Sierpinski’s Triangle PT 1
A bit like the Random Walk Algorithm, but it embellishes it a bit.
Giles McCullen-Klein
Introduction
I first became aware of the Sierpinski’s Triangle while taking Giles McCullen-Klein’s excellent ‘ Python Programmer Bootcamp ‘ on the 365 Data Science platform. The Sierpinski Triangle is named after Polish mathematician Waclaw Sierpinski, who popularized the concept in the early 20th century. Giles used the Triangle in his section on MatPlotlib to demonstrate the power of visualizations. He started with these formulas:
then explained how the formulas are chose at random to create the Triangle then followed with the formula’s equivalent in Python code (outlined in Part 2 of this post, where we build the Sierpinski’s Triangle using Python). He tried input 10 points, then 100, then 1000 where the first triangle began to take shape, albeit in outline. Finally, at a million points, we had it: The Sierpinski Triangle!
I was captivated by the appearance of the full Sierpinski’s Triangle, partly because of the ability of three simple mathematical formulas translated into Python to create such a compelling and beautiful image, but also by the design itself, with triangles inside triangles inside triangles on into infinity. I became even more intrigued when I discovered that the Sierpinski Triangle is based on fractals.
Fractals are geometric figures characterized by self-similarity: as you zoom in or out of a fractal, the pattern remains essentially the same, echoing itself over and over. What fascinated me about Sierpinski’s Triangle is that while it’s a fractal that exists in two dimensions, its fractal dimension — a measure of its complexity — is not an integer. Apparently, I am not alone: this quirk of fractal dimensions has perplexed and fascinated mathematicians since it was introduced. In the field of computer graphics, the Triangles’ algorithmic simplicity has lent itself to the generation of textures and patterns. The structure of the Triangle has relevance in fields as diverse as error correcting codes, network design, the study of dynamical systems, and the study of cellular automata.
Fractals were also found to be present in Jackson Pollock’s so-called ‘drip’ paintings of the mid-1950s. Though the concept of fractals didn’t exist until the 1970s, Pollock was discovered to have used them in his paintings, following the patterns of nature. I became aware of this when reading a plaque below his magnificent Summertime: Number 9. I can’t find the contents online, but I did find this from the University of Oregon:
In 1999, Richard Taylor and his research team published the results of their scientific analysis showing Pollock’s poured patterns to be fractal. Consisting of patterns that recur at increasingly fine magnifications, fractals are the basic building blocks of nature’s scenery. Labelled as “Fractal Expressionism,” Pollock distilled the essence of natural scenery and expressed it on his canvases with an unmatched directness. By adopting nature’s pattern generation processes, the resulting paintings didn’t mimic nature but instead stood as examples of nature. The above images compare Pollock’s fractals to those found in nature. Remarkably, the analysis revealed a highly systematic fractal painting process perfected by Pollock over a decade.
Richard Taylor, University of Oregon blog: Fractal Analysis of Jackson Pollock’s Poured Paintings
What is Sierpinski’s Triangle?
The Sierpinski’s Triangle, also known as the Sierpinski gasket, is a fractal named after the Polish mathematician Waclaw Sierpinski who described it in detail in 1915, though it’s worth noting that similar patterns were described by Italian mathematicians a few centuries earlier.
Walter Franciszek Sierpinski was born on March 14th, 1882 in Warsaw Poland. Through his long and prolific career, he made significant contributions in the field of set theory, number theory, theory of functions and topology, although he is best known for his work on set theory and the theory of numbers. Along with the triangle, he is credited with various other mathematical concepts, including Sierpinski’s conjecture, Sierpinski arrowhead curve, Sierpinski carpet, and Sierpinski constant. His work had a vital impact on the field of mathematics, and his concepts continue to be studied today. Over the course of his career, he published over 700 papers and 50 books, before passing away on October 21, 1969, in his hometown of Warsaw. That he accomplished all this while surviving through Nazi then Soviet occupation, and the leveling of Warsaw by the retreating Nazi army, makes his accomplishments all the more remarkable.
The Sierpinski Triangle is a fractal that is easy to construct and provides a clear example of self-similarity, a key property of fractals. Here’s a step-by-step guide to constructing a Sierpinski Triangle:
- Start with an equilateral triangle. This will be the base of your fractal. You can draw it on a piece of paper, or create it digitally.
- Divide the triangle into four smaller equilateral triangles. You can do this by connecting the midpoints of each side of the original triangle. This will create one triangle in the center and three triangles at the corners.
- Remove the middle triangle. This leaves you with three equilateral triangles. The shape now looks like a larger triangle made up of three smaller triangles.
- Repeat the process for each of the remaining smaller triangles. For each of the three smaller triangles, divide it into four even smaller triangles and remove the one in the center. This leaves you with nine small triangles.
- Continue this process indefinitely. Each time, you divide each remaining triangle into four smaller triangles and remove the one in the center. As you do this, the Sierpinski Triangle begins to take shape.
The Sierpinski Triangle is an example of a fractal because it is self-similar at all scales. If you zoom in on any part of the triangle, it looks the same as the whole triangle. This property is characteristic of fractals and is one of the things that makes them so interesting to mathematicians and scientists.
