M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 8
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This combination of a rotation axis and a symmetry planeproduces further symmetry planes. Their total number will be n as aconsequence of the application of the n-fold rotational symmetry tothe symmetry plane. The complete set of symmetry operations of afigure is its symmetry group.Figure 2-18 shows two flowers. The Vinca minor† has four-foldrotational symmetry and no symmetry plane. The Norwegian tuliphas three-fold rotational symmetry with the axis of rotation in asymmetry plane. The three-fold rotation axis will, of course, rotatenot only the flower but any other symmetry element, in this casethe symmetry plane, as well. The 120◦ rotations will generate altogether three symmetry planes, and these planes will make an angleof 60◦ with each other. There is though an alternative description ofthe symmetries of the Norwegian tulip.
Start with recognizing the†This plant has been used to extract physiologically important alkaloids. One of thederivatives has become an important medicine that dilates blood vessels in the brain.Cavinton has been a popular drug for improving memory.382 Simple and Combined SymmetriesFigure 2-18.
Top left: Vinca minor; Top right: Norwegian tulip; Bottom: stonecarvings along the Via Appia Antica in Rome (photographs by the authors).three symmetry planes cutting through the petals. The three symmetryplanes are at 120◦ relative to each other. Where they intersect, that lineis an axis of threefold rotation. The two flowers we chose for closerexamination have been immortalized by a Roman artist: The lowerpart of Figure 2-18 shows an ancient stone carving along Via AppiaAntica in Rome depicting two flowers that may very well representVinca minor and the Norwegian tulip.Fivefold symmetry appears frequently among primitive organisms.Examples are shown in Figure 2-19.
They have fivefold rotation axesand intersecting (vertical) symmetry planes as well. The symmetryclass of the starfish is 5·m. This starfish consists of ten congruentparts, with each pair related by a symmetry plane. The whole starfishis unchanged either by 360◦ /5 = 72◦ rotation around the rotation axis,or by mirror reflection through the symmetry planes which intersect at2.3. Combined Symmetries39an angle of 36◦ .
Fivefold rotation with coinciding mirror reflection isquite common among fruits and flowers. This symmetry is also rathercommon among molecules. On the other hand, this symmetry is usedto be considered absent in the world of crystals as will be discussed inmore detail in the chapter on crystals.Examples of n·m symmetries are shown in Figure 2-20. It is a muchfavored symmetry by designers of important buildings.2.3.2. SnowflakesIn 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 combinationof symmetries is labeled m·n:m and it is characteristic of many other(a)Figure 2-19.
(a) Starfish after Haeckel [7]; (Continued)402 Simple and Combined Symmetries(b)(c)(d)(e)Figure 2-19. (b) Starfish; (c) Carrion flower (Stapelia gigantea pallida in Honolulu,Hawaii) [8]; (d) Flower in Hawaii; (e) Apple blossom (b–e, photographs by theauthors).highly symmetrical objects, such as prisms, bipyramids, bicones,cylinders, and ellipsoids. Due to their high symmetries, these shapesare relatively simple. Some examples are shown in Figure 2-21; theyall have m·n:m symmetries: the pentagonal prism, m·5:m, the trigonal bipyramid, m·3:m, and the bicone and the cylinder, m·∞:msymmetry.One of the most beautiful and most common examples of thissymmetry is the m·6:m symmetry of snow crystals. The virtuallyendless variety of their shapes and their natural beauty make themoutstanding examples of symmetry.
The fascination in the shape andsymmetry of snowflakes goes far beyond the scientific interest in theirformation, variety, and properties. The morphology of the snowflakesis determined by their internal structures and the external conditionsof their formation. The mechanism of snowflake formation has beenthe subject of considerable research efforts. It is well known that2.3. Combined Symmetries(a)41(b)(c)Figure 2-20. Examples in architecture: (a) Eiffel Tower, Paris, from below; (b)Cupola of the Parliament building in Budapest; (c) Pentagon in Washington, DC(photographs by the authors).Figure 2-21.
