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 :)