The Platonic solids can be divided into a number of congruent
tetrahedra which share a common vertex at the centre of the solid.
The maximum number can be made when the four vertices are the
centres of the different dimensional
elements of the solid: the centre of the solid itself; a
face centre; an edge centre and a vertex (which is its own centre.)
The maximum number of tetrahedra is also the number of symmetries
of the solid, and any symmetry of the solid maps the set of tetrahedra
onto itself.
The tetrahedra have no mirror symmetry.
The following animations rearrange the symmetry unit tetrahedra,
sometimes grouped together, in various ways.
The tetrahedra are coloured black and white according to their handedness.
The animations preserve the rotational symmetry of the base polyhedron.
White tetrahedra are rotated about axes through the edge centres.
Black tetrahedra are rotated about axes through the face centres.
view animation (1.0Mb)
A cube is rearranged into a rhombic dodecahedron. The rhombic
dodecahedron has a hollow centre the same shape and size as the
original cube.
view animation (2.4Mb)
The cube faces are the bases of pyramids with an apex at the cube
centre. The pyramids travel through the cube until their bases meet and
and they form a
stellation of the rhombic dodecahedron which has half the volume of the
cube. Finally the pyramids lie on the outside of the cube forming a
rhombic dodecahedron with twice the volume of the cube (discounting the
hollow cube inside of it.)
view animation (2.9Mb)
The dodecahedron faces are the bases of pyramids with an apex at the
dodecahedron centre. The pyramids travel through the dodecahedron
until their bases coincide
with the opposite faces, pausing briefly at the halfway stage where the
bases meet.
view animation (3.0Mb)
Here are some of the interesting stages as VRML models, having
convex hulls of an
The dodecahedron faces are the bases of pyramids with an apex at the
dodecahedron centre. The pyramids travel through the dodecahedron
until their bases coincide
with the opposite faces, pausing briefly at the halfway stage where the
bases meet. The pyramids rotate through a half-turn about their axis
as they travel to the other side.
view animation (2.4Mb)
Here are some of the interesting stages as VRML models.
