Snowflake, symmetry

Objects can also possess rotational symmetry. In Figure 1.11(c) imagine an axle passing through the centre of the snowflake in the same way as a wheel rotates about an... 

As discussed previously for the snowflake and the 50p coin, molecules and crystals can also possess rotational symmetry. Figure 1.12 illustrates this for several molecules. 

Symmetry in biology is one of the many unresolved problems, but complex examples can be observed in the inanimate world. As dust specs are drifting through the wintry sky, water molecules freeze to the surface to form a delicate crystalline marvel of precisely sixfold symmetry.11 Deterministic No doubt The architecture of each of the six identical leaflets in one flake is determined in part by the nucleating surface and by the temperature gradients through which it tumbles. While it is said that no two snowflakes are alike, the sixfold symmetry is invariant. 

In addition to a rotation axis with intersecting symmetry planes (which is equivalent to having multiple intersecting symmetry planes), snowflakes have a perpendicular symmetry plane. This combination of symmetries is labeled m-n.m and it is characteristic of many other... 

One of the most beautiful and most common examples of this symmetry is the m-6 m symmetry of snow crystals. The virtually endless variety of their shapes and their natural beauty make them outstanding examples of symmetry. The fascination in the shape and symmetry of snowflakes goes far beyond the scientific interest in their formation, variety, and properties. The morphology of the snowflakes is determined by their internal structures and the external conditions of their formation. The mechanism of snowflake formation has been the subject of considerable research efforts. It is well known that... 

Returning to the snowflakes, an eloquent description of their beauty and symmetry is given by Thomas Mann in The Magic Mountain [18] ... 

The first known sketches of snowflakes from Europe in the sixteenth century did not reflect their hexagonal shape. Johannes Kepler was the first in Europe, who recognized the hexagonal symmetry of the snowflakes as he described it in his Latin tractate entitled The Six-cornered Snowflake published in 1611 [24], By this time Kepler had already discovered the first two laws of planetary motion and thus found the true celestial geometry when he turned his... 

The twofold mirror-rotation axis is the simplest among the mirror-rotation axes. There are also axes of fourfold mirror-rotation, sixfold mirror-rotation, and so on. Generally speaking, a 2 -fold mirror-rotation axis consists of the following operations a rotation by (360/2//) and a reflection through the plane perpendicular to the rotation axis. The symmetry of the snowflake involves this type of mirror-rotation axis. The snowflake obviously has a center of symmetry. The symmetry class m-6 m contains a center of symmetry at the intersection of the six-fold rotation axis and the perpendicular symmetry plane. In general, for all m n m symmetry classes with n even, the point of intersection of the //-fold rotation axis and the perpendicular symmetry plane is also a center of symmetry. When n is odd in an m-n m symmetry class, however, there is no center of symmetry present. 

Figure 4-16a shows the logo of a sporting goods store in Boston, Massachusetts. Geometrical correspondence is gone, yet we have no difficulty in recognizing the antimirror symmetry relationship. The antireflection plane relates a half-snowflake and a half-sun, symbolizing winter and summer, respectively. There are two coke machines in the picture of Figure 4-16b. There is no geometrical correspondence, but there is color reversal, and reversal of yet another, more important, property, the sugar content. This makes the two machines an example of antisymmetry with some abstraction.

A benzenoid of hexagonal symmetry belongs to one of the symmetry groups D6h and C6h. For obvious reasons these systems are called snowflakes. Sometimes it is distinguished between the D6h and C6h groups by means of the terms proper and improper snowflakes, respectively. The proper snowflakes are also said to have regular hexagonal symmetry. Snowflakes, both of D6h and C6h, occur for... 

Table 32. Numbers of classified benzenoids with regular hexagonal symmetry, D6h (proper snowflakes) ...

Fig. 30. Snowflakes all essentially disconnected benzenoids with hexagonal symmetry, Dbh (one system) or C6h, and h < 37 2 systems with h = 25 and 8 with h — 31. K numbers are given...

Fig. 32. Proper snowflakes all normal benzenoids with D6h symmetry and h < 43. K numbers are given...

All-benzenoids with hexagonal (D6h or C6h) symmetry have been referred to as all-flakes [129]. In other words, an all-flake is an all-benzenoid snowflake. We may also speak about proper (D6h) and improper (C6h) all-flakes as subclasses of the proper and improper snowflakes, respectively. 

Symmetry establishes a ridiculous and wonderful cousinship between objects, phenomena, and theories outwardly unrelated terrestrial magnetism, women s veils, polarized light, natural selection, the theory of groups, invariants and transformations, the work habits of bees in the hive, the structure of space, vase designs, quantum physics, scarabs, flower petals, X-ray interference patterns, cell division in sea urchins, equilibrium positions in crystals, Romanesque cathedrals, snowflakes, music, the theory of relativity. 

A snowflake in the present sense is a benzenoid of hexagonal symmetry [8, 57, 58], belonging to D6h or C6h. Sometimes the symmetries D6h and C6h, are associated to proper- and improper snowflakes [59], respectively. 

Symmetry is a phenomenon of the natural world, as well as the world of human invention (Figure 4-1), In nature, many types of flowers and plants, snowflakes, insects, certain fruits and vegetables, and a wide variety of microscopic plants and animals exhibit characteristic symmetry. Many engineering achievements have a degree of symmetry that contributes to their esthetic appeal. Examples include cloverleaf intersections, the pyramids of ancient Egypt, and the Eiffel Tower. 

If all possible hydrogen bonds form in a mole (Na molecules) of pure water, then every oxygen atom is surrounded by four H atoms in a tetrahedral arrangement its own two and two from neighboring molecules. This tetrahedral arrangement forms a three-dimensional network with a structure similar to that of diamond or SiOi. The result is an array of interlocking six-membered rings of water molecules (Fig. 10.14) that manifests itself macroscopically in the characteristic sixfold symmetry of snowflakes. 

A unique direction in an ice crystal that shows hexagonal rotational symmetry (and leads to identical properties along six hexagonal directions perpendicular to the [c]-axis and the familiar hexagonal symmetry of snowflakes). 

On the nanometer level, crystal structures are symmetric arrangements of molecules (bound atoms) in three-dimensional space [19]. Driven purely by energy minimization, countless manifestations of symmetry are found in nature ranging from the arrangement of atoms in unit cells and water molecules in snowflakes to the facets of crystals such as quartz and diamond [20], For a crystal constructed of identical molecules, the positions of all of the molecules in the structure can be predicted using four basic symmetry elements (1) centers of symmetry (2) two, three, four, or sixfold rotational axes (3) mirror or reflection planes or (4) combinations of a symmetry centers and rotational axes [21]. Combined with the constraint that space must be filled by the... 

But much of the behaviour of ice still remains unexplained. Consider a snowflake, its branches replicating one another in six-fold symmetry,... 

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