The Cantor Square is a fractal cluster, in which we start with a square, divide it into 9 squares, and cut out all of them except for the corner ones:
By continuing this process infinitely, we will get the Cantor Square:
The reason why this fractal is called the Cantor Square is because it can be formed using the Cantor Set. We can put two Cantor Sets on neighboring sides of a square and extend lines from all points of the sets. If we then plot the line intersections, we will get the Cantor Square.
The Cantor Square is composed of 4 identical figures, the side of each of which is 1/3 of the entire figure. Using the similarity method, we can calculate the fractal dimension to be log 4 / log 3, which is approximately 1.26.
At every iteration, we leave 4/9 of the area of the fractal. After an infinite number of iterations, the area will become 0, since the fractal is a collection of points.
The Cantor Square proved to be useful in creating simple models of galaxies.