# Fundamental Equations, Chaos, Fractals, and Leaky Faucets

It’s been a while since I’ve written here. Busy teaching. (I know, lame excuse).

So I go and watch some video on YouTube, and on the side is the list of suggested videos. For a while there’s been this one by Veritasium (Derek Muller). Now, Derek is a fantastic science educator, and is who I want to be when I grow up. One problem is he’s younger than me by over a decade. Hmm. Have to work on that somehow.

Anyway, the video that was just waiting for me to finally click on it was this one. It’s about an equation that will change how you see the world.

## The Logistic Map

The math is very simple: $x_{n+1}=r x_n(1-x_n)$ where $r$ is the growth rate. This is a very simple equation with a negative feedback loop.

When you graph $r$ by the equilibrium population, you get this:

What!?

Once the growth rate hits 3, the equilibrium population splits, and oscillates between two values. Then just after 3.4 it splits again. And very soon it becomes chaotic. Oh, and fractal. The chaotic nature was used for pseudorandom number generators.

## The Mandelbrot set

Does mentioning fractals make you think of the Mandelbrot set? If it doesn’t, then you have some research to do.

It’s probably the most famous fractal out there. Heck Johnathan Coulton has done a song about it.

But evidently if you somehow rotate the Mandelbrot Set along it’s real number axis, you get this:

Look familiar? At this point I started getting a headache, but it was one of those good, excited headaches that come from having your reality twisted about.

## Leaky Faucets

Oh yeah, leaks. Derek then mentions that if you get your faucet going drip, drip, drip, and then increase the water pressure just right, it will start doubling: drip drip, drip drip, drip drip. Push it a little more, and you get chaotic behavior.

Of course the YouTube video is so much better than my explanation. Go watch it.

# Let’s talk numbers!

Two neat things from an AskReddit post.

## One:

cubosh said: Take every planet in our solar system, line them up so they are all touching, and they will fit inside the space between earth and our moon, with room to spare.
>to which jbhall36 said: Take every planet in our solar system, line them up so they are all touching, fit them in the space between Earth and our moon, and the gravitational force would be catastrophic and likely kill everything on Earth.
>>to which Rogukast1177 said: You don’t need the “likely”
>>>to which pinkbutterfly1 said: You do need the “likely”. Because Tardigrades. http://www.bbc.com/earth/story/20150313-the-toughest-animals-on-earth

## Two:

A while ago I write about Graham’s Number, which is really migraine inducing. Another (much smaller) number is how many ways you can shuffle a standard deck of 52 cards. Here’s what VerbableNouns said (which really was written by techniforus):

One of my favorite is about the number of unique orders for cards in a standard 52 card deck.

I’ve seen a a really good explanation of how big 52! actually is.

• Set a timer to count down 52! seconds (that’s 8.0658×1067 seconds)
• Stand on the equator, and take a step forward every billion years
• When you’ve circled the earth once, take a drop of water from the Pacific Ocean, and keep going
• When the Pacific Ocean is empty, lay a sheet of paper down, refill the ocean and carry on.
• When your stack of paper reaches the sun, take a look at the timer.

The 3 left-most digits won’t have changed. 8.063×1067 seconds left to go. You have to repeat the whole process 1000 times to get 1/3 of the way through that time. 5.385×1067 seconds left to go.

So to kill that time you try something else.

• Shuffle a deck of cards, deal yourself 5 cards every billion years
• Each time you get a royal flush, buy a lottery ticket
• Each time that ticket wins the jackpot, throw a grain of sand in the grand canyon
• When the grand canyon’s full, take 1oz of rock off Mount Everest, empty the canyon and carry on.
• When Everest has been levelled, check the timer.

There’s barely any change. 5.364×1067 seconds left. You’d have to repeat this process 256 times to have run out the timer.

Now, that gives you some inkling of how big 52! is, but that’s nothing compared to Graham’s Number which I mentioned earlier.

# Large numbers are hard to understand

Lots of people have a blind spot for numbers. Especially large numbers, like a million or a billion. Lots of people think that numbers like a million, a billion, and a trillion are evenly spaced on the number line, like 1, 2, and 3 are. But the amount of space between a million and a billion dwarfs the distance from 1 to a million. A billion is a thousand millions.

Here’s another way to look at it. It takes about 12 days for 1,000,000 seconds to elapse. But it takes about 32 years for 1,000,000,000 seconds to elapse. They are so far apart. But most people just stop thinking about what these numbers really mean, and kind of lump a million and a billion together. Kind of like “they’re big numbers, and a billion is bigger than a million”. Well, yes, that’s true. But it loses how much bigger, and it’s a lot.

# One way that Math and Science are linked

Everytime math comes into my science class, the students always groan. “Why do we have to do math? We already had math class?” But math and science are linked. In fact, the math has to be there. And it can be really interesting how this happens. A number of years ago, one of my favorite youtubers did a trilogy of videos on this.

The first is on Fibonacci numbers, which seem to pop up all over the place. This then leads to one of my favorite irrational numbers: Phi (Φ). Everyone knows about Pi, but phi is pretty awesome too. Well, actually the golden ratio, which is also used all over the place, and mathematicians use phi as shorthand, kind of like they use pi for the ratio of the circumference of a circle to the diameter.

It turns out that when plants want to grow leaves, but not have the upper leaves be right above the lower leaves, they frequently put the leaves phi degrees away from the previous leaf. How do they do that? It’s not like they have protractors know about geometry or anything. It turns out that it’s really simple, as vihart gets to. It’s just growing where there’s more protein that tells the plant where to grow new leaves. This automatically ends up with the leaves being phi degrees apart. It’s really cool!

Anyway, here are the videos:

# Math migrane

So, on the way to rather large numbers, you may see some mileposts:

These, and a googolplex, are pikers compared to Graham’s Number. To get there, you first have to go up the math ladder from counting, to addition, to multiplication, to exponentiation, to tetration (this is where my math migraine kicks in), to pentation, to hexation, and wayyy beyond.

If, and that’s a big if, you can wrap your mind around Graham’s Number, … well first off, you’re lying, just admit it … but this supremely large number, where there isn’t enough space in the universe to write down all the digits (the train passed that station long ago on this math journey, just read the article at the link), is not even approaching what infinity is. This isn’t anywhere close to aleph-null (ℵ0).

## Language and Math and Zipf

### Aside

Vsauce has a really cool video on some ways that language and math are intertwined. Including how it all is interwoven with the world around us.

# Long live the work of Emma Noether

A wonderful article at arstechnica tells of the work of a woman Jew in the early 20th century. She lived in Germany in the years leading up to WWII. She had to leave, and came to America. Some of her best work is on Einstein‘s General Theory of Relativity.

Science kind of takes some symmetries for granted. For example, you should be able to perform the same experiment a year later and get the same result. Or you should be able to do it in two places and get the same result. But one of the main laws of physics — conservation of energy — seems to be broken by general relativity. It is possible for a machine to emit gravity waves, and gain energy instead of losing it.

This paradox was solved by Emma Noether who’s theorem proves a connection between symmetries and conservation laws.

It turns out that with general relativity, you may get different results depending on where you are when you perform your experiments. Here on earth, the experiments all happen in very similar circumstances. But in a strong gravitational field, the curvature of space is different, and you can get different measurements. This strange effect is predicted by Einstein’s theories, and Neother’s theorem provides the connection.

# Math homework just got easier

PhotoMath is a program that runs on IOS and Windows phones (Android coming out soon) that can solve math equations. It even shows the steps needed to get the solution. This can help students learn math, or (more likely) let them do their homework without actually learning anything.