Things to Make and Do in the Fourth Dimension A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More

365 Pages·2014·8.72 MB·English

by  Matt Parker

The author and publisher have provided this e-book to you for your personal use
only. You may not make this e-book publicly available in any way. Copyright
infringement is against the law. If you believe the copy of this e-book you
are reading infringes on the author’s copyright, please notify the publisher
at: us.macmillanusa.com/piracy.
Dedicated to Keith and Nona Pallot, my maternal grandparents, who inspired me
to make and teach
Contents
Title Page
Copyright Notice
Dedication
0 The Zeroth Chapter
1 Can You Digit?
2 Making Shapes
3 Be There and Be Square
4 Shape Shifting
5 Shapes: Now in 3D
6 Pack It Up, Pack It In
7 Prime Time
8 Knot a Problem
9 Just for Graphs
10 The Fourth Dimension
11 The Algorithm Method
12 How to Build a Computer
13 Number Mash-ups
14 Ridiculous Shapes
15 Higher Dimensions
16 Good Data Die Hard
17 Ridiculous Numbers
18 To Infinity and Beyond
n + 1 The Subsequent Chapter
Notes
The Answers at the Back of the Book
Acknowledgements
Text and Image Credits
A Note About the Author
Copyright
Zero
THE ZEROTH CHAPTER
Have a look around you and find a drinking vessel, like a pint glass or a coffee
mug. Despite appearances, almost certainly the distance around the glass will be
greater than its height. Something like a pint glass may look like it is definitely
taller than it is round, but a standard UK pint glass is actually around 1.8 times
greater in circumference than in height. A standard ‘tall’ takeaway cup from
omnipresent high-street coffee shop Starbucks is actually 2.3 times further
around, but yet they refuse my requests to rename it the ‘squat’.
Using this to your advantage is easy enough: when you are next drinking in a
pub, café or whichever drinking establishment serves the sort of beverages you
enjoy getting for free, bet someone that their drinking vessel is further around
than it is high. If there is a ‘pot’ beer glass (the ones with handles) in the pub, or
an obscenely large mug in the café, then you’re sorted: they are typically three
times as far around as they are tall, so you can dramatically stack three of them
and claim it is still further around than up. Producing a tape measure at this point
may cause your victims to question the spontaneity of the whole exercise, so use
a nearby straw, or its protective paper sleeve, as a makeshift ruler.
This works for all glasses, except for only the skinniest of champagne flutes.
If you’d like to subtly check your drinking glass without arousing suspicion, try
to wrap your hand all the way around it. Your fingers and thumb will not meet
on the other side. Now, with your thumb and index finger, try to span the height
of the glass. You will most likely succeed (or, at worst, come very close). This is
a dramatic demonstration of how much shorter glasses are than they are around.
This is exactly the sort of maths I wish more people knew about: the
surprising, the unexpected and, most importantly, the type that wins you free
drinks. My goal in this book is to show people all the fun bits of mathematics.
It’s a shame that most people think maths is just what they were subjected to at
secondary school: it is so much more than that.
In the wrong situation, maths can indeed border on the tedious. Walk into a
school at random in search of a maths class, and you’ll most likely find a room
where the majority of the students are not excited. In the least. You will shortly
be asked to leave the premises, and the police may even be called. You’re
probably on some kind of list now. The point is, those students in the classroom
are following in a long line of generation after generation of uninspired maths
students. But there will be a few exceptions. Some of them will be loving maths
and will go on to be mathematicians for the rest of their lives. What is it that
they’re enjoying which everyone else is missing?
I was one of those students: I could see through the tedious exercises to the
heart of maths, the logic behind it all. But I could sympathize with my fellow
students, and specifically, the ‘sporty ones’. At school, I dreaded football drills
in the same way that other people dreaded maths class. But I could see the
purpose of all that messing around dribbling a football between traffic cones:
you’re building up a basic repertoire of skills so that you’re better when it comes
to an actual game of football. By the same token, I had an insight into why my
sporty classmates hated maths: it’s counterproductive to make pupils practise the
basic skills needed for maths but then not let them loose into the field of
mathematics to have a play around.
That is what the maths kids knew. This is why people can make a career out
of being a mathematician. If someone works in maths research, they’re not
simply doing harder and harder sums, or longer divisions, as people imagine.
