The tetrahedral arangement of carbon atoms is what gives diamond the properties of density, hardness, and optical index of refraction(sparkle).

A tetrahedron can be made from an equilateral triangle (All angles 60 degrees)
If you start with a triangle and then bisect eact side, You will have 4 triangles. By folding each face up, you will have a tetrahedron. You can see how that even as a solid shape these can pack together perfectly. In crystals this usually means high density, and isotropic ( the same in all directions) properties.