Regular icosahedron

Imagine a shape made entirely of flat, triangular faces.

If every single triangle is the exact same size, it is called a regular icosahedron. This shape is a special kind of object called a Platonic solid. It has twenty faces, all of which are equilateral triangles. There are also thirty edges where the triangles meet. The shape has twelve corners, which mathematicians call vertices.

Because every face and corner is identical, the shape is very balanced. This balance is called symmetry. You can spin the shape in many ways and it will still look the same. It has thirty-one different axes that you can rotate it around. It also has fifteen mirror planes that divide it into matching halves.

There are many ways to build this interesting shape. One way is to start with a middle part called a pentagonal antiprism. You then attach two pointy pyramids to the top and bottom. This creates a shape called a gyroelongated pentagonal bipyramid.

Another way involves using three golden rectangles. These are special rectangles that follow a specific mathematical ratio. If you place them so they cross each other at right angles, their corners form the twelve vertices. You can even unfold the shape into a flat pattern called a net. There are actually 43,380 different ways to unfold it! The earliest drawing of one of these nets appeared in a book by Albrecht Dürer in 1525.

People have studied this shape for a very long time. The ancient Greek philosopher Plato talked about these special shapes in his writing. He believed the icosahedron represented the element of water.

Later, the mathematician Euclid wrote about these solids in his famous work, the Elements. Many other thinkers, like Hero and Pappus, studied how these shapes fit inside spheres. They even worked out how much space the shape takes up compared to a round ball. Another famous scientist, Johannes Kepler, also sketched these shapes in his studies. These thinkers helped us understand how math and shapes work together.

We can see the icosahedron in many parts of our world. In biology, some tiny things like the herpes virus use this shape for their shells.

Some tiny sea creatures called radiolarians also have skeletons shaped like this. You might also recognize the shape from a twenty-sided die used in games. These dice have been used since ancient times. One example was found in Ptolemaic Egypt. Another gold die was found in the treasure of Tipu Sultan. Today, many people use these dice for tabletop role-playing games.

This shape also helps us understand how to make maps. A man named R. Buckminster Fuller used the icosahedron to make a special map called the Dymaxion map. He used the flat triangles to show the Earth's surface more clearly. This helped him realize that Greenland is actually smaller than South America. The shape is also used in chemistry to describe how certain molecules are built. Even in physics, the icosahedron helps scientists understand how charged particles move on a sphere. It is a shape that connects many different ideas together.