Sierpiński triangle

The Sierpiński triangle is a famous mathematical fractal. It is also known as the Sierpiński gasket or the Sierpiński sieve. A fractal is a pattern that is self-similar. This means the shape looks the same at any level of magnification.

Whether you zoom in or zoom out, you see the same repeating structure. This pattern is mathematically generated and can be reproduced at any scale. It serves as a fundamental example of how complex shapes emerge from simple, repeated rules.

There are several ways to construct this shape. One common method involves the repeated removal of triangular subsets. You start with a single equilateral triangle. First, you subdivide it into four smaller, congruent equilateral triangles. Next, you remove the central triangle.

You then repeat this exact process for each of the remaining smaller triangles. If you do this infinitely, you reach the true fractal. This process is known as a finite subdivision rule.

Another method uses shrinking and duplication. You begin with any triangle in a plane. You shrink that triangle to half its original height and width. Then, you make three copies of this smaller triangle. You position them so that each corner touches the others.

This creates a central hole because the three small triangles only cover half the area of the original. You can even start with a square, and the shape will still converge toward a Sierpiński triangle. This is an example of an iterated function system.

You can also create the triangle using a method called the chaos game. This algorithm uses randomness to build the pattern.

First, label three points as the corners of a triangle. Pick a random starting point anywhere in the plane. Then, randomly select one of the three corners. Move half the distance from your current position toward that chosen corner and plot a new point. If you repeat this many times, the points will eventually form the Sierpiński gasket. Even if you start outside the triangle, the points will converge on the shape.

The history of this shape is quite interesting. The Polish mathematician Wacław Sierpiński described it in his 1915 article.

His original goal was to demonstrate a specific type of continuous curve, called a Cantorian curve. This is also known as the Sierpiński arrowhead. However, this pattern is not new to mathematics. Similar decorative patterns appeared in art many centuries before Sierpiński's formal mathematical work.

The Sierpiński triangle appears in many different mathematical systems. If you look at Pascal's triangle, you can find it hidden in the numbers. If you color the odd numbers black and the even numbers white, the pattern emerges.

It also appears in cellular automata, such as Rule 90. In the Towers of Hanoi puzzle, the possible moves form a graph. This graph can be represented as the Sierpiński triangle. These connections show how one mathematical idea can link many different fields.

This fractal has unique properties regarding its dimension and area. For a standard 2D shape like a square, doubling the side length creates four copies. For the Sierpiński triangle, scaling it by a factor of two creates exactly three copies. Because of this, its Hausdorff dimension is log 3 divided by log 2. As you perform more iterations, the area of the shape actually shrinks. At each step, the remaining area is three-quarters of the previous area. In the limit of infinite steps, the total area tends to zero.

You can even extend this idea into three dimensions. This creates a shape called a Sierpiński tetrahedron, or a tetrix.

You build it by repeatedly shrinking a regular tetrahedron and combining four copies. While the surface area of a tetrix remains constant during construction, its volume is halved at every step. Like the 2D version, the volume eventually approaches zero. It is a complex, connected structure that exists between dimensions.