Doubling the tetrahedron
A regular tetrahedron of side two units contains four regular tetrahedra of side one unit and a hole. A close look shows that this hole has eight faces which are equilateral triangles. Hence it is a regular octahedron.
What is the relationship between the volume of a doubled tetrahedron and the volume of a single tetrahedron?
How many cubes of side 1 is there in a cube of side 2? (you can try with dice)
What is the ratio between the volume of a doubled tetrahedron and the volume of a single tetrahedron?
Consider regular tetrahedra and a regular octahedron, all of the same side. If the volume of the octahedron plus the volume of four tetrahedra is eight times the volume of a single tetrahedron, how many tetrahedra are there in an octahedron?
A YouTube playlist implementing this with origami.