A realisation... also Euclid is a boss

So I now realise that the likelihood of me posting daily is quite low. Moreover, if I set that as my bar, I will probably see trying as too hard to bother. So where does this leave my blog?

I think the blog will change a little from now. I will post once a week, on friday nights (the night is still to be decided upon, but for now fridays are good).

The content and style won't change. However, I will most likely make a second maths blog. In this blog I will be testing out ideas for how to structure a course for Maths Methods 3/4 (the mid-range year 12 maths subject in Victoria), because I intend to put together a short electronic textbook for the subject, to address the woeful inadequacy of the textbooks currently on the market. Potentially more on that later (certainly I'll post a link to that blog once it's underway), but if anyone's interested or wants to help out or anything, please let me know :)

Now that that's out of the way, a quick thing on primes.

Lets start by defining some stuff. A number F is a factor of N if when it is divided N, the result is a whole, positive number (also known as a natural number). We say, then, that N is divisible by N. So for example, 6 is divisible by 1,2,3 and 6, and 5 is divisible by 1 and 5, but not 3. Obviously F can't be greater than N, because then we would get fractions (and they aren't whole numbers, which breaks the definition of a factor)... for example, 6/12=1/2 (in this case N=6 and F=12... the half is just a half. Get over it).

A prime number is a number which has no factors other than 1 and itself. So 2, 3 and 5 are prime, but 4 and 6 are not (because 2x2=4, and 3x2=6).

What I'm going to do is follow in the steps of Euclid (the Greek father of geometry) and prove that there is an infinite number of primes. This isn't necessarily obvious - why SHOULD there be an infinite number? Why can't all the potential gaps eventually be covered up? Well, here's how the explanation goes.

What I'm going to do is this: I will assume we have a finite number of ALL of the primes, and prove that I can find another prime that's not on the list. Because my technique will work for any finite list of primes, then I will have proven that no matter what you do, your list won't cover all of the primes, so there's an infinite number of primes.

But I'm getting ahead of myself. Let's start by assuming that we have a list of all of the primes: 2,3,5,7,11... all the way up to some last number P, which is our 'last prime'. What I'm going to do is multply them together.

So we have some really big number - 2x3x5x7x11x..xP. Let's call it Bob. So Bob is all of the numbers, all the way up to P, all multiplied together. Let's ask a question: what are the factors of this number? Well, we have 2,3,5,...,P... all of the prime numbers are its factors. So clearly Bob is not prime, because it has a bunch of factors other than 1 and Bob.

But what if I add 1? Then, we get Bob+1. Is 2 a factor? No, because Bob is divisible by 2, so Bob plus 1 can't be (imagine 20 and 21 or whatever... even +1 is odd). Is 3 a factor? No, because Bob is divisible by 3, so Bob +1 isn't (again, imagine 18 and 19). In fact, none of the factors we had before are factors. What does this mean?

Well I'm so glad you asked. What it means is that this new number only has 2 factors - 1, and itself (Bob +1), which means it's a new prime, one that wasn't on the list! I know there can't be any others, because if there were, they would have been on the list in the first place... after all, I did assume it covered ALL of the primes. So no matter which list of primes you give me, I can simply multiply all the terms together and add 1 and hey presto! I have a new prime!

So this means that no matter which list of primes you give me, I can find another prime. Imagine this were a game - you give me a list, I find a new one. We could play that game literally forever - you thinking up an ever growing list of numbers, me always finding one not on your list. It's not your fault that you're losing though - you're losing because the game is rigged! So, since we could play the game forever, this means there's an infinite number of primes to be found! Which is what we wanted in the first place. Yay!

This is widely regarded as one of the simplest, beautiful proofs in mathematics. Certainly, G.H Hardy thought so. It shows a simple truth that is otherwise not necessarily so obvious, but when you look at it more closely, you eventually realise (through a simple explanation) that it has to be true! There is an infinite number of primes!

I love playing with buckyballs and figuring out how many I'm holding by using prime factors - I'll explain that one some other time, but suffice to say that buckyballs (or any other equivalent version of the same thing) are pretty cool to play with, in both a 'normal' way, and of course the much more fun mathsy/physicsy way. Get some and muck around!

So I think we have come to several conclusions today.

1) I will be blogging regularly again! Yay!
2) There are an infinite number of primes
3) I like Buckyballs (not this kind, which is also pretty cool... I mean the toys)
4) Euclid is a boss

More to come on Friday (potentially on primes, but maybe not... we'll see). Bye for now :)

Damn camera

I don't like my computer much. Apparently it doesn't like my camera :/ anyway I had a post lined up about toroidal polyhedra (remember, donuts with flat shapes making the sides) but noooo it didn't happen thanks to silly computer. Anyway that'll happen tomorrow if i can coerce this damned machine into working.

On a separate note, I have a little bit of fun with cubes and fruit. The idea is not Vi Hart's, but she does a good job explaining it in her own blog. While her pictures are pretty and nice, it's much more fun to do this yourself.

So what is this? Well basically, you get an apple, and cut it into a cube. That bit's pretty easy. Then, you need to cut it in half, but not just any way. You need to cut it THIS way (see right).

