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