Sierpiński triangle

Imagine a big triangle.

You can make a special pattern with it. First, you cut the big triangle into four small ones. Then, you take away the middle one.

You can do this again and again. Each time, you take away more small triangles. This makes many tiny holes. The pattern looks the same even if you zoom in. A man named Wacław Sierpiński named this shape. It is a very cool pattern to see.

Imagine a large triangle with three equal sides.

You can turn this shape into a special pattern. One way is to cut it into four smaller triangles. Then, you remove the middle one.

You can repeat this step over and over. Each time you do this, you create more tiny holes. This pattern is a fractal. A fractal is a shape that looks the same no matter how much you zoom in.

A man named Wacław Sierpiński described this in 1915. But people used this pattern for art long before him. You can even make it using a game of chance. This is called the chaos game. You pick a random point. Then, you move halfway toward one of the three corners. If you do this many times, the pattern appears. You can also find this shape in math puzzles. For example, it relates to the Towers of Hanoi puzzle. Some people even make a 3D version. This is called a Sierpiński tetrahedron.

It looks like a pyramid made of many smaller pyramids.

A fractal is a special kind of shape that never seems to end. One of the most famous examples is the Sierpiński triangle.

This shape is a fractal because it is self-similar. This means the pattern looks the same no matter how much you zoom in. If you look closely at a small part, it looks like a tiny version of the whole thing. It is a mathematical pattern that can be made at any scale. You can see it in tiny details or in huge drawings.

There are many ways to build this pattern. One way is to start with a large equilateral triangle.

You can divide it into four smaller triangles. Then, you simply remove the middle triangle. You repeat this same step for every new triangle that is left. Another way is to shrink the original triangle and make three copies.

You place these three copies so their corners touch. This leaves a hole in the center. You can even start with a square instead of a triangle. Eventually, the shape will still turn into a Sierpiński triangle.

A mathematician named Wacław Sierpiński described this shape in 1915. His original goal was to show a special kind of continuous curve. However, this beautiful pattern appeared in decorative art many centuries before he studied it. It is also found in math patterns like Pascal's triangle. If you color the odd numbers black and the even numbers white, the shape appears. This shows how math patterns hide in many different places.

You can even create this shape using a game of chance. This method is called the chaos game. You start with three points that form a triangle. Then, you pick a random point inside. You choose one of the three corners at random. You move halfway toward that corner and draw a new dot. If you do this many times, the triangle emerges from the dots. It is amazing how random moves can create such a perfect pattern.

This shape connects to many other interesting ideas. It is related to the Towers of Hanoi puzzle. The moves in that puzzle can be mapped to this triangle. You can also build a three-dimensional version.

This is called a Sierpiński tetrahedron or a tetrix. It looks like a pyramid made of smaller pyramids. As you add more layers, the volume of the shape actually shrinks toward zero. It is a wonderful way to see how math works in different dimensions.

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.