Dividing a Regular Tetrahedron into Four Congruent Pieces

A regular tetrahedron is divided into four congruent pieces, each of which is bordered by three large and three small quadrilaterals. The quadrilaterals are kites, which have two pairs of adjacent sides of the same length. Each piece is a distorted cube. The new vertices are the centers of the faces of the tetrahedron at a third of the height of the face triangles and the center of the tetrahedron at a fourth of its height.