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.

:)

Reuleaux's triangle... and then some

So what makes a circle so important? Firstly, it is the shape with the highest possible area to volume ratio... in other words, it has the highest area for the least volume. This is one thing that makes circles unique. What I'm going to talk about is something else, that the circle has constant width. That means that there is a set maximum distance between any 2 points, and you can make that distance given any starting point. At first glance, this also seems like something which is unique among shapes since the circle has rotational symmetry. The green lines are other lines which have a total length of less than the red line. The red line represents the maximum width.

The awesome thing is that there are other shapes with constant width, called Reuleaux polygons. These were made famous by Franz Reuleaux, a German engineer. The shape below, the Reuleaux triangle, is the most famous example.

The shape in between the circles is the shape we're looking for. If you pick any spot on the triangle's perimeter, it has a constant width as discussed before.

 

The trick behind the shape is that, as illustrated in the second picture, each corner is actually the centre of a circle, and each side a part of a circle's perimeter. This means that all of the orange lines are the same length, since they are each a radius of one circle. Imagine an orange line sweeping across from one corner to another, and you should be able to visualise it. The pink lines show shorter lengths that you can find. The other side to this is that if you pick any point which isn't a corner, the longest distance across is always between that point and the opposite corner.

This is pretty cool. While there are an infinite number of other shapes which do it (by performing the same trick of rotating a radius of a circle around corners), this is the most famous. The question now is, what's the use of this? Well for starters, manholes need to have constant width so that the lid can't fall down the manhole. Reuleaux triangles can do this. Trust San Francisco to have a Reuleaux triangle manholes :D (above).

The last thing I'll show you is a 3d equivalent of a Reuleaux triangle. A Reuleaux triangle-shaped manhole would roll smoothly, like a circle, due to its constant width. If you get a reuleaux triangle and rotate it around a central axis (as below), you get the shape below. The red lines show the axis of rotation.

If you want to see these shapes in action (and i recommend it highly, it's pretty cool) check out this youtube video. I've had a play with 3d printed versions before and they're mindblowing. Also, check out the other videos this guy's made, they really mess with what you think you know about shapes.

I guess the point of all this stuff is that there's more to geometry than meets the eye. There's certainly more behind these shapes than I could possibly write about in one post (look at the size of this monster already). I expect my house will be full of mathematical curiosities like this by my thirties... certainly more interesting to have on your coffee table than a bunch of fashion magasines and barely touched books about the antarctic.

The universe

I should probably mention at this point that the universe is pretty big. Put it this way. Relative to electrons, buzzing around in atoms, grains of salt are enormous. I mean, they're unfathomably huge. An electron is about 5.7*10^-15 metres across (so that's 0.0000000000000057 metres), while a grain of salt is about half a millimetre across. That means that the grain of salt is almost 88 billion times the size of an electron.

88 BILLION TIMES?!?!?! WHAT DOES THAT EVEN MEAN?!

Well, let me explain. It's so much bigger that even if the electron were aware of its existence, fully appreciating the intricate sodium chloride lattice, which a key to understanding the nature of salt crystals, would be nigh on impossible. From our point of view, a salt crystal is only 1/340th our length, and even with our (relatively) high level of intelligence, we took ages to come close to understanding it. But we don't fully understand electrons.

Well, at least most physicists would agree. My thoughts today rested on this issue... if by being 15 orders of magnitude bigger than something we found it hard to understand it... shouldn't be the same when we're dealing with things bigger thant us? The Milky Way galaxy (our galaxy) is around 10^18 times out size... that's 1000000000000000000 times our size. Once again, unfathomably big. What this means is that any insights which we manage to glean about the galaxy as a whole are incredible. Maybe one day we'll know whether or not there's a big black hole in the centre. Who knows? But it's a testament to the work of countless astronomers, theoretical physicists and mathematicians that we know anything at all, let alone that we know it to the amount of detail that we do.

Long live science! :)

Apocalypse?

So I stayed up til 2 last night watching this lecture. I seriously recommend taking the time (about 80 minutes) to watch it. It's ridiculously important AND it involves very basic mathematics. Just in case you'd prefer to spend your 80 minutes pruning your facebook account or doing a minimal amount of homework in a half-assed way, just take the time, maybe before bed or something, and watch it. It shook up my views on overpopulation, and I expect it will do the same to you.

If I get any decent amount of time sometime soon, or someone requests it or something, I'll summarise the important bits but for now... time to chill after a long day of study :)

First post

As a first post, my role is to be meaningless and unoriginal.

Seriously though, if you're interested in maths, you should have some fun following this thing. At this stage everything will be in layman's terms, with minimal mathematical jargon, so that people who aren't immersed in maths can still enjoy it. It won't be fully active until mid to late November, as until then I have exams to contend with, but once that's done, this will hopefully be an active blog.

Until then, I'll post something cool that my maths lecturer showed me.

What's the sum of all of the natural (read whole numbers greater than 0) from 1 to n? Up to, say, 10, we can do without too much effort but when adding to 256 (my favourite number) or 2034986 this becomes more difficult.

We were asked to find the sum of the numbers from 1 to 100. I paired numbers that added to give 100, and it's not too hard to see that there are 49 of these (1 and 99, 2 and 98... all the way to 49 and 51). 49*100 is 4900. Then you add the 50 and the 100 that you've skipped out on and voila, 5050 is spat out. This took me around 20 seconds, but Gauss managed this in similar time when he was 6. This is supposed to be something like what he did.

Lets say we want to add the numbers from 1 to 4. The answer is obviously 10. Let's stack them in a triangle.

A good estimation for the number of dots is the area of the triangle made by them. This has base length 4 and height 4. Using the formula A=1/2*base*height, we find that the estimated area is 8. It's close, but it's not quite right. Here's the cool part. What happens if you put another one of these triangles on top of the first one?

Turns out what you get is a rectangle of area 5*4=20. Remember that we want to find the area of half the rectangle, which is of course 20/2=10.

This can, of course, be generalised into something easy to use. Turns out that, if you let n=the highest number you add to, if you add up all the numbers from 1 to n, you can calculate this using the formula below.

Sum of all numbers from 1 to n =
Using this formula, we can fund out the sum of all the numbers up to 256.

256*257/2=32896

Well, that's the first of many I expect. There will probably be a few on infinite series to come, since we spent the last little while doing that at uni. Until then, salut :)