When I was young I used to love visualizing equations (hey, I lived in rural Shropshire! It was that or watch badgers).

This is something I played with when I was about 17.  I still remember it, but I didn’t have the processing power or tools to make a video out of it.  Now I do. The video’s at the bottom of the page, but bear with the maths stuff – I promise it’s REALLY easy.

Initial explanation

I had to write programs to work this stuff out. Fortunately Excel makes it all a bit easier now.

Create a spreadsheet with a sequence of numbers in row 1, and fill in column 1.  This’ll be our x and y axis.

For the values we’ll use y√x.

So:

We’ll then drag that across to fill in our table.

Now unsurprisingly the numbers get bigger as x and y increase.

Also whenever x (the first row) is a square number (1,4,9,16 etc) then y√x is a whole number.

If we were to plot this as a graph we’d get a really boring graph.

Now the interesting bit

Let’s just concentrate on the decimal bits – let’s throw away the whole numbers.

Okay, something is going on, but it’s difficult to see what.

The codey bit

Instead I’m going to plot the graph using code. 

I’m going to use white for close to a whole number and black for not near a whole number.

Also, because I want 1 pixel to be 1 number, I’m going to concentrate on a different range.

I’m going to look at the x axis between 2500 (50*50) and  2601 (51*51).  Which will give me a strip 101 pixels wide.

We get some sort of spirally thing going on.

I’ve always thought that it looks a bit like smoke.

It’s actually easier to see on for the smaller values (this is used on my twitter profile http://twitter.com/AdamJTP click the image below for a better look).

So we can sort of see that the squares change as they move up.

Woo-hoo processing power!

I now have much more powerful computers than I had 20 years ago, so I was finally able to get my computer to spit out an animation of how the cells evolve and break up.

So now I’m going to take slices from that chart on the top right and animate them (I’ve never uploaded to YouTube – this could get messy).

YouTube – animated graph of decimal part of y sqrt(x)

Heh – I typed the wrong description in when I upload this image. Oh well.

For me maths isn’t a spectator sport. It’s a game to play, even if you don’t play it well, and I try to teach my children this too. 

This is what we did after Saturday lunch. I’ve put this here for my children to refer back to, and in case anybody else finds it useful.

On Saturday I was talking to my children about population dynamics.

We started talking about populations that doubled in number after each generation:
1,2,4,8,16,32,64,128,256,512,1024 …

Then Fibonacci’s rabbits:
0,1,1,2,3,5,8,13,21,34,55,89,144,…

…and we all agreed that if rabbits really did breed like this we’d be up to our armpits in rabbits by now.

So I asked them to grab their laptops and fire up Excel.

This is a famous (and simple) population model.  All you need to know is the number of "rabbits" will vary between 0 (no rabbits) and 1 (too many rabbits i.e. complete environmental destruction by rabbits!).

If you’re a reading sort of person then Wikipedia and the web will explain it better than I can – http://en.wikipedia.org/wiki/Logistic_map

Whereas here are the "just play with it" instructions for eight year olds. 

There are applets that do this on the web – but I wanted my nippers to create the spreadsheet themselves.

Creating the experiment

1. Open Excel
2. In cell A1 type the number 2

3. In cell B1 type the number 0.01 (this is the starting population. Remember 0 is none, 1 is too many).

4. In cell B2 type “=A$1*B1*(1-B1)”

4a. We talked a bit about what that equation meant – but anyway, who cares…

5. Click on cell B2, grab the little square on the corner and drag it down for at least 100 cells.

6. Let go and scroll back up to the top

7. Click on the B at the top of the page to highlight all of column B

8. From the insert menu, choose line chart

9. Drag the corners to make the chart bigger

Now – playing with the numbers

See that number 2? That’s our “breeding factor”, change it to numbers between 1 and 4.  This is the whole point of the experiment to be able to change these numbers ourselves.

Cell A1 = 2

Gives us the graph below – the graph goes up and stabilizes at 0.5.  My children seemed to think this was the “right amount” (presumably because it’s between 0 and 1).

Cell A1 = 2.6

Ooo unexpected! This overshoots a bit before stabilizing at 0.615.  The children already like this model – the number of rabbits is limited (we seem stuck with the rabbit metaphor – damn you Fibonacci!)

Cell A1 = 3

Hmmm… that’s tricky, the number bounces about a bit.  Maybe it would get down to a single value if left long enough.

Cell A1 = 3.2

Oh no – there’s too many rabbits then the population crashes (then presumably the grass grows back) and then loads of rabbits again

Cell A1 = 3.5

High, low, not so high, not so low, high, low, not so high, … funny looking graph.

3.8

Now it gets interesting.  The graph is all over the place!  It swings about, seems to find a level then it’s off again.

but what about….

3.8000000001

We’ve changed the breeding factor by a tenth of one billionth and got a different graph.  Okay, they’re both spiky but it is clear that the difference is more than a tenth of a billionth.

Further investigation

This is then a good point to talk about the butterfly effect (how a small change in a number lead to a big change in the graph).

And then I left them exploring different numbers. However, navigating is always easier with a map.  If you’re trying this yourself – see if you can work out how the map below relates to the graphs above.

Or here’s a video from the excellent sixty symbols website.

…and then…

…we ended up setting up a model to explore the likelihood of “assembling a 747 in a junkyard using a tornado” using a set of wargaming dice.  Okay, we didn’t make a 747 (using wargaming dice?  C’mon that would be weird!), but we did beat godzillions-to-one odds.  But I’ll blog about that another day (unless we find a dead rabbit to mummify first – because that would be VERY cool).