Sierpinski Gasket -- Math Art -- Fractals

 

Today I am going to share some fun math art looking at a famous fractal--the Sierpinski Gasket or the Sierpinski Triangle. It is a perfect fractal to have kids create and goes well with geometry lessons. A fun way to introduce it and create it is the Chaos Game. Here is a video showing the Chaos Game with a triangle, square, and pentagon. With the triangle the Sierpinski Triangle will appear with enough iterations of the game. The rule as explained in the video is to begin with a random point. Then randomly choose a vertex. Connect your point to the vertex and find the midpoint. (Erase the line.) The midpoint is your new starting point. Repeat. This is a game you could easily play in a class as well. Don't watch the video first though. 

Typically the triangle used in the Sierpinski Gasket is an equilateral triangle, but any triangle can be used. I made this paper one using an isosceles triangle punch. I cut the punched triangles into the fourths. I glued the paper triangles onto a black poster board. I love how it came out. 

This fractal is easy to draw as well. It can be used to practice measuring with a ruler or can be constructed (with a straight edge and compass and not measuring tool). To construct an equilateral triangle start with a line segment.

Then use a compass by putting the point of the compass on one end of the segment and the pencil point on the other. Draw an arc above the segment. Then switch the point of the compass to the other endpoint of the segment and repeat. 

Where the arcs intersect is the third vertex of the triangle. Connect it to the endpoints of the line segment and you will have an equilateral triangle.

The next step in creating the Sierpinski Gasket is to find the midpoint of each of the sides of the triangle. To construct the midpoint the easiest way is the construction for the perpendicular bisector. Remember we are using a straightedge and not a ruler (or at least not measuring with the ruler). To find the midpoint open the compass more than halfway across the segment or side but not all the way across. 

With the compass point on an endpoint of the segment or vertex of the side, swing an arc above and below the line. Repeat from the other endpoint. You will have a football shape for one line. Since we need all three midpoints it is easier to draw almost a full circle from each vertex. Be sure not to change the amount the compass is open though.

The points where the arcs intersect are on the perpendicular bisector of the segment/side. Lining the straightedge between them and mark the midpoint. If you draw the line, you will have the perpendicular bisector. Do this for each side of the triangle. 

Now you will draw the midsegments of the triangle. This is of course part of a geometry lesson. The midsegment connects the midpoints of two of the sides of the triangle. It will be parallel to the third side as well as half the length of the third side.

You need all three of the midsegments of your triangle.

For stage 1 of the fractal you would remove the middle triangle and we will remove it by coloring black.

If you are working on fractions, it is perfect to ask what fraction of the triangle is the black triangle. It is 1/4 of the whole triangle as well as 1/4 of the area. The area of the white triangles is 3/4 of the whole and the perimeter

For the next stage we will need the midpoints of the new triangles. We repeat the constructions of the smaller triangles. I did my arcs before coloring the center black.

We will again connect the midpoints to get the midsegments.

We will color the middle ones in black again. But first let's look at the fraction of the smaller triangles.

The purple triangle is 1/16 of the entire triangle or 1/12 of the white triangles (not including the black triangle).

To create stage 3 we need the midpoints of the smaller triangles. You can see I did constructed them before coloring the triangles black.

As I kept going I realized I only needed the midpoints of the outer triangle sides. I could then use my straightedge to create all the lines I needed. 

Removing or coloring black the center triangles we get stage 4.

I tried to do stage 5 as well. Some of my midpoints and triangles were slightly off and it showed in this stage.

Now of course the Sierpinski Gasket as a fractal these steps are repeated infinitely. You can see that more on a computer. There are some interesting facts about the Sierpinski Gasket. I am sharing two videos here that explain them well.

First it has a dimension that is not a whole number. I always find this fascinating. Here is a video that explains it.

Another fun fact about the Sierpinski Triangle is it has an area of 0 and a perimeter of infinity. Here is a video that shows this. Note: it does talk about limits so there may need to be some explanation to kids not in a calculus class about limits of a fraction to the power of n and when it goes to 0 and when it goes to infinity.

Now I have mentioned that I want to bring origami into my classroom this year. When I took a fractal course for teachers at Yale we created the Sierpinski Tetrahedron out of envelopes. I don't remember who we made the tetrahedrons, so I looked for an origami tutorial. I used this one. I did however glue the two pieces together because I couldn't get it to stay. I then hot glued the tetrahedron together to form the various stages of the Sierpinski Pyramid.

I am going to put a string on it and hang it in my classroom. I love how easy it is to make the Sierpinski Gasket or Pyramid and how easy it is to explain the concept of self-similar and look at fractions, area, dimension, etc. You can see more math and more about the Sierpinski Gasket and fractals here.