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 :)