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 Amy C. EdmondsonA Fuller Explanation Chapter 8, Tales Told by the Spheres: Closest Packingpages 117 through 119

Icosahedron

 Fig. 8-14. Removal of nuclear sphere.Click on thumbnail for larger image.

We return to the initial twelve-around-one cluster, to try a new twist (literally). Imagine that we have thirteen spherical balloons—instead of Ping Pong balls—so that we can we reach through with a long pin to puncture the nuclear balloon, causing it to slowly deflate. As the nuclear balloon disappears, the twelve outside balloons shift, closing in symmetrically toward the empty space in the center. They can't go far-only enough for the six unstable squares of the VE to "tighten up" into two noncoplanar (or hinged) triangles. Squares were stable in the VE only because the nuclear sphere held them in place; the array was thus stabilized by triangulation in radial directions, producing alternating tetrahedra and half octahedra. Now, without the nucleus, the surface must be stable by itself; it therefore must be triangulated (Fig. 8-14).

The VE configuration has twelve "vertex" spheres. What shape do these twelve spheres become when fully triangulated? The six squares transform into two triangles each, which (added to the original eight) make twenty triangles in all. Twenty triangles and twelve vertices? That is none other than our friend the icosahedron, largest of the regular polyhedra, one of the "three prime structural systems in Universe."

We can create higher-frequency versions of the icosahedron, but they will always be single-layer shells. The icosahedral configuration arose as a result of removing the VE nucleus; the remaining spheres move in, partially filling the gap, and thus their positions no longer allow a space-filling array of spheres. Because icosahedral clusters are completely triangulated, they cannot be extended either inwardly or outwardly; they lack the necessary alternation of tetrahedra and octahedra.

Such 0mni-triangulated sphere-packing shells can have any frequency, despite being restricted to single-layer construction. Let's see what happens. Consecutive higher-frequency icosahedral shells cannot surround a previous layer as they do in the VE—which of course starts with a nucleus and continually surrounds it with layers. Icosahedral shells simply do not nest together. Instead, progressively larger, or higher-frequency icosahedra must be built one by one, each with one more sphere per edge, and always single-thickness.

 Fig. 8-15Click on thumbnail for larger image.

What will happen to the relationship between frequency and number of spheres on a given shell? It turns out that—while both shape and volume change considerably—the number is unaffected by this transition from the VE's fourteen faces to the twenty icosahedral triangles. We verify this fact through the following observations. Notice in Figure 8-15 that a square pattern of spheres can be compressed into a rhomboid (diamond) shape without changing the number of spheres. The resulting diamond is more tightly packed than the square and consists of two triangles of the same frequency as the original square, sharing one edge, that is, the row of spheres that used to form the diagonal of the square. Figure 8-15 shows how the spheres of two triangles on an icosahedral shell correspond to one square face of a VE of the same frequency. And therefore, because the spheres on an icosahedral shell are all as closely packed as possible (as opposed to the VE, which alternates triangles with the more loosely packed square faces), a smaller, denser shell is produced, with the same number of spheres as a VE shell of the same frequency. l0 f ² + 2 therefore also applies to icosahedra.

The icosahedron contains as much interior volume relative to surface area as is possible with only one type of face. Ever economical, nature therefore chooses icosahedral symmetry for the construction of a shell made of identical units; requiring a minimum of effort, this arrangement can arise automatically. Maximum volume, minimum material. It is thus easy to account for the icosahedral symmetry detected in the isometric virus capsid, the tough protein shell created by nature to house and protect the more fragile genetic material within, which is the source of the virus's instructions.3 Nature consistently exhibits elegant solutions to design problems, because she finds the most efficient, or least energetic, way to operate. She has no choice but to adhere to the constraints of space. The example of the spherical virus shell is worth our brief attention, for it provides an elegant illustration of the "design science" of nature at work.

Let's examine the criteria: (1) A container must be constructed out of a large number of identical constituents (protein molecules), (2) for reasons which will be explained below, the shell must be able to self-assemble—that is, build and rebuild itself automatically, (3) maximum symmetry is advantageous to minimize the energy required for attractive bonds between the capsid molecules, and (4) the arrangement must be stable, which means triangulated.

The elegance of the relationship between structure and function is well documented in modern biology, and the isometric virus is no exception; its structure must be suited to its specific functions. A tough shell is to completely enclose minute amounts of genetic material—quantities necessarily insufficient for carrying detailed bonding instructions—yet it must easily disassemble and reassemble itself in order to release the viral genetic material into a host cell. The overall structure must therefore be dictated by properties of the subunits and by the constraints of space itself—both criteria also establishing a built-in check system.

A sphere, which maximizes the volume-to-surface-area ratio, is the key to an efficient solution. Interconnected molecules, which can only approximate that theoretical sphere, will achieve a spherical distribution most efficiently through icosahedral symmetry. Observations of isometric viruses have consistently revealed icosahedral patterns, thus reconfirming that nature chooses optimal designs. (4) We shall study this configuration in more detail in Chapter 15.

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