In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the T-square is a two-dimensional fractal

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

. As all two-dimensional fractals, it has a boundary of infinite length bounding a finite area. Its name follows from that for a T-square

T-square

A T-square is a technical drawing instrument used by draftsmen primarily as a guide for drawing horizontal lines on a drafting table. It may also guide a triangle to draw vertical or diagonal lines. Its name comes from the general shape of the instrument where the horizontal member of the T slides...

.

Algorithmic description

It can be generated from using this algorithm

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

:

1. Image 1:
   1. Start with a square.
   2. Subtract a square half the original length and width (one-quarter the area) from the center.
2. Image 2:
   1. Start with the previous image.
   2. Scale down a copy to one-half the original length and width.
   3. From each of the quadrants of Image 1, subtract the copy of the image.
3. Images 3–6:
   1. Repeat step 2.

The method of creation is rather similar to the ones used to create a Koch snowflake

Koch snowflake

The Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described...

 or a Sierpinski triangle

Sierpinski triangle

The Sierpinski triangle , also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set named after the Polish mathematician Wacław Sierpiński who described it in 1915. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral...

.

Properties

T-square has a fractal dimension

Fractal dimension

In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...

 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
  Sierpinski carpet
  The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions . Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional...