I’m a children’s book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart’s essay “The Mathematician’s Lament,” and found that I, too, lament the uninspiring quality of my elementary math education.
I want to make a book that discredits the notion that math is merely a series of calculations, and inspires a sense of awe and genuine curiosity in young readers.
However, I myself am mathematically unsophisticated.
What was the first bit of mathematics that made you realize that math is beautiful?
For the purposes of this children’s book, accessible answers would be appreciated.
This wasn’t the first, but it’s definitely awesome:
This is a proof of the Pythagorean theorem, and it uses no words!
For me it was the Times Table of $9$.
We are usually forced to memorize the multiplication tables in school. I remember looking at the table for $9$, and seeing that the digit in ten’s place increased by one, while the digit in the one’s place decreased by one.
$$
\begin{array}{r|r}
\times & 9 \\
\hline
1 & 9 \\
2 & 18 \\
3 & 27 \\
4 & 36 \\
5 & 45 \\
6 & 54 \\
7 & 63 \\
8 & 72 \\
9 & 81 \\
10 & 90
\end{array}
$$
After this, I realized that I could always add $10$ and subtract $1$ to get the next result. For a $7$ year old, this was the greatest discovery ever made.
And that your hands could give you the answer immediately: $7 \times 9$ = hold down your $7$th finger, leaves $6$ fingers on left of held down finger, and $3$ on right: $63$.. works all the way up to $9\times10$, beautiful.
Whether this is ‘simple’ enough is debatable… the method to generate the Mandelbrot set is likely to be far too complicated for the book in question, but the mathematical expression that’s at its heart couldn’t be much simpler.
$z_{n+1} = {z_n}^2 + c$
After implementing the Mandelbrot set I learned about the Buddhabrot, which is basically a way of rendering the stages of the Mandelbrot algorithm, and after some considerable processing time I had a render:
I then tweaked my input parameters to ‘zoom in’ on a particular area, and when I saw the result my jaw hit the floor. This is when I saw the true beauty in mathematics beyond ‘nice’ results. Again, it’s probably too advanced for your book because of the steps involved in creating the visual, but maybe it’d make for a nice final hurrah to inspire further exploration? It still boggles my mind to see such amazing results from something so simple.
I used to love naughty $37$.
$37 \times 3 = 111;$
$37 \times 6 = 222;$
$37 \times 9 = 333;$
$37 \times 12 = 444;$
$37 \times 15 = 555;$
$37 \times 18 = 666;$
$37 \times 21 = 777;$
$37 \times 24 = 888;$
$37 \times 27 = 999;$
I found it completely amazing that the angles in a triangle always added up to 180 degrees. No matter how you drew a triangle, you could measure the angles with a protractor and they always add up to about 180 degrees, like magic. Even more amazing when I realized it wasn’t some rule of thumb or approximation, but true in some deeper sense for the ideal, platonic triangle.
The first “math thing” that just blew my mind was the identity
$$
e^{i\pi} = -1
$$
Namely the fact that the two independently discovered transcendent numbers and imaginary one so simply and elegantly bound.
In the marginally rearranged form
$$
e^{iπ}+1=0
$$
it uses absolutely nothing but nine essential concepts in mathematics:
I remember being very pleased at an early age, perhaps five or six, by the following bits of calculator tinkering, among others:
12345679
× $n$ × 9 = nnnnnnnnn
.When I got a little older, I loved that I could find an $n$th-degree polynomial to pass through $n+1$ arbitrarily chosen points, and that if I made up the points knowing the polynomial ahead of time, the method would magically produce the polynomial I had used in the first place. I spent hours doing this.
I also spent hours graphing functions, and observing the way the shapes changed as I varied the parameters. I accumulated a looseleaf binder full of these graphs, which I still have.
As a teenager, I was thrilled to observe that although the number “2 in a pentagon”
in the Steinhaus–Moser notation is far too enormous to calculate, it is a trivial matter to observe that its decimal expansion ends with a 6
.
