The Icosahedron

Recall that Dr. Sarah uses the icosahedron and a mathematical process to turn a 2-sphere into a triangular shaped pillow with diameter approximately Pi/4.82.  This is the smallest space that can be obtained from a 2-sphere, and thus is important in her research, which has applications to crystallography and string theory.  Here is some more info on icosahedrons.

From http://www.maths.uq.oz.au/~infinity/feb98/warts.htm

It's rude to stare, but .... next time you see someone with a wart on the end of their nose, stop and take a close look. If you look close enough, you will see that the human wart virus looks like an icosahedron.

An icosahedron is a highly symmetrical solid; each of it’s 20 faces are equilateral triangles. In the third century BC, a Greek mathematician by the name of Euclid showed that there are precisely five solids where all the faces are regular polygons (tetrahedron, cube, octahedron, dodecahedron and icosahedron). The ancient Greeks believed they were the foundations of the four elements: fire, earth, air and water. We now know that most crystals take the shape of regular solids and the smallpox, polio and herpes virus take the form of an icosahedron.

From http://www.uct.ac.za/depts/mmi/stannard/virarch.html

The complex arrangements of macromolecules in the virus shell are minute marvels of molecular architecture. Specific requirements of each type of virus have resulted in a fascinating apparent diversity of organization and geometrical design. Nevertheless, there are certain common features and general principles of architecture that apply to all viruses.

In 1956, Crick and Watson proposed on theoretical considerations and on the basis of rather flimsy experimental evidence then available, principles of virus structure that have been amply confirmed and universally accepted.

They first pointed out that the nucleic acid in small virions was probably insufficient to code for more than a few sorts of protein molecules of limited size. The only reasonable way to build a protein shell, therefore, was to use the same type of molecule over and over again, hence their theory of identical subunits.

The second part of their proposal concerned the way in which the subunits must be packed in the protein shell or capsid. On general grounds it was expected that subunits would be packed so as to provide each with an identical environment. This is possible only if they are packed symmetrically.  These predictions were soon confirmed and it became evident that the occurrence of icosahedral features in quite unrelated viruses was not a matter of chance selection but that icosahedral symmetry is preferred in virus structure.

Domes have been around for centuries. What makes geodesic domes different?

Efficiency. A sphere is already efficient: it encloses the most volume with the least surface. Thus, any dome that is a portion of a sphere has the least surface through which to lose heat or intercept potentially damaging winds.  A geodesic dome uses a pattern of self-bracing triangles in a pattern that gives maximum structural advantage, thus theoretically using the least material possible. (A "geodesic" line on a sphere is the shortest distance between any two points.)

This picture should remind you of the spherical icosahedron from Dr. Sarah’s research:

Local loads are distributed throughout the geodesic dome, utilizing the entire structure. Geodesic domes get stronger, lighter and cheaper per unit of volume as their size increases--just the opposite of conventional building.  Bucky cooled critics by erecting enormous geodesic domes of many different designs, very quickly--sometimes in mere hours instead of months or years. Serving atop mountains, sheltering Arctic radar installations, and even covering the South Pole, they have proved to be the strongest structures ever devised. Earthquakes cannot damage them unless the ground opens up and swallows the foundation (or it is undermined, as the South Pole dome has been.) There has been no report of hurricane damage of a properly designed geodesic dome; indeed, they are demonstrations of more-with-less, or "ephemeralization," as Bucky liked to say. The best ones are proportionally thinner than a chicken egg shell is to the egg. More volume is sheltered by Bucky's domes than by the work of any other architect.

On the next page, cut out along the outer edge, fold all edges away from you, and tape together to form an icosahedron.  I find it easiest to place tape on the other side of the flap (the inside of the icosahedron), then look from the outside to match up the triangle edges, and then press down the tape in the desired location.