T-square (fractal)

This article is about a two dimensional fractal in mathematics. For other uses, see T-square (disambiguation).

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.

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. From each of the quadrants of Image 1, subtract the copy of the image.
3. Images 3–6:
   1. Repeat step 2.

T-square.

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

Properties

T-square has a fractal dimension of ln(4)/ln(2) = 2.^{[citation needed]} 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 $\textstyle{\frac{\log{3}}{\log{2}}=1.5849...}$.

See also

• List of fractals by Hausdorff dimension
• Sierpinski carpet
• The Toothpick sequence generates a similar pattern