Updated: Feb 14
Hi everybody! I'm back after winter break, and we're starting off 2020 on the right foot. We're looking at some of my favorite mathematical objects, fractals! Fractals are patterns that exist somewhere between the finite and infinite. As we'll see, they even have fractional dimensions (hence the name fractal) because they exist somewhere between integer dimensions! We'll look at how these seemingly impossible shapes exist when we allow ourselves to extend to infinity, in the third part of my infinity series (as promised)!
Before we start, let me define for you more clearly what a fractal is. A fractal is any shape that has infinite self-symmetry, meaning if you zoom in forever, you'll get a repeating pattern. Here's an example of what this zoom looks like:
Do you see what I mean by infinite self-symmetry? With each successive zoom, we see the same triangles! This infinite zoom is where infinity comes in! It turns out this makes for some pretty wacky properties, so let's look at some of my favorite fractals.
The first fractal we'll be looking at is Koch's Snowflake:
It turns out that this snowflake has an infinite perimeter but a finite area!
Here's how we construct it. We start with an equilateral triangle and with each successive step, we "poke" out the middle third of each exposed side into another equilateral triangle. As we reach infinity, the shape is made up of an infinite number of line segments, so it becomes a curve!
If you've had a bit of background with geometric series, I can explain to you why this shape has a finite area but an infinite perimeter. If you haven't, just skip on to the next section.
Let's look at the perimeter first. Let's say our equilateral triangle has sides of length 1.
Then, the perimeter of Step 1 is 3.
In Step 2, we're replacing each side of length 1 with four line segments of length 1/3, so each side is increasing by 1/3. Our total perimeter is then 3+3(1/3)=4.
In Step 3, we're replacing each side of length 1/3 with four line segments of length 1/9, so each side is increasing by 1/9. The number of sides has been multiplied by 4, so our total perimeter is then 4+12(1/9) = 4+(4/3)=16/3
In Step 4, we're replacing each side of length 1/9 with four line segments of length 1/27, so each side is increasing by 1/27. The number of sides has been multiplied by 4, so our total perimeter is then 16/3+48(1/27) = 16/3 + (16/9) = 64/9
Continuing this reasoning, in total, the perimeter is going to be the sum of the series 3+1+(4/3)+(16/9)+(64/27)+... which is a geometric series with ratio 4/3, which is greater than 1 (if you disregard the 3), so it's going to snowball to infinity.
Now let's look at the area. This time, let's assume our triangle has an area of 1.
In Step 1, the area is 1.
In Step 2, we're adding three triangles that have sides 1/3 of the original triangle, so their areas are 1/9 of the original triangle. Then, our total area is 1+3(1/9)=4/3
In Step 3, we're adding 12 triangles (remember the number of exposed sides are multiplied by 4 each time) that have areas 1/9 of the previous triangle or 1/9^2 = 1/81 of the original triangle. Then, our total area is 4/3 + 12/81 = 40/27
In Step 4, we're adding 48 (12*4) triangles that have areas 1/9 of the previous triangle or 1/9^3 = 1/729 of the original triangle. Then, our total area is 40/27+48/729 = 1128/729.
Continuing this reasoning, in total, the perimeter is going to be the sum of the series 1 + (3/9) + (12/81) + (48/729)+... which is a geometric series with ratio 4/9, which is less than 1, so it's going to converge on a value.
This value is going to be 1+ (1/3)/(1-(4/9)) = 1 + 9/15 = 8/5. (Here we use the a/(1-r) formula for infinite geometric series, so look it up if you don't know it!)
What all those calculations tell you is that the Koch snowflake is miraculous! It's almost impossible to consider a finite area and infinite perimeter, and yet, this snowflake embodies that. I absolutely love this fractal because it reveals some of the wonder of math, so yes, there's a reason it's my profile picture on here! :)
I want all my members to always be curious. Don't just accept the fact that this bizarre property is true for the Koch Snowflake; work it out! You'll learn more if you allow yourself to question and figure things out for yourself, and that's the reason I included these calculations here! We'll be looking at a couple other fractals that have similar crazy properties, and I advise you to work out why on your own! All you need is geometric series and a little bit of skill in setting it all up.
Before we move on, let me show you one other cool thing about the Koch snowflake: It can tessellate the plane! This means that if we place different sizes of snowflakes down, we can actually cover an entire surface with no gaps, which seems crazy for infinitely curved shapes. Here are two Koch patterns, but so many exist, so search up "koch snowflake tessellations" for more!
The second fractal we'll be looking at is the Sierpinski Triangle, a fractal you've seen both at the top of this article and in the GLeaM logo! Though I thought it best we start with the Koch snowflake, this one is actually my favorite because it shows up in such surprising places.
Let's look at how this fractal is constructed.
We begin with an equilateral triangle, and then we cut out the middle triangle of the overall triangle, and then we cut out the middles of those triangles, and then those... and you get the picture.
You can also create it by starting with your triangle and adding the successive triangles to the right and up from it, making it bigger and bigger instead of adding holes.
But, it turns out there are also some nonstandard ways of constructing the Sierpinski triangle! Here, we get the Sierpinski triangle from a square which we repeatedly reduce to 1/4 of its original on the top. It turns out you can use any other shape in a similar way too to get the Sierpinski triangle!
Here, we construct the triangle not from a bunch of shapes but out of one continuous line, as we repeatedly add trapezoidal arcs! This is called the Sierpinski arrowhead.
Here's one neat aspect of the Sierpinski triangle. It doesn't lie in a typical dimension. Here's how we'll think about this.