I'm sure that everyone is familiar with the works of Escher; you know the back to back geckos, the blokes walking up and down the square staircase that only goes up...or down. It's all good stuff.
Anyway, while sitting in an otherwise extremely boring meeting I started doodling the impossible triangle. At first I drew it picturing in my mind the corner perspectives - I'd drawn it several times before. I got a reasonably crude version done, and then I started to notice the rotational symmetry of the thing. It is essentially a spiral! Well, if you think about it long enough like I have (by the way, don't bother) it makes complete sense and what's more, it will work for any two dimensional closed polygon. Even a two sided one (ie a straight line) but that looks kinda silly...
Right, let me show you.
1. Draw an equilateral triangle: That means all the sides are the same length (well duh). No real easy way, just fudge it so it looks about right. 60 degrees per angle, as we all know, because the sum of all the angles in a triangle add to 180 degrees...
2. Initiate the spiral out: Draw some little "extensions" on your triangle, ie extend the line. Go clockwise or anticlockwise, it doesn't matter. How far you extend is how wide the triangle is.
3. Spiral out: Draw lines parallel to the original triangle from the tips of the extensions continuing in your clockwise or anticlockwise direction. Go beyond the extensions by the same amount you extended the first time (what the...? ;))
4. Keep spiralling: from the tips of where you just ended off. This time, don't go beyond the extensions, instead fall a bit short of them.
5. Close off the shape: The "closures" should be parallel to the opposite side of the triangle. Get it?
6. Shading: If I'm using a pencil that's easy, if I'm using a pen, colour one and stripey line another. You have to shade the thing, it accentuates the impossibility of it all. You're done, sit back and be amazed! Show your friends how clever you are!
Here in East Perth they actually built an impossible triangle. How you may ask? Well, when viewed from the right spot at a distance the sun shining creates the shading and everything lines up. Up close it is made of stainless steel box section. The bottom joins to one of the triangle sides coming forward and the other side going backward - see it isn't really a triangle, it's just an odd bit of bent metal. The sides still angle at 60° to each other; but the z-axis forward/backward gives the different sun reflections, and when viewed straight on you can't see the z-axis - but you can see the different shading effect giving the illusion of a real impossible traingle. Cool huh!