding water.

londoncloudy

Cloudy weather in London, UK

The temperature of the environment affects the humidity: the higher the temperature, the more water vapour the air can hold (and vice versa). But there is a threshold: for a fixed temperature (say 20 C) the air cannot absorb and hold unlimited amounts of water. The concept of relative humidity (RH) is basically the ratio between the current amount of water vapour present in the air and the maximum amount of water vapour that the air could hold at a given temperature, expressed as a percentage. For instance, if the forecast of a sunny day reports a relative humidity of 60%, the air can still hold more water (at least 40 percentage points more). If we see a relative humidity of 95%, it means that the air is almost full of water, and can hold just five percentage points more. Low temperatures tend to give high values of RH, and high temperatures tend to give medium and small values of RH.

canadianforest

Strathcona Park on Vancouver Island, Canada

The formation of snowflakes requires high values of relative humidity, meaning that it is necessary to have too much water vapour in the environment; or, in fewer words, we need clouds, as this is the place where snowflakes are born.

However, we still need something more. In the figure below, we can observe the average maximum and minimum temperatures and RH registered during 2016 in two cities: London, UK, and Toronto, Canada. We observe that London always has high values of RH (hence the cloudy weather), and in winter seasons these values tend to increase even more. Unfortunately (or fortunately, depending on your point of view), London rarely experiences snowfall, because the temperatures aren’t usually low enough (around 4–6C). On the other hand, the city of Toronto is a snowy place and, although it has RH values similar to London’s, its average temperatures are expected to drop below 0C.

avconditionslo

Average conditions for the cities of London, UK and Toronto (the provincial capital of Ontario), Canada in 2016. AM stands for antemeridiem (before noon). Information taken from BBC Weather

Supersaturation

crystalforest

Crystallisation: a product of supersaturation

So, we need high values of RH and low temperatures, but this is still not enough to form snowflakes: a particular unstable condition in the clouds has to be present: supersaturation, occurring when there is more water vapour in the air than the ordinary limit of 100%.

This phenomenon has a cause: imagine that the temperature is 4C, and RH reported is around 96% (which is almost the maximum capacity of air). If the temperature decreases below 4C, the water vapour-holding capacity of the air will decrease (the RH will increase), which is an unstable situation. The air has to do something in order to return to the stable state: the excess water vapour crystallises out, either into water droplets or directly into ice, depending on the temperature of the cloud.

Formation of a snowflake

As we saw, supersaturation leads to the formation of water droplets. The life of a snowflake begins as a tiny droplet of supercooled water that freezes in the clouds to create an ice crystal. The droplets might freeze if the temperature of the cloud is cold enough. If the clouds are warmer, the crystal formation can begin around a nucleus such as an impurity (a dust or pollen particle) in a process known as nucleation.

diagramsnow

Simplified diagram that shows the formation of a snowflake

The figure on the right describes the formation process of a snowflake:

a) A single particle of dust is floating in a cloud. The dust will provide a solid surface where the freezing process will start.
b) Water vapour will stick to the particle of dust, condensing onto its surface.
c) This stage is one of the most important: the water will turn into ice. The arrangement of the molecules of a substance determines the way it crystallises: water freezes into a hexagonal structure. This hexagonal formation allows water molecules (each with two atoms of hydrogen and one of oxygen) to form together in the most efficient way. This tiny crystal can be seen as a seed from which a snowflake will grow.

d) The six corners of the hexagonal structure of the crystal grow faster because they stick out a bit farther into the humid air, so they collect water faster than anywhere else on the crystal, causing branches to sprout. And as the conditions around the snow crystal are almost constant, the arms grow out at roughly the same rate (symmetry).

e) Randomness takes over here: the conditions might favour the growth of new plates, the formation of arms/branches on the new plates, and so on. The snowflake grows as water molecules in the air join the ice crystal, until it becomes heavy enough to fall to the ground. The temperatures still have to be cold enough, otherwise the snowflake will melt and the result will be wet snow.

The stage e) mentioned above and what comes after it are, overall, a complex dynamical process: since the snow crystals fall through the cloud, they experience different temperatures, humidities, air currents and physical processes along their descent, and thus the snowflake structure will always be changing with time. The final result is the complex and unique pattern of a snowflake.

