Relevant examples from Escher's work:
- Smaller and Smaller. Also see the geometric scaffolding in Visions of Symmetry, pg. 252.
- Regelmatige vlakverdeling, Plate VI
- Square Limit. Also see the geometric scaffolding in Visions of Symmetry, pg. 315.
- Print Gallery
- Fish and Scales
- Sketch #101 (Division)
Definition: Fractals are objects that are self-similar under magnification.
Fractals show up in the work of artists as varied as M.C. Escher, Salvador Dali, Max Ernst, and Jackson Pollock. Fractals are also used in Computer Science. If an object is self-similar, then data storage becomes more efficient. All you would need to store is the basic shape, and how many times you use a magnification.
The Sierpinski Triangle is one of the best-known fractals. One method of creating fractals is through a process called iteration. Iterating an operation simply means that we perform the operation over and over again. To create the Sierpinski triangle we take a triangle, find the midpoints of all the sides, and use those to create four smaller triangles. We throw away the central triangle, and repeat the process.
Above you see 3 iterations of the division process. On the left we divided the triangle in four smaller triangles and threw away the middle one. The figure in the middle shows the second iteration of the operation. We took the remaining three triangles, and divided them into four triangles and threw away the middle one. The triangle on the right shows what we get if we repeat the process five times. The real fractal is what we get if we repeat the process infinitely often. This means that we can think about a fractal, but we can only draw stages of the fractal as it is being produced. We know that a line is one dimensional, and a triangle is a two dimensional object. We will see later that the dimension of the Sierpinski Triangle lies somewhere between one and two. This means that the dimension of this object is not a whole number! This is where the term fractal comes from: The dimension of a fractal is some whole number plus a fraction of one.
Koch Edge The middle third of a segment is replaced by an equilateral “bump” consisting of two new segments. Recursively repeating on these new segments (and the remaining two segment thirds of the initial segment) results in the Koch Edge.
The Koch Snowflake refers to the object you get if you apply the iteration to all three sides of a triangle:
This snowflake is the result of only 4 iterations, but clearly shows the fractal nature of the Koch Snowflake. If you think about the construction of the Koch edge, then you will realize that at every stage you are adding more line segment than you are removing. This means that the length of the object is steadily growing. Remember that a real fractal is the result of iterating the process infinitely often. This means that the fractal has infinite length. Also note that the area bounded by the snowflake remains finite. This means that we have a finite area bounded by a curve of infinite length. This may seem counter intuitive at first, but we clearly have an example here that shows that this is possible.