The Sierpinski's Gasket is like many fractals which are repeated over and over again. However, in this case the shape being repeated is a triangle. This fractal's special dimensions is log 3 / log 2 = 1.5849 and it lies between a plane and a line. Every iteration, the fractal's area moves towards 0 because of its infinite length and its dimensions.
How is it made?
There are many ways to make the Sierpinski's Triangle using equations and a bit of randomness.
One method it to take an equilateral triangle, and divide it into four similar triangles, then remove the one in the center of the triangle. Repeat on each of the similar triangle sections until infinity.
One method it to take an equilateral triangle, and divide it into four similar triangles, then remove the one in the center of the triangle. Repeat on each of the similar triangle sections until infinity.
Another way the Sierpinski’s Triangle can be generated is using Pascal’s triangle. If you take Pascal's triangle and then eliminate all of the even numbers, the the structure is Sierpinski’s triangle!
Also, the Sierpinski's Triangle can be generated using the Chaos Game.
Select three points randomly in a plane and color them. These three points will be the vertices of the triangle and they will remain the vertices of the triangle for the entire creation of the fractal. Color the vertices. The fractal normally looks better if the points form an acute triangle. Then, select a fourth point randomly in the plane to be the starting point. Color it. Choose one of the three vertices of the triangle randomly. Then, find the midpoint between the starting point and the vertex of the triangle. Color this point. This midpoint then becomes the starting point and the whole process is repeated. From the new starting point (the midpoint) randomly select a vertex of the triangle, and find the midpoint.
Select three points randomly in a plane and color them. These three points will be the vertices of the triangle and they will remain the vertices of the triangle for the entire creation of the fractal. Color the vertices. The fractal normally looks better if the points form an acute triangle. Then, select a fourth point randomly in the plane to be the starting point. Color it. Choose one of the three vertices of the triangle randomly. Then, find the midpoint between the starting point and the vertex of the triangle. Color this point. This midpoint then becomes the starting point and the whole process is repeated. From the new starting point (the midpoint) randomly select a vertex of the triangle, and find the midpoint.
Sierpinski's Triangle Activity
Another way to make the Sierpinski's Triangle is to use your TI-84 calculator Use these steps to make it one your calculator!