Sierpinski's Triangle

Sierpinski's triangle
Fields: Geometry and Fractals
Image Created By: Unknown

Sierpinski's triangle

Sierpinski's triangle is a simple fractal created by repeatedly removing smaller triangles from the original shape.

Basic Description

Creation of the triangle

Sierpinski's triangle starts as a shaded triangle of equal lengths.

We split the triangle into four equal triangles by connecting the centers of each side together and remove this central triangle.

We then repeat this process on the 3 newly created smaller triangles.

This process is repeated several times on each newly created smaller triangle to arrive at the picture displayed. However, this picture is not a true Sierpinski's triangle, but rather the first few iterations of creating a Sierpinski's triangle. A Sierpinski's triangle is created by infinitely repeating this construction process.

The animation on the right depicts what would happen if you keep zooming in on a true Sierpinski's triangle.

Notice that by zooming into a corner of the triangle produces an image of another identical triangle; thus Sierpinski's triangle is self-similar. This animation also shows that Sierpinski's triangle has infinite detail, since we may zoom to infinite magnification with no change in the triangle. By showing Sierpinski's triangle to be infinitely self-similar, we have established Sierpinski's triangle to be a fractal.

While the Sierpinski triangle is generally depicted as made up of equilateral triangles, any triangle can be used to make a Sierpinski triangle. Click the links to see examples.

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Chaos Construction

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In this method, we start with only the vertices of an equilateral triangle. We pick a random point within the space defined by these three vertices and a random vertex. The median between these points is the point we define on the triangle. Repeating this method several hundred times will produce an outline of Sierpinski's triangle, while repeating this process several thousand times will give us a clear image of Sierpinski's triangle. This construction method is a simple example of an iterated function.

The triangle was first described by Waclaw Sierpinski in 1960.

Interactive Applet

A More Mathematical Explanation

[Click to view A More Mathematical Explanation]

Knowledge of high school math and a basic knowledge of calculus would be helpful here.

==Number of E [...]

[Click to hide A More Mathematical Explanation]

Knowledge of high school math and a basic knowledge of calculus would be helpful here.

Number of Edges

Sierpinski's triangle begins with 3 sides. As we can count from the images above, each iteration increases the number of sides by a factor of 3. We define a side as a line that is the boundary of a triangle.

We can thus relate the number of sides to the number of iterations in the equation $N=3^{n}\,$, where $N$ is the number of edges and $n$ is the number of iterations.

The Sierpinski's triangle has an infinite number of edges. The pictures of Sierpinski's triangle appear to contradict this; however, this is a flaw in finite iteration construction process. All the images of Sierpinski's triangle have a finite number of iterations while in actuality the triangle has an infinite number of iteration. From an algebraic viewpoint we are increasing the number of edges by a factor of 3 each time and we can evaluate this as a limit where x goes to infinity.

$\lim_{x \to \infty}(3)^{x} = \infty$

Perimeter

The perimeter of the triangle increases by a factor of 3/2 as we can see from the images above. Thus we can express the total perimeter of the triangle as a function of number of iteration, as shown below. $P_n=P_0*(3/2)^n\,$

From this expression we can see that the total perimeter length of a Sierpinski triangle is infinite. We can verify this by taking the limit of our perimeter function.

$\lim_{x \to \infty}(3/2)^{x} = \infty$

Area

Each iteration of the construction process reduces the area by 1/4. This is clear to see as the creation process splits each triangle into 4 congruent parts and by lightening the central one, removes 1/4 of the area.

We assume the original congruent triangle to have total area of $A_0\,$. The area of subsequent iteration could be expressed as $A_n=A_0*(3/4)^n\,$

The Sierpinski's triangle has total area of 0 (defining area as the shaded region). The pictures of Sierpinski's triangle appear to contradict this, however, this is a flaw in finite iteration construction process. From an algebraic viewpoint we are decreasing the area by a factor of 3/4 each time and we can evaluate this as a limit where x goes to infinity.

$\lim_{x \to \infty}(3/4)^{x} =0$

Fractal Dimension

The fractal dimension of Sierpinski's triangle can be calculated as follows. $D = \frac{log(N)}{log(e)} = \frac{\log 3}{\log 2}\approx 1.585.$

Pascal's Triangle

This section assumes general familiarity with Pascal's triangle, for more information please click the following link. Pascal's triangle presents a third way of constructing Sierpinski's Triangle. We shade all odd numbers and keep even numbers lightened. As Pascal's triangle is enlarged, it is apparent that the Sierpinski's triangle is formed.

Sierpinski's Triangle in 3 Dimensions

Sierpinski's Triangle can be expanded into 3 dimensions, making it into a Sierpinski tetrahedron, as above. It is formed by randomly picking a point within the tetrahedron and then drawing the median between the point and a random vertex. Infinitely iterating this process produces the complete tetrahedron.

It is interesting to note that the surface area of a Sierpinski's tetrahedron remains constant, regardless of the iterations taken. Volume is scaled down by a factor of 4 /5. The fractal dimension of a Sierpinski Tetrahedron is 2.

Teaching Materials

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Future Directions for this Page

-Some sort of interactive animation where the user can click randomly within a triangle to create Sierpinski's triangle by way of chaos construction.

-A 3d interactive version of Sierpinski's tetrahedron that you can move and watch several iterations get created.

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