excavated polyhedra

We have seen that it is easy to create new polyhedra, not necessary convex, by "augmentation" (we add one or several polyhedra on one or several compatible faces): this is the case for most of the convex deltahedra.
In the same way we can excavate polyhedra on compatible faces; thus we get non convex polyhedra.

Here are two polyhedra obtained from the cuboctahedron by excavation of 8 regular tetrahedra (on the left) and of 6 square pyramids (on the right). These two polyhedra have the same 13 vertices, the same 48 edges and 24 identical "triangular faces" (in light blue); only the faces in dark blue (respectively 6 squares and 8 triangles) are different. Each is an assemblage of the pyramids excavated to create the other.

These two uniform polyhedra (the cubohemioctahedron and the octahemioctahedron) have four regular hexagonal faces (the 4x6=24 triangles in light blue) which are self intersecting and pass through the center.

Of course excavation also allows to create tunnels; here is a cube from which two regular pyramids (equilateral triangular faces) have been excavated from opposite faces.

Excavation is thus an other technique which opens new perspectives in polyhedra creation.

the rhombic dodecahemioctahedron

Described by Guy Inchbald, this space filling polyhedron can be obtained in two ways:
•   from the cube, by excavation of pyramids on two opposites faces and augmentation with pyramids on the four other faces,
•   from the rhombic dodecahedron by excavation of two oblate octahedra (the dodecahedron is an assemblage of six such octahedra).
Besides the twelve faces of the exterior ring (4 rhombuses and 8 isosceles triangles) four self intersecting non regular hexagons going through the center build the eight faces of the two excavations.

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updated 04-10-2004