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14 15 symmetries in 3d spatial isometries Just as the sphere is the threedimensional equivalent of the perfect symmetry of the twodimensional circle, the transformations of figures in space correspond with that of the regular division of the plane that we saw earlier, and similar isometric principles are involved 16 . If we look to the ways in which space can be symmetrically partitioned, the most elementary divisions follow from the regular planefilling figures. So, just as the equilateral triangle, square and hexagon fill 2dimensions, the prisms based on these will completely fill space 7. When it comes to spacefillers that are regular in all directions the options are rather less obvious, but include the cube, the truncated octahedron 5, the cubeoctahedral system 8 and the rhombic dodecahedron 9. The three spherical symmetrical systems 10 have a particular bearing on the regular solid figures. Interestingly, among the huge variety of regular figures, nature consistently chooses one family above all others, namely, the pentagonal dodecahedra. These shapes, made up of hexagons and pentagons, are adopted by forms as diverse as the Fullerene molecule, sootparticles, radiolaria and viruses below. The intriguing aspect of these shapes, and perhaps the key to their usefulness in nature, is that while hexagons themselves cannot enclose space, any number can be enabled to do so with the addition of just twelve pentagons. 1. 3d symmetry along a line 2. 3d rotation about an axis 3. 3d reflection about a mirrorplane 4. 3d pointgroup symmetry 5. 3d spacegroup symmetry 6. 3d dilation symmetry 7. Spacefilling prisms 8. Cubeoctahedral system 9. Rhombic dodecahedron 10. The three spherical systems of symmetry tetrahedral, octahedral, and icosahedral.
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