Watch fascinating real movies of growing snowflakes here.

The morphology diagram

Modelling snowflake formation mathematically is a real challenge, because it involves lots of different physics (liquid-vapour phase transformation, thermodynamics, energy and mass transfer) is dependent on time and on the meteorological conditions, and so on. Researchers have studied this process for a long time. One of the most famous researchers in this area is a Japanese physicist called Ukichiro Nakaya, who was the first person to carry out snow crystal formation experiments under controlled conditions in 1930. He observed that the morphology of the snowflakes depends on the temperature and the conditions of supersaturation. All his observations and results can be summarised in the snow crystal morphology diagram.

The diagram provides information about the different types of snow crystals that grow in air at atmospheric pressure as a function of the temperature and water vapour supersaturation (grams of water per cubic metre). At small values of supersaturation and temperatures greater than -5C, small plates will be obtained. It is surprising how the morphology of the snowflakes changes drastically at even lower temperatures: from columns and solid prisms (-5C) to solid plates (below -10C). The temperature mainly determines whether snow crystals will grow into plates or columns.

morphologydiagram

The morphology diagram of snow crystals

When the values of humidity are low, the crystals will grow slowly and simple forms will be obtained. When the humidity is higher, the crystals will grow rapidly and then branched and more complex forms appear. In few words: higher saturations will produce more complex structures. The information and observations of this diagram have been proven and confirmed experimentally during the last years, and it has been extended to temperatures of -70C.

dendritesnow

Dendrite snowflake

The most famous snowflake structure is the dendrite structure, whose name means tree-like. It is easy to identify due to its branching structure. Small dendrites are obtained at temperatures around -3C, but bigger ones are seen in colder temperatures (-20 to -25C), and this is because the atmosphere contains more moisture.

 

snowcrystal1

Sectored plate snowflake

Another beautiful and recognisable snowflake is the sectored plate which forms a hexagonal pattern with a star-like shape in the centre, and this structure is seen at almost similar temperature conditions for the dendrites, but with lower values of humidity.

snowthinplate

Thin-plate snowflake

 

The thin-plate snowflakes cannot obtain sufficient moisture to form branches. These crystals tend to form at warmer temperatures.

snowflakesinlab

Snowflakes designed in Prof. K. Libbrect’s laboratory

Some researchers around the world are modelling the structures of snowflakes, others are simulating the formation of a snowflake using complex models, others are attempting to apply chaos and fractal theory to explain the formation of the branches and the complex patterns of the snowflakes. There are still some mysteries that remain to be answered about this wonderful phenomenon.

If you want to see more amazing, colourful photographs of
real snowflakes, click here.

Special thanks to Prof. Kenneth G. Libbrecht  for allowing us to use his pictures. Visit his page Snow Crystal  for further information: it is full of wonderful and interesting things about snowflakes.


Attributions:

[Pictures: 1- Pictures of snowflakes taken by Prof. Kenneth G. Libbrecht. All the pictures belong to him. For personal and web uses, please read copyright information   2 – adapted from Flickr.com –  london by Roberto TrombettaCC-BY 2.0; 3 – adapted from Flickr.com – Battleship Lake by James WheelerCC-BY 2.0; 4 – adapted from Flickr.com – Crystal Forest – Forêt de cristal by Monteregina (Nicole), CC-BY 2.0 ; 5 – Pictures of snowflakes (1931) by Wilson Bentley belongs to public domain;  6 – The morphology diagram by Ukichiro Nakaya belongs to the Public Domainother pictures by Chalkdust]

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Catching criminals with maths

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As a result of decades of empirical research, crime science has emerged as the leading multidisciplinary approach to develop new ways to tackle crime and terrorism. As opposed to traditional criminologists, crime scientists commonly use a broad spectrum of different disciplines and sciences to achieve their aim of cutting crime. Using knowledge from chemistry, geography and physics, to architecture, public health, psychology and information technology, crime science has been able to offer new solutions to the most pressing issues that impact on the health and security of millions of people. Among all the fields and disciplines used, applied mathematics, statistics and econometrics are perhaps the most common tools used by crime scientists.  Continue reading

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