Fractals are amazing. Their beauty can be created by following a few simple rules that anyone can do. Many of the shapes in nature can be categorized as fractals. Defined, a fractal is a geometric shape that is created by subdividing a shape according to some iterative process. We see fractals in snowflakes, ferns, mountains, erosion patterns, and other places in nature. There are many types of fractals, including legal circles, dragon curve, Sierpinski carpet, and Sierpinski triangle to name a few. In previous posts, I described the Legal Circles Fractal and Dragon Fractal. In this post, I will discuss a few Sierpinski shapes.
Constructing the Sierpinski carpet fractal
The Sierpinski family of shapes are easy to construct. The first shape we’ll look at is the Sierpinski carpet. To begin, draw a square. It is easier if the sides are multiples of 3 (3, 9, 12, etc). Let’s choose 3 inch sides.
Next, divide the square into 9 equal parts, each 1 inch square. Your square should look like this:
Now, fill in all the squares except the middle (the one marked with a 1). That is the first iteration. Basically, each iteration is a solid square with the middle taken out. Here is the result after that first iteration:
For the next iteration, there are two ways to proceed:
- Copy the first iteration 8 times and build a square out of them, leaving the middle blank
- Remove the centers from the 8 remaining squares from the first iteration.
I chose to do #1 because in my experience, it was the simplest using the tool I was using, CorelDraw, a vector drawing program (you could also use Inkscape which is free) . The result after 2 iterations should look like this:
Now you are finished with the second iteration. Continue following these two rules:
- Copy the previous iteration 8 times
- Make a square out of the 8 copies, leaving the center blank
The size will quickly become overwhelming; if you started with a 3 inch square, after the second iteration, you would have a 9 inch square, then 27 inches, etc. To keep the size manageable, scale the copy you make in step #1 back down to 3 inches each time. Continue until you reach the complexity you want. Here is what it looks like after 5 iterations:
Congratulations, you have just created a Sierpinski carpet. On to the next shape.
Constructing the Sierpinski triangle fractal
Now that we have looked at the Sierpinski carpet, let’s go ahead and make a Sierpinski triangle. It is created in a very similar way to the Sierpinski carpet; by copying the previous iteration and using it as the basis for the next iteration.
First, start with an equilateral triangle (one with all three sides the same length). Let’s pick 1 inches for each side. Go ahead and fill it in with some color (I used black). Most software drawing programs help you create proportional shapes such as equilateral triangles and squares so if you are using software, create an equilateral triangle with a width of 1 inches.
If your software does not, no worries. Borrowing a little math from Pythagoras, we can determine that if the width of the equilateral triangle is 1 inches, the height should be about 0.866 inches. If the numbers I picked seemed strange, don’t worry, they will make a little more sense in a second.
Next, copy the initial triangle 2 times for a total of 3 triangles. Place each triangle in a triangle shape where they only touch another triangle at 1 point. The overall procedure is as follows:
- Starting with a triangle, make 2 copies, ending up with 3 triangles
- Place the triangles in the shape of a larger triangle with only the tips touching the others
The end result will look like a larger triangle with the center removed. Here is what the first iteration looks like:
You should have a triangle with 2 inch sides. For the next iteration, reduce the size of the first iteration by 50% (or back to the original dimensions of 1 inch wide by 0.866 inches tall). Copy that reduced triangle 2 times and form them into a larger triangle. Here is what the second iteration looks like:
Continue those 2 steps until you reach the complexity you want. Here is what it looks like after 5 iterations:
Congratulations, you just created a Sierpinski triangle. Since most things are better with laser engraving, here are a few Sierpinski shapes made into earrings. What do you think?
I hope you have learned something new about fractals today. As technology gets better and better, it is becoming easier and easier to construct your own fractals and experiment. If you haven’t made one yet, get busy and try your hand at a few Sierpinski shapes. You’ll be glad you did.