# Mathematical Impressions: An Exploration of Symmetric Structures [Video]

Objects with icosahedral symmetry occur in nature only at microscopic scales, including quasicrystals, many viruses and some beautiful protozoa in the radiolarian family

Image: Simons Foundation Mathematical Impressions

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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.

Credits:

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

Quasicrystal images from Wikipedia and Stanford University.

Virus images from Virusworld.

Related:

More videos from the Mathematical Impressions series.

Reprinted with permission from Simons Science News, an editorially-independent division of SimonsFoundation.org whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the computational, physical and life sciences.

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1. 1. restarter 06:12 PM 3/22/13

I just wanted to say what a delightful video this was. I had no idea the Platonic solids occurred so frequently in nature, nor that they formed three families of duals.

I had a candle burning beside me when the video announced that carbon buckyballs were being produced. Made me look twice at the flame, and I peered at it as if being a foot away was the thing preventing me from seeing them.

Lovely work.

2. 2. secretagent3180 09:21 PM 3/22/13

It makes me sad though that Earnest Haeckel resorted to fraud in his efforts to promote evolution. Ontology does not recapulate phylogeny.

3. 3. secretagent3180 in reply to secretagent3180 09:23 PM 3/22/13

mis-spelled, meant recapitulate.

4. 4. Ungolythe 04:34 AM 3/24/13

Don't be sad secretagent3180, the fraud allegations were most likely totally unfounded to begin with and the theory of Evolution is as strong as it ever was.

5. 5. jtdwyer 06:53 PM 3/24/13

Actually, I clicked on this article accidentally, but it is very interesting and enjoyable. I had no idea that candle soot contained buckminsterfullerene particle-waves!

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