Mathematical!

At the beginning of this year I was sitting at work, waiting for the hours to tick by, when I began to wonder what I would do if I discovered I was immortal. Many scientists have believed for some time that the first person to live to be 200 has already been born, so it may not even be as impossible a situation as it sounds. I figured the key thing for an immortal would be capitalising on long-term investment. Money is certainly a nice thing to have a lot of, but surely the novelty would soon wear off and the eons of malaise would set in.

I would need a good hobby to stay occupied throughout the millennia, so I figured I would learn everything. Take every course under the sun and become a specialist in every field. The more I thought about it, the more it made sense, and then another though occurred. What if I set out on this quest for ultimate smarts and then, only later, discovered I was not immortal? Then I would have spent a lifetime of optimistic work, chasing a dream. Life is probably longer than it seems, so why not live like it’s endless?

I decided to go back to uni and study something new. As I have a degree in visual-arts, I thought I’d go the other way and begin again in science.

Maths was an interesting option. I’d liked maths as a kid, but by high school it had started to seem like an endless chore. Maybe there was more to it. And as an immortal I could afford to give it another look.

Unfortunately maths at ANU is not exactly child’s-play. Taking an English class is not easy, but if you don’t speak the language it’s practically impossible. Having not studied tertiary maths I did not speak the language, and the serious courses required a double-major in college maths. Also, I happened to be considering all of this just two weeks before the beginning of classes.

I submitted a very late application for a bachelor of science at ANU and turned to the internet to see if it was possible to learn maths in a fortnight.

Fortunately, Khan Academy.

That guy will learn you brains you didn’t know what were.

I spent about 10-12 hours a day working through the courses. My application for uni was successful and I signed up for maths. The first thing the course provided was a general test designed to outline what you should know. I passed, and the course began. I also took psychology, biology and physics. The first two were simple enough because they didn’t require a lot of pre-knowledge, but physics needed both maths and physics experience. The courses started at the beginning anyway, but maths and physics took off at a blistering pace. I had to work harder than I’ve ever worked on anything, but I kept up. I never really got ahead of the curve in physics but maths started to fall into place, and I wound up getting a HD.

For the first time I saw math teachers who actually cared about maths, and I started to get a glimpse of the depth of the subject. The biggest change, though, came from reading Paul Lockhart’s book, which shows the fundamentally aesthetic nature of maths, and how poorly this tends to be communicated in school.

I’m definitely hooked on this approach now, and I thought I’d attempt to explain through a problem that I came up with myself.

Introducing; Simon’s Marvellous Math Problem of Fantastical Wonder.

The problem follows as such:
If you take two standard A4 sheets of paper, and put them next to each other with the short sides at the top, this makes an A3 sheet of paper. The interesting thing about this is, the A3 sheet is the same shape as the A4, just a scaled up version.  So the question is: What’s the deal with that?

The first thing to notice is that there can only be one particular rectangle that has this property. Two squares next to each other make a rectangle, and vice versa. So at some specific point in between a 1x1 square and a 1x2 rectangle there is a magic shape that is half of itself.

The easiest way to describe a rectangle is by the two lengths of the sides, but in our case, it doesn’t matter what the size of the rectangle is, only the shape. So the thing we’re looking for, the thing that doesn’t change with the scale of the shape, is the ratio of the sides.

A 1x2 rectangle is the same shape as a 2x4 rectangle because ½ = 2/4 = 0.5

So how do we find the ratio of the sides of an A4 sheet?

We could try looking up the measurements.

International A4 size is 210 x 297mm. So the ratio of the short side to the long side is 210/297 or  0.70707070707(repeating).

That’s cool, but what about an A3 sheet. It should be the same ratio, right?

International A3 size is 297 x 420mm. So the ratio is 297/420 or 0.70714285714...

What we’ve discovered is that the shape of A4 paper is very close to having the property we described, but not perfect, because 0.70707070707 is not exactly the same as 0.70714285714. So is it possible to have a perfect ratio, and what is it, and why?!?

