Mathematical Impressions: Symmetric Structures

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Symmetric Structures

Mathematicians classify objects by their symmetries. If you turn a five-armed starfish a fifth of a revolution, it looks unchanged, so it has a five-fold rotational symmetry axis. Objects like a soccer ball, which has five-fold rotation axes (through the black pentagons) and three-fold rotation axes (through the white hexagons), are said to have “icosahedral symmetry.” The arrangement of rotations which leave the objects looking unchanged is the same as that of a regular icosahedron.

It is an unexplained fact that objects with icosahedral symmetry occur in nature only at microscopic scales. Examples include quasicrystals, many viruses, the carbon-60 molecule, and some beautiful protozoa in the radiolarian family. Luckily, we humans can make our own human-scale examples, so everyone can see and appreciate this lovely symmetry group. However, nature’s radiolarian examples are the most stunning instances of icosahedral symmetry and well worth a careful look.


Radiolarian images from Ernst Haeckel’s “Art Forms in Nature,” 1899–1904.

Quasicrystal images from Wikipedia and Stanford University.

Virus images from Virusworld.


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  • Quasi crystals have been observed in a variety of macroscopic systems. Search for example for quasi crystals and Faraday waves. Macroscopic scale quasi crystals can be visually observed on the surface of fluid layer vibrated along the normal to the surface at rest.

  • I enjoy this Web site enormously as a selftough digital artist as of now a work wlth Power Point and Paint Programs. Thank You

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