I realize that your book wants to discredit the notion that math is merely a series of calculations, but I have always been fascinated by calculation, and I sometimes think, as the authors of Concrete Mathematics say in the introduction, that we do not always give enough attention to matters of technique. Calculation is interesting, for theoretical and practical reasons, and a lot of very deep mathematics arises from the desire to calculate.
Adding to LaceySnr’s answer, I’d like to mention fractals in general. While fractals will probably count as a higher application of maths, they are very often very visually beautiful. So you could easily show a picture of a fractal and explain that there is just a simple formula behind it all.
Some more examples:
This isn’t what did it for me, but it’s fairly simple and quite nice:
$$0.9999999999\ldots =1$$
Here is my favorite classic illustrating the power and beauty of mathematical argument. Consider the question:
Question: Can an irrational number raised to an irrational number be rational?
Answer: One of the classic answer goes as follows. Consider the number $x=\sqrt{2}^\sqrt{2}$. If $x$ is rational, we are done. If $x$ is irrational, then consider $x^{\sqrt2}$, which is $2$ and now we are done.
The number of pennies stacked in a triangle $(1,3,6,10,\cdots)$ is along one diagonal line of Pascal’s Triangle. The number of spheres stacked in a tetrahedron $(1,4,10,20,\cdots)$ is the line next to it. The next line is the number of hyperspheres in a pentachoron.
I was about $10$ and living in a hotel and home sick from school, stacking up pennies and “red hots” in pyramids, etc. I made a table of these numbers. Noticing the simple addition rule in the table, I extrapolated to the $4$th, $5$th, dimensions. Later when I learned of Pascal’s triangle that moment was probably my biggest joy of mathematics, realizing I’d run into that years before.
As a child, the Fibonacci numbers
$$1,\; 1,\; 2,\; 3,\; 5,\; 8,\; 13,\; 21,\; 34,\; 55,\;\ldots$$
were very fascinating to me.
They are named after the the Italian mathematician Fibonacci, who described these numbers in his 1202 book Liber abaci modeling a growing rabbit population:
Formally, the Fibonacci numbers $F_n$ are defined recursively by $$F_1 = 1, \quad F_2 = 1, \quad F_{n+2} = F_{n+1} + F_n$$
It was a lot of fun to compute them, one after the other, and to collect the results in ever-growing tables:
$$F_3 = F_2 + F_1 = 1 + 1 = \mathbf{2}\\F_4 = F_3 + F_2 = 2 + 1 = \mathbf{3}\\F_5 = F_4 + F_3 = 3 + 2 = \mathbf{5}\\F_6 = F_5 + F_4 = 5 + 3 = \mathbf{8}\\F_7 = F_6 + F_5 = 8 + 5 = \mathbf{13}\\\vdots$$
At some point, I asked myself the question: To compute $F_{10}$, do I really have to compute all the Fibonacci numbers up to $F_9$ beforehand? So I tried to figure out some formula where you can plug in $n$, do some basic arithmetics, and get $F_n$ as a result.
I’ve spent a lot of time on this. However no matter how hard I tried, I didn’t succeed.
After a while I found the closed form
$$F_n = \frac{1}{\sqrt{5}} \left(\left(\frac{1 + \sqrt{5}}2\right)^{\!n} – \left(\frac{1 – \sqrt{5}}{2}\right)^{\!n}\right)
$$
in some book. I was paralyzed.
How can it happen that such an easy recurrence formula needs to be described by such a complex expression? Where do the square roots come from, and why does the expression still always evaluate to an integer in the end? And, most importantly: How on earth can one find such a formula??
My son loved this when he was little – patterns everywhere:
(Copy from http://mathforum.org/library/drmath/view/57919.html)
There is a well known story about Karl Friedrich Gauss when he was in
elementary school. His teacher got mad at the class and told them to
add the numbers 1 to 100 and give him the answer by the end of the
class. About 30 seconds later Gauss gave him the answer.
