Geodesic structures

It is useful to test our understanding of determinate structures by considering the simple, yet powerful class of 3-d objects called geodesic structures. These are structures that can be mapped onto the surface of a sphere. There is a beautiful structural theorem about geodesic structures that we will now prove. 

Theorem A geodesic structure is determinate if and only if all of its faces are triangles. 

A geodesic structure with all triangular faces is determinate. 

A geodesic structure with any non-triangular faces is underdetermined, since adding the struts needed to make all faces triangles will add constraints. 

In order to clarify the central arguments and to minimize conceptual problems in this initial development, we assume that the model universe is stationary in the sense that the overall statistical properties of the material distribution do not evolve in any way. Whilst this was intended merely as a simplifying assumption, it has the fundamental effect of making the development inherently nonrelativistic (in the sense that the system evolves within a curved metric 3-space, rather than being a geodesic structure within a spacetime continuum). 

The discovery of fullerenes, carbon molecules with geodesic structures, has prompted a number of studies, both theoretical and experimental, devoted to the possible existence of non-carbon analogues of these species [18-21]. From a theoretical point of view this discovery opens the way toward the preparation of new forms of known substances. In addition to theoretical interests, such polyhedrons could fecilitate a much more diverse chemistry than has been possible so far with carbon fullerenes. 

The discovery of perfect geodesic dome closed structures of carbon, such as C o has led to numerous studies of so-called Buckminster fullerene. Dislocations are important features of the structures of nested fullerenes also called onion skin, multilayered or Russian doll fullerenes. A recent theoretical study [118] shows that these defects serve to relieve large inherent strains in thick-walled nested fullerenes such that they can show faceted shapes. 

Buckminsterfullerene (Chapter 11 essay Carbon Clusters Fullerenes and Nanotubes ) Name given to the Cgo cluster with structure resembling the geodesic domes of R Buck minster Fuller see front cover... 

The C50 molecule contains 12 pentagons and 20 hexagons. This type of hexagonal-pentagonal structure closely resembles the geodesic domes developed by the architect and engineer R. Buckminster Fuller, after whom the molecule is named. In the Csn molecule each carbon atom is bonded to three... 

If you re a sports fan, you ve almost certainly seen this structure before. It is that of a soccer ball with a carbon atom at each vertex. Smalley and his colleagues could have named this allotropic form of carbon "carbosoccer" or "soc-cerene," but they didn t. Instead they called it "buckminster fullerene" after the architect R. Buckminster Fuller, whose geodesic domes vaguely resembled truncated soccer balls. 

Chemists were greatly surprised when soccer-ball-shaped carbon molecules were first identified in 1985, particularly because they might be even more abundant than graphite and diamond The C60 molecule (10) is named buckminsterfullerene after the American architect R. Buckminster Fuller, whose geodesic domes it resembles. Within 2 years, scientists had succeeded in making crystals of buckminsterfullerene the solid samples are called fullerite (Fig. 14.32). The discovery of this molecule and others with similar structures, such as C70, opened up the prospect of a whole new field of chemistry. For instance, the interior of a C60 molecule is big enough to hold an atom of another element, and chemists are now busily preparing a whole new periodic table of these shrink-wrapped atoms. 

Figure 4.19C shows Ceo, which is one type of fullerene discovered in 1985. It was given the name buckminsterfullerene because it resembles the geodesic-domed structure designed by architect R. Buckminster Fuller. Also known as buckyballs, Ceo is just one of several fullerenes that have been discovered. Others have been shown to have the formula C70, C74, and C82. Because of their spherical shape, researchers have speculated that fullerenes might make good lubricants. 

Kahler structures are easy to construct and flexible. For example, any complex submanifold of a Kahler manifold is again Kahler, and a Kahler metric is locally given by a Kahler potential, i.e. uj = / ddu for a strictly pseudo convex function u. However, hyper-Kahler structures are neither easy to construct nor flexible (even locally). A hypercomplex submanifold of a hyper-Kahler manifold must be totally geodesic, and there is no good notion of hyper-Kahler potential. The following quotient construction, which was introduced by Hitchin et al.[39] as an analogue of Marsden-Weinstein quotients for symplectic manifolds, is one of the most powerful tool for constructing new hyper-Kahler manifolds. 

In C60 fullerene-type carbon allotrope, there is only one structure in which all the pentagons are nonadjacent and this is icosohedral symmetry-I (Fig. 4.11). This structure is often referred to as backyball to reflect on its full name buckminster-fullerene (after Buckminster Fuller who popularized the geodesic dome as an architectural form). 

Cgg was named buckminsterfullerene, in honor of the visionary American architect Richard Buckminster Fuller (1895-1983). Fuller is known for developing and promoting the geodesic dome, which resembles. (Buckminsterfullerene molecules are also sometimes called buckyballs.) Later, researchers discovered this molecule belongs to a family of related carbon structures, which have become known as fuller-enes. The smallest fullerene is containing 20 carbon atoms. 

This polymorph of carbon was only discovered in 1985 by Sir Harry Kroto at the University of Sussex while looking for carbon chains. It is made by passing an electric arc between two carbon rods in a partial atmosphere of helium. Kroto was awarded the Nobel Prize in chemistry in 1996, along with two American researchers (Robert F.Curl Jr. and Richard E.Smalley). The molecule has the formula Ceo and has the same shape as a soccer ball—a truncated icosahedron it takes its name from the engineer and philosopher Buckminster Fuller who discovered the architectural principle of the hollow geodesic dome that this molecule resembles (a geodesic dome was built for EXPO 67 in Montreal). The structure is depicted in Figure 6.14. 

---