The first clue to working it out is that both the A4 and A3 sheets share a measurement: 297mm. This is because the long side of the A4 becomes the short side of the A3.

Knowing this we can do something tricky. If we arbitrarily decide that the short side of the A4 is 1 unit in length (it doesn’t matter what the unit is; 1cm or 1billion cm), then the long side is the unknown length, which we’ll call Z. And by studying the picture below we can see something interesting:

If the A4 shape is 1/Z, then the A3 shape can also be described using those units.  It’s short side is Z, and it’s long side is the same as 2 A4 short sides, which makes it’s ratio Z/2.

For the shapes to be the same, the ratios of the lengths must be identical, which means:

The question has become; “what number must Z be to make this statement true?”

To find this, we can alter the way we express this statement, and “solve for Z”

Because the A3 side is “Z divided by two”, if we double everything this comes out as

Did we get anywhere? Now Z equals two divided by Z. To get rid of the fraction we can do the same thing again, except this time we multiply everything by Z.

Now there’s only one Z in the equation but it’s squared.  That’s not a problem though, if 2 equals Z x Z, then that just means Z is a number that if you multiply it by itself you get 2. Which we can express as

Now we have it. It turns out that the only ratio of lengths that our perfect A4 rectangle can have is

So what is the square root of two?

1 * 1 = 1

1.5 * 1.5 = 2.25

So it must be between those two numbers. If we type it into a calculator it comes out as 1.41421356237. Fair enough. Is that the end of the story?

But what if we calculate 1.41421356237 * 1.41421356237?

This comes out as 1.99999999999.

So this can’t really be the square root of 2. The calculator must have not given us the precise value because it can only display a limited amount of numerals after the decimal point. So how many numerals would be required for the precise value that is equal to the square root of two?

The answer is infinite.

Just like the value pi, which is the ratio of a circle’s circumference to it's diameter, the value equal to the square root of two cannot be described with numerals. This type of number is called “irrational”.

Pythagoras and the some other ancient Greek lads discovered this and came up with a proof by showing that the number could not be even, but also could not be odd. Pythagoras actually tried to hide this fact because his troop had developed a kind of number-based religion that stated that all values could be expressed as fractions.

Much later down the track, a mathematician discovered that there are actually more irrational numbers than rational ones. In fact there are so many more irrational numbers that if you threw a dart at a number-line, and with infinite care, found the exact value that the dart hit, the chance of this value being a rational number (1 or 2 or ½ or 118739182/19723649872634 etc.)

Is actually zero!

So we have answered the question of what the perfect A4 shape is, but we haven’t really worked out why this is.

There is, of course, no complete answer to this question, but still, we might be able to find some interesting clues.

One interesting thing to note is that if you have a square with sides of length 1, the length of the diagonal is also the square root of two (which is essentially where we get the term “square root”)

This doesn’t seem like an accident. There must be some intrinsic connection between the two questions “How long is the diagonal in a square?” and “How long is longer side of a shape that is equal to half of itself?”

Now, we are in the domain of mathematics. This is what maths is about. Maths doesn’t care what the ratio of two lengths are, it cares about why that ratio is the way it is.

In school we’re taught algebra and the like as though calculation is the purpose of maths. This couldn’t be more wrong. Discovering the intrinsic nature of the realms of our thoughts is what maths is about.

But the thing is, knowing a bit of algebra enabled us to go from a question about the shape of an A4 page to a deeper question about the connection between two shapes and an irrational number. Without algebra and other concepts, that journey, and the journey on from there into more interesting questions, would be essentially impossible.

The tools of maths need to be learnt in order to ask the interesting questions, but too many maths teachers have forgotten or have never learnt that the tools are the means to studying maths, not the subject itself.

As to the question of the connection between our shape and the diagonal of a square, if you have read this far you’ve proven yourself able to take up this question yourself. But I’ll share two things that I discovered from various ponderisations.

First, if you take a square with sides of root 2, the diagonal is of length 2.

And, second, when you go from an A4 to an A3 size, the area doubles. Similarly the area of a square doubles when the side length becomes the diagonal length.

