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:
   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.

T-square.

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

Properties

The T-square fractal 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 .

See also

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

References

• Hamma, Alioscia; Lidar, Daniel A.; Severini, Simone (2010). "Entanglement and area law with a fractal boundary in topologically ordered phase". Phys. Rev. A. 82. doi:10.1103/PhysRevA.81.010102. 
• Ahmed, Emad S. (2012). "Dual-mode dual-band microstrip bandpass filter based on fourth iteration T-square fractal and shorting pin". Radioengineering. 21 (2): 617. 

Fractals
Characteristics	Fractal dimensions Assouad Box-counting Correlation Hausdorff Packing Topological Recursion Self-similarity
Iterated function system	Barnsley fern Cantor set Dragon curve Koch snowflake Menger sponge Sierpinski carpet Sierpinski triangle Space-filling curve T-square
Strange attractor	Multifractal system
L-system	Space-filling curve
Escape-time fractals	Burning Ship fractal Julia set Filled Lyapunov fractal Mandelbrot set Nova fractal
Random fractals	Brownian motion Brownian tree Diffusion-limited aggregation Fractal landscape Lévy flight Percolation theory Self-avoiding walk
People	Georg Cantor Felix Hausdorff Gaston Julia Helge von Koch Paul Lévy Aleksandr Lyapunov Benoit Mandelbrot Lewis Fry Richardson Wacław Sierpiński
Other	"How Long Is the Coast of Britain?" Coastline paradox List of fractals by Hausdorff dimension The Beauty of Fractals (1986 book)