x(n) = x(n-1) + cos f
y(n) = y(n-1) + sin f
f(n) = f(n-1) + d
d(n) = d(n-1) + c

Strictly speaking, this probably only counts as a fractal (with detail appearing at an infinite range of length scales) when the input number (c above as a fraction of Pi) is irrational. An irrational number is one that can't be expressed as a fraction, and to write out its value exactly would take an infinite number of non-repeating digits. Pi, the ratio of the circumference of a circle to its diameter, is an irrational number; so is the square root of two, which is the ratio of the diagonal of a square to one of its sides (in fact, the square root of every prime number is irrational). The existence of irrational numbers was reputedly the deepest mystical secret of the Pythagorean order in ancient Greece.

The impossibility of writing out an irrational number in a finite number of digits, and the fact that only irrational numbers lead to a true fractal here, both have the same fundamental reason behind them: that every rational number is a fraction. This means that if you multiply a rational number by one of the right whole numbers you get another whole number; in the case of the Curlicue Fractal this leaves the Turtle pointing in exactly the same direction it started at, poised to repeat just the same set of steps all over again. In most cases, the Turtle will be in a different spot from where it started, leading ultimately to an infinitely long line of finite length; however, for some numbers it winds up precisely where it started so that the whole form is bounded in a finite area. Might any identifiable irrational numbers lead to similarly bounded forms? Would they all fill the plane if extended to infinity? I don't know.

There is much more about this, and a 3D version using OpenGL (written in C++) here.

(discuss this)

To a page about irrational numbers (which goes some way towards explaining the repeating-decimal/fraction identity)

To the code for this applet

Marijke van Gans's machine code implementation of more-or-less the same mathematics, which shows up its fractalishness better but in a more austere sort of a way.

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