T-square (fractal)

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In mathematics, the T-square is a two-dimensional fractal. As all two-dimensional fractals, it has a boundary of infinite length bounding a finite area. Its name follows from that for a T-square.

[edit] Algorithmic description

It can be generated from using this algorithm:

1. Image 1:
   1. Start with a square.
   2. Subtract a square half the original length and width (one-quarter the area) from the center.
2. Image 2:
   1. Start with the previous image.
   2. Scale down a copy to one-half the original length and width.
   3. The previous image's square has four equal quadrants. From each of the quadrants, subtract the copy of the image.
3. Images 3-6:
   1. Repeat step 2.

The method of creation is rather similar to the ones used to create a Koch snowflake or a Sierpinski triangle.

[edit] Properties

T-square has a fractal dimension of log(4)/log(2) = 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 limit curve is a fractal line, of fractal dimension 2.

[edit] See also

• List of fractals by Hausdorff dimension
• Sierpinski carpet