The other kids were adding the numbers like this:
$$
1 + 2 + 3 + … + 99 + 100 = ?
$$
But Gauss rearranged the numbers to add them like this:
$$
(1 + 100) + (2 + 99) + (3 + 98) + … + (50 + 51) = ?
$$
If you notice every pair of numbers adds up to 101. There are 50
pairs of numbers, so the answer is
$$
50 * 101 = 5050
$$ Of course Gauss
came up with the answer about 20 times faster than the other kids.
In general to find the sum of all the numbers from 1 to n:
$$
1 + 2 + 3 + 4 + … + n = (1 + n) * \bigg(\frac{n}2\bigg)
$$
That is “1 plus n quantity times n divided by 2”.
When I was a kid my parents explained basic arithmetic to me. After thinking for a while I told them that multiplying is difficult because you need to remember if $a \cdot b$ means $a+a+\ldots + a$ ($b$ times) or $b + b + \ldots + b$ ($a$ times). I was truly amazed by their answer.
I always thought cycles in decimal fractions were magic, until I realized I can easily create whichever cycle I wanted:
I failed a number theory exam because the professor did not know this trick and said I needed to prove it.
I don’t remember what the first beautiful piece of math I encountered was, but here are a couple of candidates:
Proof that the square root of 2 is irrational
Euclid’s proof that there are infinitely many prime numbers
The fact that you can always divide something by two. That is an amazing discovery my dad tells me I made as a toddler.
I think that ever since I remember abstract mathematics was a fascination of mine, even before I knew what it was (because it was obvious school mathematics wasn’t that).
Another fact I stumbled upon as a teenager and fascinated me was that if you hold a magnifying glass over a ruled paper the parallel lines bend, and eventually meet at the edge of the glass. That, in a nutshell, is a non-Euclidean geometry.
For me, it was the discovery that the sum of the digits in all multiples of three are themselves multiples of three, and you can recursively sum them to get to 3, 6, or 9 (i.e. an ‘easy’ multiple of three)
E.g.
The sum of the digits in $13845$ is $21$,
The sum of the digits in $21$ is $3$
Edit: Should probably add that what made this useful to me was that numbers that are not multiples of three do not have this pattern.
When I was a child, I spent the whole summer at a camping at the coast of Catalonia. There I was always around my grandfather. He himself had no proper education and never finished school. Nevertheless he liked to read books on his own, about many things, grammar, the French language, mechanics, mathematics…
I remember he taught me many things. He was the first to explain me, as I fell asleep in his arms, under the starry night, that the Earth was a ball, and that there were people underneath the ground where we stood, on the other side of the planet, who were standing upside down without falling, because we were all attracted to the center of the ball. I did not understand, at that moment, how was that possible. But I trusted him and knew that there were many things I did not understand about the world.
One particular thing related to mathematics that he told me and that got me thinking, making myself questions and reaching the boundaries of my mind, was that one frog could try to jump her way across a puddle (we also went together to catch frogs), jump first to the half of it, then to the half of the remaining half and so on, and that after an infinite number of jumps she would arrive at the other shore.
This was, I think, one of the first things that made me feel that the world or that reality itself was infinitely bigger, more complex and more beautiful that anything we could understand or even begin to grasp. I guess this sense of real magic is what makes me have a special love for mathematics.
Who remembers magic squares? Those sparked my interest in mathematics.
Here are some things that I found interesting back when I was in junior high school. I hope they are not too advanced for young children:
A few things come to mind:
Sir Francis Galton’s bean machine, which demonstrates the
Central Limit Theorem, is quite remarkable.
Source: Wikipedia’s Entry on the Bean Machine
Here’s a video of this device in action: http://youtu.be/xDIyAOBa_yU
I mentioned this in comments elsewhere, but factorization diagrams are fascinating.