Synthetic II: 1 - 20

Russelling-up Prescience

The following is a quote by Daniel Dennett in his introduction to a book by Bertrand Russell called The Conquest of Happiness.

“Bertrand Russell’s technical work in logic created the field mathematical logic (laying the foundations on which Alan Turing and others created the computer) and posed the central issues that have preoccupied analytic philosophers in universities ever since. His three-volume work with Alfred North Whitehead, Principia Mathematica, consists of hundreds of pages of formal proofs designed to establish all of mathematics on a firm logical foundation, and is one of the most unreadable great books ever written. But he was also a passionate antiwar activist in the public square, who went to prison during World War I for his pacifism, was deeply critical of Communism (since a meeting with Lenin in 1920, when he visited Russia to investigate the revolution), supported the war against Hitler, an evil greater than war itself, but later was an ardent antinuclear polemicist and outspoken critic of the Vietnam War. His appointment to a professorship at City College New York was reversed by the college administration, which declared him “morally unfit” because of his boldly expressed views on sexual morality.”

Here a few of my favourite Bertrand Russell quotes:

“Ethics is in origin the art of recommending to others the sacrifices required for co-operation with oneself.”

“Humankind has become so much one family that we cannot ensure our own prosperity except by ensuring that of everyone else. If you wish to be happy yourself, you must resign yourself to seeing others also happy.”

“Religion is based, I think, primarily and mainly upon fear.”

“Obscenity is whatever happens to shock some elderly and ignorant magistrate.”

The point I’d like to make, however, is from the second half of Dennett’s introduction

“The Conquest of Happiness is a fascinating time capsule, a mixture including perennial observations that speak as clearly today as they did to its initial readers, and antiquated issues and attitudes that by today’s standards are offensive when they are not amusing. …Perhaps the moral to draw from this confrontation is that we should probably expect our grandchildren to be as uncomfortable with some of our attitudes as we are with some of Russell’s.”

It’s a fascinating and a wonderful thing to see the way that even the most prescient of social thinkers will always pass on instances of what will become the embarrassingly out-dated. For anything written by the most morally and rationally progressive geniuses to now appear to our culture as politically backwards is testament to the direction that our society is moving in.

It also raises an interesting question as to which of our current views our grandchildren will find embarrassing.

Yet another quote by Russell:

“When you meet with opposition …endeavour to overcome it by argument and not by authority, for a victory dependent upon authority is unreal and illusory.

This, I think may be the key to determining where the cracks in our ideas will show. The history of science is littered with embarrassments caused by appeals to what is obvious, such as the centrality of the earth, and the superiority of the white-skinned male to name only two.  Challenging even the most obvious ideas is the greatest tool we have for illumination and sanitation. And hopefully even this will one day seem like the crack-pot ravings of an almighty bigot.  

Bossed in Translation

I haven't done a proper oil painting in quite a while, so recently I decided to dust off one of my half-started canvases, and paint a picture of Bill Murray from Lost in Translation.

I have a couple of posts where I break down the process for making a picture, so I thought this time I'd focus on the endgame, and give a basic overview of some of the cheating you can do with digital.

The best way to see the changes is probably to click on the first image below and then use the thumbnails to move to the next ones.

Hopefully it might help your paintings to not get... 

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Moving along.

When cropping it's best to go into Free Transform, then hold cntrl or command and click a corner to stretch the canvas into shape.

All I used for colour editing was Levels.

This was a process of checking the photo very carefully and pushing the face into place.

There are three tiers of the picture now with a lot of Gaussian Blur on the background and a little on the seats in the foreground.

The tiers are also useful for putting a bit of colour around the edges behind the foreground.

I started painting the shirt.

Now the hands get some paint.

The colour change between these two is due to exporting as a regular jpg, rather than a web jpg.  

I hate that it makes such a difference.

Hard to tell what's going on here.  Probably the owl.

The hair gets done here, and the ears.

More head stuff.

And we're done.  The smoother quality of the last image is just due to being higher resolution.

For relaxing times, make it Suntory Time.