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