Source: The Math Less Traveled (Post 1, Post 2)
Here’s a beautiful JavaScript demo of these graphs being generated: http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/
Even as an adult, I think continued fractions and generalized continued fractions are amazing. One of the simplest is the golden ratio:
$$\varphi = 1 + \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}$$
And this identity is downright incredible:
$$ \frac{\pi}{2} = 1+\cfrac{1}{1+\cfrac{1}{1/2+\cfrac{1}{1/3+\cfrac{1}{1/4+\ddots}}}}$$
I should stop myself now… But math is really filled with astounding phenomena like I’ve mentioned above…
I always had a peripheral understanding that there was something more to maths than working out how your change or divvying up sweets with your siblings. But the day I really, truly understood was when I learned about $\pi$.
$\pi$ was magical to me. For one thing, it’s a funny-looking Greek letter with a funny-sounding name. But, more captivatingly, it introduced me to an epiphany: that somewhere, on some level, the fundamental structure of reality itself could be understood through mathematics.
Let’s assume your childen understand what a circle is, and how to measure things with a measuring tape. Introduce them to circumference and diameter. Give them a table with three columns—circumference, diameter and “the secret of circles”—and a big tape measure. Tell them to go out and measure as many circles as they can find: plates, car tires, stop signs, plant pots, lines on a basketball court… anything so long as it’s circular. Let them loose.
Later, once they’re done measuring everything in the neighbourhood, hand them a calculator and tell them to go through each of their circles and divide the circumference by the diameter, and write the number they get in the mystery third column. Tell them that a pattern will start to appear, and they need to see if they can spot it.
Once they’re done, you can explain to them that the reason the first couple of numbers is the same is because there’s a number, a magical number, that tells us a secret about every circle in the universe—from rings we wear on our fingers to the sun and moon in the sky and the whole planet Earth. No matter how big or how small, how grand or how humble, every single circle is a bit more than three times bigger around than it is from one side to the other. This number is so special that it has its own name, pi, and its own special letter, $\pi$. It’s not three and it’s not four—it’s somewhere just after three, and we can’t write down exactly where because it goes on forever. Luckily, we only really need to know the first few numbers most of the time, so we can use this magical number whenever we need it.
The sense of revelation that came from knowing that every circle in the universe is connected by this strange, special number stayed with me for a long time, and is at least partly responsible for my love of mathematics in later life.
The game of Nim and its solution are pretty cool. The proof might be a bit difficult, but I think kids would love to learn a game like that and how to beat their parents at it.
There’s a lot of other fun mathematical games like that too. But I think the first thing I learned that turned me towards mathematics was the existence of multiple infinities, and things like Hilbert’s infinite hotel.
Everybody loves fractals. I think this one – The Dragon Curve – is particularly easy to explain, and it is very surprising and aesthetically pleasing:
Here’s a video I’ve seen which explains how it comes about:
The Dragon Curve from Numberphile
http://en.wikipedia.org/wiki/Donald_in_Mathmagic_Land
Disney, as it was long time ago.
The Golden ratio
It was presented to me like this: There exists a number that you can square, subtract itself, and you’ll get 1. Or, you can inverse the number, add 1, and you’ll get the number back. What a beautiful number, I thought. Of course, I later realized the number was just a solution to:
$$x^2 – x – 1 = 0.$$
However, I was really impressed when later I learned this number also shows up in nature in the patterns of plant growth. Wow! Who would have thought?
I first understood the difference between zero and none when thinking about thermometer readings. If you had a ton of thermometers scattered around the world, and you collected their readings periodically and put them in a database, what would you do if any thermometer was broken? If you just put a zero reading, you’ll screw up your averages, but if you put a null value, you can handle broken thermometers easily.
That made me realise what a number is.
I was in elementary school, drawing 3D shapes in class while bored. I drew cubes by drawing two overlapping squares and connecting the vertices, like the top row in this image:
Then I thought, what if I did the same procedure, but to a cube? So I drew four squares and connected the vertices, like this:
I was struck by the beauty of the resulting image, with its intricate structure of star-like patterns. Here’s a static version:
It was years later that I discovered, to much fascination, that this was in fact the four-dimensional analogue of the cube: the hypercube. Hence my username.
Edit: Another thing I remember thinking about when I was younger was that I could not always draw a straight line through three points, but was surprised to find that it would always work for two points.