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]

T-Square fractal (evolution)

T-square, evolution in six steps.

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 fractal (8 iterations)

Golden Square fractal with T-branching 8

Golden squares with T-branching

Golden Square fractal 6

Square branches, related by the 1/φ

Half square fractal 5

Squares branches, related by 1/2

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]


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 \( \textstyle {{\frac {\log {3}}{\log {2}}}=1.5849...} \).

Using mathematical induction one can prove that for each n ≥ 2 the number of new squares that are added at stage n equals \( {\displaystyle 4*3^{(n-1)}} \).
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:

V4 ban1 inc2

Randomly chosen v[i] ≠ v[i-1] + 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:

V4 ban1

Randomly chosen v[i] ≠ v[i-1] + 0

V4 ban1 inc1

Randomly chosen v[i] ≠ v[i-1] + 1
See also

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


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.

Undergraduate Texts in Mathematics

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Graduate Studies in Mathematics

Mathematics Encyclopedia



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