T-square (fractal)


In mathematics, the T-square is a two-dimensional fractal. It has a boundary of infinite length bounding a finite area. Its name comes from the drawing instrument known as a T-square.

Algorithmic description

It can be generated from using this algorithm:
  1. Image 1:
  2. # Start with a square.
  3. Image 2:
  4. # At each convex corner of the previous image, place another square, centered at that corner, with half the side length of the square from the previous image.
  5. # Take the union of the previous image with the collection of smaller squares placed in this way.
  6. Images 3–6:
  7. # Repeat step 2.
The method of creation is rather similar to the ones used to create a Koch snowflake or a Sierpinski triangle, "both based on recursively drawing equilateral triangles and the Sierpinski carpet."

Properties

The T-square fractal has a fractal dimension of ln/ln = 2. The black surface extent is almost everywhere in the bigger square, for once a point has been darkened, it remains black for every other iteration; however some points remain white.
The fractal dimension of the boundary equals.
Using mathematical induction one can prove that for each n ≥ 2 the number of new squares that are added at stage n equals.

The T-Square and the chaos game

The T-square fractal can also be generated by an adaptation of the chaos game, in which a point jumps repeatedly half-way towards the randomly chosen vertices of a square. The T-square appears when the jumping point is unable to target the vertex directly opposite the vertex previously chosen. That is, if the current vertex is v and the previous vertex was v, then v ≠ v + vinc, where vinc = 2 and modular arithmetic means that 3 + 2 = 1, 4 + 2 = 2:
File:V4 ban1 inc2.gif|thumb|none|200px|Randomly chosen v ≠ v + 2
If vinc is given different values, allomorphs of the T-square appear that are computationally equivalent to the T-square but very different in appearance: