Fractals
From EscherMath
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
- Division
- Whirlpools
- Path of Life I
- Path of Life II
- Path of Life III
Contents |
[edit] Explorations
Begin learning about self-similarity and fractals with:
- Self-Similarity : A first look Sierpinski Triangle, Koch Snowflake etc.
- Self Similarity in Advertisement Exploration Self similarity in some logos. Short exploration which looks at "The Laughing Cow" (Cheese) and Droste Cacoa.
- Self-Similarity Exploration Escher and Dali's use of self-similarity
- Iteration Exploration Three iteration type problems.
- Escher Fractal Exploration Escher's fractals based on some prints in Visions of Symmetry
- Fractal Dimension Exploration Computing the dimension of a fractal.
- Golden Ratio Exploration
[edit] Self-Similarity
- Fractals
- A fractal is a figure or object that contains copies of itself at smaller scales.
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.
Common Fractals
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 4 iterations of the division process. First the triangle is divided into four smaller triangles and then the middle triangle is removed. In the next step, the remaining three triangles are each divided into four triangles, discarding the middle ones. The final image shows what we get if we repeat the process six 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.
- Self-Similarity : A first look The Sierpinski Triangle
- Self-Similarity Exploration Escher's Artwork
[edit] Fractals
A fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole" according to Benoît Mandelbrot. [1]. Another way to describe fractals is to note that they are irregular objects which show self-similarity or something close to it.
A famous example of a fractal is the Mandelbrot set (image on the left), named after the afore mentioned Benoit Mandelbrot. Upon magnification we can see several images of this set appear (image on the right).
Escher created several fractals as well. In Visions of Symmetry, page 91, there are copies of his notebooks. Escher named these sketches "Regular Division of the plane by similar figures of which size and content rhytmically diminish in size, receding towards the center." We can see some of the same patterns in Smaller and Smaller. In this print by Escher we see that the artist created a print where we see small copies of the print sitting inside the bigger picture. As we move to the center of the image we see smaller and smaller versions of the lizards. After 1955 Escher switched from experimenting with congruent shapes to experimenting with similar shapes. Escher at first created these images because he was trying to capture the concept of infinity. He was apparently not quite happy with the Smaller and Smaller print. We can see infinity in the cenetr of the print, but at the outside edge there would be room to add increasingly larger images of lizards.
In Fish and Scales we see black and white fish of different sizes. This print shows two metamorphoses. The scales of the large black fish develop into fish and we see the same occurring from left to the right in the lower half of the print.
[edit] Exercises
[edit] Related Sites
- Fractals class at Yale, by Michael Frame, Benoit Mandelbrot, and Nial Neger.
- Escher and the Droste Effect at Universiteit Leiden.
- Droste Effect Gallery by Jos Leys.
- Jackson's Fractals An article about the fractal nature of the painting by Jackson Pollock.
