What Fits in a Cube

Notice the tetrahedron has six edges and that a cube has six faces. This is not just a coincidence, but a geometric relationship between the tetrahedron and the cube. When you slide the tetrahedron into the cube there will be exactly one edge on each face of the cube.
When you slide the tetrahedron into the cube, the four corners of the tetrahedron occupies four corners of the cube – leaving the other four corners free. The Kepler Star is made by intersecting two tetrahedrons, so it has eight corners that can occupy all the corners of the cube.
Finally, the cubeoctahedron is made from eight triangle faces and six square faces. When you slide the solid into the glass cube, each square face will meet each face of the cube. In fact, the cubeoctahedron is made by snipping off the corners of a cube.

The tetrahedron is called a Platonic solid. These solids have identical regular faces (like, triangles, squares and pentagons), and the same number of faces at each corner. There are only five Platonic solids, the tetrahedron, cube, octahedron, icosahedron and dodecahedron. They are named after the ancient Greek philosopher Plato, who associated these solids with the classical elements of fire, earth, air, water and “quintessence”.
The Kepler Star is named after the famous astronomer Johannes Kepler, although he called it a stella octangular or octagonal star.
The cubeoctahedron is called an Archimedean solid, which are made with two or more types of regular faces, with the same number of faces at each corner. There are 13 Archimedean solids.

Geometry is one of the oldest subjects in mathematics. Properties such as length, volume and weight are essential to engineering and architecture.

Some of the first ever proofs in mathematics are about the properties of shapes. Learning how to prove things about shapes laid the foundations of modern mathematics.