An icosahedron is any three-dimensional shape with twenty faces. However, we usually use the word to mean a regular, convex icosahedron, and this is the one that I like the most. You can see what it looks like in the picture, courtesy of Wikipedia:
This kind of icosahedron is made up of twenty equilateral triangles the same size. If you look at the vertices (corners) of the shape, you can see how each one is made up of the points of five triangles. They seem to meet in a kind of pentagon-star.
This is why the regular icosahedron has so much symmetry. Each vertex has the rotational and reflective symmetry of a pentagon, and all the vertices are the same, so the shape can be rotated or reflected to reposition a given vertex on any of the nineteen others.
The icosahedron has a kind of mathematical brother. If you take twelve regular pentagons the same size, and place one on each of the twelve vertices of the icosahedron, then spin each pentagon to be positioned in the right way, the pentagons will all fit together to make a regular dodecahedron. What’s even more fascinating is that if you take your dodecahedron and pace an equilateral triangle on each of the twenty vertices, you will get your icosahedron back! This quality between two shapes is called duality. Like a magician turning your 50p coin into a rabbit inside his hat, duality is a two-way magic trick between two very special shapes.
You can see the transition between the two shapes best by holding one inside the other:
Can you think of any other 3D shapes that have duality between them? Remember, the number of vertices on one shape must be equal to the number of faces on the other, and the number of faces that meet around a point on one shape must be equal to the number of sides on each of the 2D faces that make up the other shape.
If you like the icosahedron as much as I do, you might like to find out about some of the other four Platonic solids.