One of the more curious things about Sierpinski’s Triangle is that if you zoom in on the triangle, you will see the same pattern repeating over and over again, indefinitely. Despite this, the Sierpinski Triangle fits within an infinites space. Possibly, this is due to the nature of its construction: as we repeatedly remove the middle triangle from each smaller triangle in the figure, the total area of the triangles that make up the Sierpinski Triangle decreases, even as the number of triangles increases.
If we were to continue this process indefinitely, the total area of the Sierpinski Triangle would approach zero, even as it contains an infinite number of triangles. This is how the Sierpinski Triangle, and fractals generally, can exhibit complexity while remaining within an infinite space.
The mathematical principle behind the Sierpinski Triangle involves concepts from fractal geometry, particularly the idea of fractal dimension.
The Sierpinski Triangle is a fractal with a Hausdorff-Besicovitch dimension, also known as fractal dimension, of log(3)/log(2), which is approximately 1.585. This value indicates that the Sierpinski Triangle is more complex than a one-dimensional line (which has a fractal dimension of 1) but less complex than a two-dimensional shape (which has a fractal dimension of 2).
The area of the Sierpinski Triangle decreases with each iteration because with each step, we remove triangles, thereby reducing the total area. If we start with a triangle of area 1, after the first iteration, we remove the middle triangle, leaving 3/4 of the area. After the second iteration, we remove additional triangles, leaving (3/4)² of the area, and so on. So, with each iteration, the area of the Sierpinski Triangle is (3/4) to the power of the number of iterations, which approaches zero as the number of iterations goes to infinity.
On the other hand, the perimeter of the Sierpinski Triangle increases with each iteration. With each step, we add more edges, thereby increasing the total length of the boundary. If we start with a triangle of side length 1, after the first iteration, each side is divided into two segments of length 1/2, so the total perimeter is 3 21/2 = 3. After the second iteration, each side is divided into four segments of length 1/4, so the total perimeter is 3 4 1/4 = 3, and so on. So, with each iteration, the perimeter of the Sierpinski Triangle remains constant, but because there are an infinite number of iterations, the total perimeter is infinite.
This paradoxical situation, where the area approaches zero while the perimeter goes to infinity, is one of the fascinating properties of fractals like the Sierpinski Triangle.
Sierpinski’s Triangle has many parallels in art and nature.
In nature, certain types of ferns exhibit a fractal pattern similar to the Sierpinski triangle. The leaves of the fern are self-similar, with each leaflet being a smaller copy of the whole leaf. This self-similarity is a key characteristic of fractals.
In terms of culture, the triadic structure found in the triskelion symbol, which is common in Celtic and Greek art, is reminiscent of the Sierpinski triangle. The triskelion consists of three interlocked spirals or three bent human legs, and its recursive, triadic structure is similar to the recursive, triadic structure of the Sierpinski triangle.
In the 1990s, physicist Richard Taylor used computer analysis to study Pollock’s paintings and found that they contain fractal patterns. According to Taylor, Pollock’s drip paintings achieved a level of complexity in their fractal dimensions that is similar to those found in natural landscapes.
While Pollock’s paintings may not contain specific fractal shapes like the Sierpinski triangle, the overall fractal nature of his work does draw a parallel with the properties of the Sierpinski triangle and other fractals. Both Pollock’s paintings and the Sierpinski triangle demonstrate how simple rules and processes can generate complex and infinitely detailed patterns.
It’s important to note that while this analysis provides a fascinating intersection of art and mathematics, Pollock himself likely did not consciously incorporate mathematical fractals into his work. Rather, his intuitive process and the physical properties of the paint and canvas resulted in patterns that have fractal-like properties.
Pollock’s work has been analyzed from various scientific perspectives, and parallels between his paintings and fractal geometry have been suggested. Fractal geometry, which includes figures like the Sierpinski Triangle, is a branch of mathematics that deals with complex patterns that are self-similar across different scales.
Art critics and scientists have suggested that Pollock’s drip paintings exhibit fractal properties. Richard Taylor and his team initially proposed their theory that Jackson Pollock’s paintings may contain fractal patterns in the late 1990s. Their research, published in the scientific journal “Nature” in 1999, argued that Pollock’s drip paintings exhibited fractal properties, and that Pollock seemed to intuitively grasp these complex mathematical principles, even if he did not consciously understand them as fractals.
This intersection of art and mathematics demonstrates how concepts from one field can resonate in an entirely different one. However, this interpretation of Pollock’s work remains somewhat controversial, with some critics suggesting that Pollock likely wasn’t consciously employing mathematical principles in his work. Regardless of the controversy, the suggested link between Pollock’s paintings and fractal geometry, like the Sierpinski Triangle, offers an intriguing perspective on his unique artistic process and its outcomes.
This research has led to an ongoing and intriguing dialogue between art and science, and while not everyone in the scientific or artistic community agrees with Taylor’s conclusions, it certainly opened up new ways of analyzing and understanding abstract expressionist works.
Part 2: We make Sierpinski’s Triangle
More on fractals in Jackson Pollock’s work: https://www.arthurimiller.com/colliding-worlds-blog/how-fractals-science-and-technology-helped-to-resolve-a-major-art-scandal/
The facts about Pollock’s fractals: https://blogs.uoregon.edu/richardtaylor/2017/01/04/the-facts-about-pollocks-fractals/
Originally published at https://tim-beckett.com on July 29, 2023.
