updated 04-10-2004
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.
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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. |
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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. |
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the rhombic dodecahemioctahedronDescribed 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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convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects | January 2004 updated 04-10-2004 |