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. (The black square in the image)
2. Image 2:
   1. 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.
   2. Take the union of the previous image with the collection of smaller squares placed in this way.
3. Images 3–6:
   1. Repeat step 2.

File:T-Square fractal (8 iterations).png

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. 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