Examples of m·n:m symmetries: from the left, pentagonal prism; trigonal bipyramid; bicone; cylinder.422 Simple and Combined Symmetriesthe internal hexagonal arrangement of the water molecules producedby the hydrogen bonds is responsible for the hexagonal symmetryof snowflakes. This does not explain yet the countless number ofdifferent shapes of snowflakes, and why even the smallest variationsfrom the basic underlying shape of a snowflake are repeated in all sixdirections.According to L. L. Whyte, the translator of Kepler’s New Year’sGift, the snowflake is “an important clue to the shaping agencies ofnature. .
.” [9]. As the puzzling questions concerning snowflakes arerelated to their morphology rather than to their internal structures,these questions will be discussed at some length in the present section.The process of solidification of fluids into crystals has been simulatedby mathematical models. These simulations showed that crystals withsharp tips grew rapidly and had high stability, while crystals withfat shapes grew slowly and were less stable. However, when theseslowly growing shapes were slightly perturbed, they tended to splitinto sharp, rapidly growing tips.
This observation led to the hypothesis of the so-called points of marginal stability [10]. According tothis model, the snow crystal may start with a relatively stable shape.The crystal may, however, be easily destabilized by a small perturbation. A rapid process of crystallization from the surrounding watervapor ensues. The rapid growth gradually transforms the crystal intoanother semi-stable shape.
A subsequent perturbation may then occurresulting again in a new direction of growth with a different rate. Themarginal stability of the snowflake makes the growing crystal verysensitive to even slight changes in its microenvironment.The uniqueness of snowflakes may be related to the marginalstability. The ice starts crystallizing in a flat six-fold pattern of watercrystals so it is growing in six equivalent directions. As the ice isquickly solidifying, latent heat is released which flows between thegrowing six bulges.
The released latent heat retards the growth inthe areas between these bulges. This model accounts for the dendriticor tree-like growth. Both the minute differences in the conditions oftwo growing crystals and their marginal stability make them developdifferently. “Something that is almost unstable, will be very susceptible to changes, and will respond in a large way to a small force”[11].
At each step of growth slightly new micro-environmental conditions are encountered, causing new and new variations in the branches.However, it is assumed that each of the six branches will encounter2.3. Combined Symmetries43exactly the same micro-environmental conditions, hence their almostexact likeness.The marginal stability model is attractive in its explanation of thegreat variety of snowflake shapes. It is somewhat less convincingin explaining the repetitiveness of the minute variations in all sixdirections since the micro-environmental changes may occur alsoacross the snowflakes themselves and not only between the spacesassigned to different snowflakes.In order to explain the morphological symmetry of the dendriticsnow crystals, D.
McLachlan [12] suggested a mechanism decadesago, which has not yet been seriously challenged. He posed the veryquestion already mentioned above: “How does one branch of thecrystal know what the other branches are doing during growth?”McLachlan noted that the kind of regularity encountered among thesnowflakes is not uncommon among flowers and blossoms or amongsea animals in which hormones and nerves coordinate the development of the living organisms.McLachlan’s explanation for the coordination of the growth amongthe six branches of a snow crystal is based on the existence of thermaland acoustical standing waves in the crystal.
As the snowflake growsby deposition of water molecules upon a small nucleus, it undergoesthermal vibrations at temperatures between 250 and 273 K. The watermolecules strike and bounce off the nucleus and those which stay addto the growth. Branching occurs at points with high concentration ofwater molecules. If the starting ice nucleus has the hexagonal shapeshown in Figure 2-22a and the conditions favor dendritic growth,then the six corners would be receiving more molecules and wouldbe releasing more heat of crystallization than the flat portions. Thedendritic development evolving from this situation is shown in Figure2-22b. The next stage in the development of a snowflake is the production of a new set of equally spaced dendritic branches determined bythe modes of vibration along the spines of the flake.
The long spinesof Figure 2-22c are thought to be particular molecular arrays whichcorrespond to the ice structure. The molecules are vibrating and theenergy distribution between the modes of vibration is influenced bythe boundary conditions. When one of the spines becomes “heavilyloaded” at some point, then nodes are induced along this spine. Thesenodes will eject dendritic branches that are equally spaced as indicated in Figure 2-22d–f. The question of how the standing waves in442 Simple and Combined SymmetriesFigure 2-22. McLachlan’s selection [13] of Bentley’s snowflake images [14] toillustrate the coordinated growth of the six branches of a snowflake based on thestanding wave theory.one of the six branches are coupled with those in the other branchesis answered by considering the torque about an axis through theintersection point.
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