One nice way of making it easy for yourself is to mark halfway down each of the sides which the slice goes through, then line it up and go for it! My picture kinda gives away what the cross section will look like, but it's still a lot of fun to do.

Vi Hart recommends slicing one corner off and then slowly slicing bits off that corner, and that way you eventually find the hexagon in the middle. Check out her working of this, it's probably more entertaining than mine thanks to a certain computer which is prejudiced against perfectly nice cameras.

Anyway one of the best parts about it is that eventually you get to eat your sliced cube. Given that the last post and most of the coming ones are all on platonic solids and their awesomeness, this fits nicely because it shows you a side of the cube (a platonic solid) you never thought was there. I look forward to posting more about tasty fruit when i start talking about conic sections (soon... ) because fruit is tasty and i like tasty things. Also maths.

Adios :)

I'm back!

I've been putting off getting back into this for no good reason for too long, and now is as good a time as any to start again. I have a bunch of really cool stuff to blog about in the coming weeks, and luckily for my armies of loyal followers I have a ton of time on my hands right up until new years and then again until late February. I plan to blog at least once a day.

Just to give you guys a taste of what's to come: I'm going to deal with the shape lights make on walls, snakes (a la Vi Hart), fractals and lots and lots of stuff on those shapes people make with those things in the image on the right. Actually, I plan on buying a huge box of those things (called Geoshapes, it seems) to play with... I'll make platonic solids, Archimedean solids, Johnson solids and (probably most excitingly) make some hyperbolic planes with them. It will be awesome.

Anyway I had a request a few hours ago from an entire half of my followers to do something on the platonic solids. There will be more to come on these, but I thought I'd do a teaser on it as an intro.

Ok, so for starters, what are the platonic solids? They are a set of 3d objects which follow a set of rules about how they're made. The rules are as follows:

1) Each side must be a flat, regular shape (this means only equilateral triangles, squares, regular pentagons etc.)

2) Only one shape can be used for sides on any one solid (so you can't, say, make a square based pyramid, because then you're using squares and triangles)

3) Each vertex (corner) must be identical (This rules out putting 2 triangle-based pyramids together like on the right, because some vertices have 3 triangles and some have 4)

4) The shapes must be convex. Wikipedia's telling me that convex means 'curving out or bulging outward, as opposed to convex'. This means the shapes can't have any 'hollowed in' bits. It also means that they can't have holes in them like toroidal polyhedra (the technical term is '3d shapes with flat sides and holes in them', but i like to avoid jargon... basically if it looks like a donut or 3 mushed together, you're doing it wrong).

That's it! Now with these simple rules (only flat, regular shapes for sides, one side-shape per solid, identical vertices and it must be convex), how many can shapes can we make? More importantly, what are the shapes?

You can see why those Geoshapes thingies would be nice to have around at this stage. Personally, I love hands-on maths. It only really comes alive when you have the shapes in your hands, or when you've got some huge whiteboard or A2 piece of paper to work with or something.

I'll start by giving you a simple shape that fits the rule: the cube. Each side is a square, each vertex has 3 squares touching it and it's nice and convex. Good work Jake.

I'm going to give you the answer to the first question, and leave the second one for a few days for you to ponder. There are only 5 platonic solids - think about why that might be. There are an infinite number of side possibilities (like an octagon, a 13-gon or a 256-gon), but only 5 will fit together nicely to make a platonic solid.

Anyway, find some shape-toys like the ones I've referred to and put them together. See if you can find all 5 shapes! Soon, I'll do some more stuff on the platonic solids, showing why these shapes are so special and are seen by many mathematicians to be the most elegant, simple and beautiful shapes in existence.

Good luck :)

I'll be back

Well this blog has been fun. I need to spend less time on the internet so that I can spend some decent time studying. Point is that this is my last blog post until sometime in the middle of November.

But until then, my loyal followers can feast their eyes on a brilliant little infinite series (ie adding an infinite number of numbers together).

The question was on an early high school maths competition paper that my friend did. He only figured out how to do it years later, but the idea behind it is pretty simple. You get shown an image, and the question is, if the red squares keep reducing in size to infinity, what proportion of the total square do the red parts make up?

The first thing to notice is that an infinite number of things added together don't necessarily sum to infinity. The total red part is obviously less than the entire square. So how can we find the answer?

The 'proper' (by which i really mean more rigorous) way to do it is to use the geometric series, which is fairly simple. However, in order to use it, I'd want to show how it works first. I don't have the time, so I'll show you something a little more intuitive.

Let's divide this into an infinite number of chunks. Each chunk is surrounded by green. The largest chunk is on the right.



So how does this help? The idea with this is that we're turning the hard problem into an easy one we can do. That problem is this... What proportion of the area of the chunk on the right is red? The answer is obviously 1/3. What about the next chunk? Well it's exactly the same as the previous chunk, only smaller. So it's 1/3 of that chunk.

This means that each separate chunk is 1/3 red. If 1/3 of every chunk is red, that means that 1/3 of all of the chunks combined is red. What this means is that the red area makes up 1/3 of the area of the entire square.

