# 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.[1]

## 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, "both based on recursively drawing equilateral triangles and the Sierpinski carpet."[1]

## 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 .

Using mathematical induction one can prove that for each n ≥ 2 the number of new squares that are added at stage n equals .

## The T-Square and the chaos game

The T-square fractal can also be generated by an adaptation of the chaos game, in which a point jumps repeatedly half-way towards the randomly chosen vertices of a square. The T-square appears when the jumping point is unable to target the vertex directly opposite the vertex previously chosen. That is, if the current vertex is *v*[i] and the previous vertex was *v*[i-1], then *v*[i] ≠ *v*[i-1] + *vinc*, where *vinc* = 2 and modular arithmetic means that 3 + 2 = 1, 4 + 2 = 2:

If *vinc* is given different values, allomorphs of the T-square appear that are computationally equivalent to the T-square but very different in appearance:

## See also

- List of fractals by Hausdorff dimension
- The Toothpick sequence generates a similar pattern
- H tree

## References

- Dale, Nell; Joyce, Daniel T.; and Weems, Chip (2016).
*Object-Oriented Data Structures Using Java*, p.187. Jones & Bartlett Learning. ISBN 9781284125818. "Our resulting image is a fractal called a T-square because with it we can see shapes that remind us of the technical drawing instrument of the same name."

## Further reading

- 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.