That would be like a professional footballer merely getting faster at dribbling up
the field. A professional mathematician is using the skills they’ve learned and
the techniques they’ve honed to explore the field of mathematics and discover
new things. They might be hunting for shapes in higher dimensions, trying to
find new types of numbers, or exploring a world beyond infinity. They are not
just practising arithmetic.
Herein lies the secret of mathematics: it’s one big game. Professional
mathematicians are playing. This is the goal of this book: to open up this world
and give you the freedom to play with maths. You too can feel like a premier-
league mathematician, and if you were already one of those kids who embraced
maths, there are still plenty of new things to discover. Everything here starts
with things you can really make and do. You can build a 4D object, you can
dissect counter-intuitive shapes and you can tie unbelievable knots. A book is
also an amazing piece of technology with a state-of-the-art pause function. If
you do want to stop and play around with a bit of maths for a while, you can.
The book will sit here, the words static on the page, waiting for you to return.
All the most exciting bits of cutting-edge technology are mathematical at
heart, from the number-crunching behind modern medicine to the equations that
help carry text messages between mobile phones. But even technology which
relies on bespoke mathematical techniques still ultimately rests on mathematics
that was originally developed because a mathematician thought it would be fun
to try to solve a puzzle.
This is the essence of mathematics. It is the pursuit of pattern and logic for
their own sake; it is sating our playful curiosity. New mathematical discoveries
may have countless practical applications – and we may owe our lives to them –
but that’s rarely why they were discovered in the first place. As the Nobel Prize-
winning physicist Richard Feynman allegedly said of his own subject: ‘Physics
is a lot like sex; sure it has a practical use, but that’s not why we do it.’
I also hope to bring the maths you did learn at school into focus. Without it,
all the other interesting bits of maths would not be within reach. Every student
has vague memories of learning about the mathematical constant pi (roughly
equal to 3.14), and some may recall that it is the ratio of the diameter to the
circumference of a circle. It is because of pi that we know the distance around a
glass is over three times greater than the distance across it. And it is the distance
across which most people use when judging how big a glass is, forgetting to
multiply by pi. This is more than memorizing a ratio, this is taking it for a test-
run in the real world. Sadly, very little school maths focuses on how to win free
drinks in a pub.
The reason that school maths cannot be completely dismissed is that the more
exciting bits of mathematics rest on the less exciting bits. This is partly why
some people think maths is so hard: they’ve missed a few vital steps along the
way, and without them the higher ideas seem impossibly out of reach. But if they
had tackled the subject one step at a time, in the correct order, it would have
been fine.
No one bit of mathematics is that hard to master, but sometimes it’s
important that you do things in the optimal order. Sure, getting to the top of a
very high ladder may take a lot of effort from start to finish, but each individual
rung is no more work to reach than the last one was. It’s the same with
mathematics. Step by step, it’s great fun. If you understand prime numbers, then
exploring prime knots is much easier. If you get to grips with 3D shapes first,
then 4D shapes are not that intimidating. You can imagine all the chapters in this
book as a structure, where each one rests upon several of the previous chapters.
You can even choose your path through the chapters, as long as, before
tackling a later chapter, you’ve read all the ones that support it. As the book goes
on, the chapters cover more advanced mathematics, the sorts of things you
generally won’t hear about in a classroom. This can be daunting at first glance.
But as long as you pass through them in the right order, by the time you reach
the far-flung corners of mathematics, you’ll be fully equipped to enjoy all the
delights and surprises there are on offer.
Above all else, remember that the motivation for climbing this structure
should be merely to enjoy the view as you go. For too long, maths has been
synonymous with education; it should be about fun and exploration. One puzzle
at a time, one maths game after another, and soon we’ll be at the top, enjoying
all the maths most people never know even exists. We’ll be able to play with
things beyond normal human intuition. Mathematics allows access to the world
of imaginary numbers, to shapes that exist only in 196,883 dimensions, and
objects beyond infinity. From the fourth dimension to transcendental numbers,
we’ll see it all.