T-square (fractal)

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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, evolution in six steps.

Algorithmic description[edit]


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.

Golden squares with T-branching
Square branches, related by the 1/φ
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.[citation needed] 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 .

See also[edit]


  1. ^ a b 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[edit]

  • 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.