12 is a magic number

The statement 12 is a magic number could be expressed as; how many sides to a cylinder? Or how I do minimize the amount of geometry to use in a game and still get great results. This is a little tutorial that I decided to write to explain why 12 is a magic number.

First let us understand something, so we are all on the same page, have you ever wondered why there are Twelve inches in a foot? Twelve items make a dozen. There are twelve months in a year? How many function keys on your keyboard? Twelve. Did you know there are twelve cranial nerves. Twelve numbers on a clock. There are twelve zodiac signs. Twelve apostles. Twelve furlongs in a mile. For the mathematically inclined; twelve is a composite number, the smallest number with exactly six divisors.

What is it about 12 that it so often appears? So what has this got to do with animation? Nothing, it has got to do with modeling.

You are making a cylinder (or anything else) and the 3d application's dialog asks you how many sides for this cylinder? What do you use? 8, 10, 16? 12 is the answer and here is why. Let us assume that you firstly want a smooth cylinder, not a faceted one. So maybe you'll smooth it, or probably subdivide. Well that is why 12 is the best number to have, it makes it smoothest geometry, with the lowest amount of sides.

The reason for that is to subdivide an object, it (the software) has to create a curve from one point to the next. Now if your creating a curve from one point to the next, it is always best if you have two other points to join to, so you get 3 points. As your curve will be a 3rd degree curve, e.g. a Bezier curve, and thus it will be smooth, if the points are equidistant, and have the same tangency.

Now if you are creating a cylinder and want it to be round, it would be best if the points were equally spaced. Again let us repeat, "The points to have the same tangency at every point."

Why? Because this means that the rate of change of the curve, the tangent at any point along that curve, can be expressed by a first order differential equation, the solution of which is linear, and thus the solution of the second order differential becomes a constant. This reasoning is also used in road building, or race track building, and it is what great grand prix drivers know, it's all hyperbolic's (including that Ashtekar formulation stuff, that's a math joke, Gerry). Also the 1st differential can be represented as velocity, then 2nd differential is acceleration. So when you understand these tangency and curve equations and the relation to speed, well, you can go around corners very fast.

Back to modelling, to aid subdivision, or smoothing, we need to make these points lay on a perfectly uniform curve.

Ok, so how do we do all of this? How do we have equally spaced points, one before and one after the point in question, all the points laying on the curve, and have them equidistant? Click on Magic 12 to start the easy tutorial.