Snowflakes hexagonal symmetry

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... 

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...

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. 

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). 

The magnificent hexagonal 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 shap>e and symmetry of snowflakes goes far beyond the scientific interest in their formation, variety, and properties. The morphology of 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 the internal hexagonal arrangement of water molecules... 

Kepler suggested that the hexagonal symmetry of the snowflakes is due to the regular packing of constituent particles. ... 

Note that the combination of the bent structure of individual water molecules and the linear or almost linear nature of hydrogen bonds, H X—H (where X = O in water), leads to an ice structure characterized by rather large hexagonaUy shaped holes. The shape of snowflakes, one example of which is shown in Figure 11.4, reflects this overall hexagonal symmetry on the molecular level. When ice melts, some of the water molecules break off from the ice structure and fill the hexagonal... 

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. 

The reason why double-kernelled peaches and apricots are harmful to people is that the flowers of these trees are properly speaking five-petalled yet if they develop with sixfold (symmetry), twinning will occur. Plants and trees all have the fivefold pattern only the yellow-berry and snowflake crystals are hexagonal. This is one of the principles of Yin and Yang. So if double-kernelled peaches and apricots with an (aberrant) sixfold (symmetry) are harmful, it is because these trees have lost their standard rule. 

Many molecules and other objects have multiple rotation axes. For example, snowflakes (Figure 4.2) exhibit complex shapes that are nearly always hexagonal and approximately planar. The line through the center of the flake perpendicular to the plane of the flake contains a twofold (C2) axis, a threefold (C3) axis, and a sixfold (Cg) axis. Rotations by 240° (C ) and 300° (Q ) are also symmetry operations of the snowflake. 

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