Problem solved! Well, once exams are done I'll do something on the geometric series (which is awesome), but until then, this will have to do.

Adios, amigos :) (until November 18, anyway :P)

Maths pun

The other day I came across OCTOPUNS! For those who like puns, they're brilliant. Anyway, one of their puns is maths related, so I thought I'd post it here.

If I find any more maths-related humour, I'll post it. It certainly makes my life just that little bit easier... :)

Pythagoras... proved in an awesome way

Yeah, yeah, for both of the people who follow this blog, here is something pretty cool. Most people who do maths up to year 8 have seen pythagoras' theorem, but they have no idea how we know it works. Well here's one way of proving that it works. You can actually do this with paper, but because im so kind im going to do it on paint for you.

Ok so start with a piece of paper. Fold it into 4 so that you can cut a triangle off one of the corners. What happens is that you get 4 equally sized right angles triangles. Now arrange them into a square as shown below.

What you get is a neat little square inside a larger one. If we label one of the outside edges of the identical green triangles 'a' and one 'b', we can see that we get a big square of side length a+b. Also, if we label the hypotenuse (the longest side of the triangle) c, we can see that the area of the blue square in the middle is c²

Now, draw a square around the big square with a pen. What this proof relies on is that, no matter how much you move the green triangles around, the area inside the big square you drew with a pen will remain the same. If you're having trouble visualising it, get out the pen and paper and do it yourself, it really helps.

Ok so no matter how much you move the green triangles, the area inside the big square remains the same. Also, the blue area will remain the same (no matter how much you move the green triangles, the remaining area will remain constant). This means that i can rearrange the triangles. One way to do so is shown above.

Remember that before, the blue area was c². Now, the area is made up by 2 triangles of side length a and b, so the blue area is a²+b². Since the area of the blue parts remains constant, this means that the old area=the new area, or, written more nicely in a larger font and centred for your reading pleasure...

a²+b²=c²

Well, when i was shown this, I thought it was pretty cool. I mean, finally, a simple non-rigorous proof of something in maths! Of course, there's much more there, but at this stage im only just scratching the surface of it. I'll post some more when I have some more :)

Life in 4d

My guess is that anyone who reads this has at some stage heard some nerdy person talking about stuff in 4d, and not been able to understand. Honestly, the ideas aren't that hard. The trick when dealing with these sorts of problems is to phrase them in terms that people understand.

So lets start simply. What does 0 dimensions look like? A 0 dimensional shape has no length, no width, nothing. It's just a point. My incredible paint skills allow you to see one of these for your self. Aren't I kind?

Aren't I kind. I guess the question to ask is this... Now that I've got 0 dimensions, how can I make 1? One way to do it is to make 2 0-dimensional points, and connect them.

And now we have a 1 dimensional thingamajiggy. I'm going to call it a 'line'. How do we make a 2 dimensional shape? Connect 2 lines, of course.

The red lines can be the one that are connected to the pre-existing blue ones.

I hope you can all see the pattern. More than that, I expect many of you have drawn cubes in this way... if not, you should try it. It's pretty cool.

So again, we've started with 2 2-dimensional shapes,joined them at the corners, and we get a 3d shape. Now that the pattern is firmly established, let's try using it to gain insight into what a 4d shape could look like

At this point I should probably explain why this is an insight, and not a full understanding. We can visualise shapes and bodies in dimensions up to 3 dimensions, but our world is not 4d. 1 way to see this is to imagine what a sphere would look like in a 2d world. The 2d version of a sphere is, predictably, a circle. When a sphere travels through a 2d plane, you can't really understand it as a sphere, rather as a circle whos radius increases and decreases.


This is my attempt at drawing a sphere passing downwards through a horizontal, flat plane. If you can't see the image in your head, get an orange and cut it anywhere. You'll always get a circle in the flat part. Now imagine cutting circles in it every 5cm upwards... you'll get bigger and smalle circles every time, depending how close to the middle you cut.

So imagine you lived in that 2d world. You're able to see the 2d equivalent of the 3d shape, but you'll never really understand what the sphere looks like because you can't imagine the extra dimension necessary. That's the basic issue we face. That said, we can still move our 4d cube (called a 4d hypercube) around in 3d (well, 2d actually. It's a computer screen, not a hologram... yet), to see what it looks like.

Here are 2 different rotations of the 4d hypercube, in 2d.


AAAH! These may look complex but when you look at it in terms of what you know, this becomes easy. I'll start with the one on the right. Can you see the cubes on the inside and outside? So it's all the same as the other ones. The one on the left looks more all over the place, but it's not so hard to visualise either. If you pick, say, 8 of the closer dots, you can probably make a slanted cube. Choose another few dots! There are actually 8 cubes to be made.

Sadly, this link is where my ability to draw ends, and my linking to other peoples' work begins. Imagine this as a pair of cubes, looking around eachother.

That's all i've got to say on this post. I'll leave you with a question. How many vertices (corners) does a 4d hypercube have? edges (lines)? faces? 3d chunks?! Look for a pattern that occurs when increasing your number of dimensions, and the problem becomes easy.

:)