M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 7
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Also, leaves often have bilateral symmetry, butit may be only accidental for a tree. Generally, trees and many otherFigure 2-3. Animals of bilateral symmetry (photographs by Zoltán Bagosi,Budapest Zoo, used with permission).2.1. Bilateral Symmetry29Figure 2-4. Flowers and leaves of bilateral symmetry (photographs by the authors).plants have radial, cylindrical, or conical symmetries with respectto the trunk and stem. Although these symmetries may occur in avery approximate way, they can be recognized without any ambiguity(Figure 2-5).Figure 2-5. Conical and radial symmetries of trees (photographs by the authors).302 Simple and Combined SymmetriesThe symmetry plane of the human face is sometimes emphasizedby artists (Figure 2-6a–c) while other artists idealize the faces theypresent (Figure 2-6d–f).
Of course, there are minute variations, oreven considerable ones as we age, between the left and right sidesof the human face (see, e.g., Figure 2-7). Differences between the leftand right hemispheres of the brain have been the subject of intensivestudies [5].Bilateral symmetry has outstanding importance in man-madeobjects due to its functional role.
The bilateral symmetry of variousvehicles, for example, is determined by their translational motion.On the other hand, the cylindrical symmetry of the Lunar Moduleis consistent with its function of vertical motion with respect to the(a)(d)(b)(e)(c)(f)Figure 2-6. Human faces in artistic expression. (a) Henri Matisse, Portrait ofLydia Delektroskaya (reproduced by permission from the State Hermitage Museum,St.
Petersburg); (b) Jenő Barcsay, Woman’s head (used with permission from Ms.Barcsay); (c) George Buday, Miklós Radnóti, wood-cut, 1969 (used with permission from George Buday, R. E.); (d) Buddha sculpture in Japan; (e) St. Peter at theSt. Peter’s Square, Rome; (f) Bust of D. I. Mendeleev in front of Moscow StateUniversity (d; e; f, photographs by the authors).2.1. Bilateral Symmetry(a)31(b)(c)Figure 2-7. Professor Alan L. Mackay’s face; (a) Right-side composite; (b) Original; (c) Left-side composite (photographs by the authors, used with permission ofAlan L. Mackay, London).moon’s surface.
Examples of cylindrical symmetry, related to the preferential importance of the vertical direction are the stalactites andthe stalagmites in caves (Figure 2-8), formed of calcium carbonate.The occurrence of radial-type symmetries rather than more restrictedones necessitates a spatial freedom in all relevant directions.
Thus,for example, the copper formation in Figure 2-9a has a tendency toform cylindrically symmetric structures. On the other hand, the solidified iron dendrites obtained from iron-copper alloys, after dissolvingaway the copper, display bilateral symmetry in Figure 2-9b.Both folk music and music by master composers are rich in symmetries. Figure 2-10 shows two examples with bilateral symmetry.
Thefirst example is from Bartók’s Microcosmos series written specifi-Figure 2-8. Calcium carbonate stalactites and stalagmites in a cave in southernGermany (photographs by the authors).322 Simple and Combined Symmetries(a)(b)Figure 2-9. (a) Electrolytically deposited copper, magnification × 1000. Courtesyof Maria Kazinets, Beer Sheva; (b) Directionally solidified iron dendrites from aniron-copper alloy after dissolving away the copper, magnification × 2600. Courtesyof J. Morral, Storrs, Connecticut.(a)(b)(c1)(c2)Figure 2-10. (a) Bartók: Microcosmos, Unisono No. 6. The vertical dasched lineindicates the plane of reflection; (b) Bartók: Microcosmos, Unisono No.
1; (c) Drawings inspired by the Unisono No. 1 by early teenagers, Komló Music School (courtesy of Mária Apagyi, Pécs, Hungary).2.2. Rotational Symmetry33Figure 2-11. Double-headed eagles in (left) Madrid; (center) Prague; and (right)Moscow (photographs by the authors).cally for children. Figure 2-10a of Unisono No. 6 illustrates a mirrorplane which includes a sound. The introductory piece of the Microcosmos is depicted in Figure 2-10b. It has only approximate bilateralsymmetry though the two halves are markedly present.
When someschool children in their early teens were asked to express their impressions in drawing while listening to this piece of music for the firsttime, they invariably produced patterns with bilateral symmetry. Twoof the drawings are reproduced in Figure 2-10c.Weyl calls bilateral symmetry also heraldic symmetry as it is socommon in coats of arms [6]. Characteristically, the eagles of theHabsburgs (Figure 2-11a and b) and the Russian Romanovs (Figure2-11c) were double headed.2.2.
Rotational SymmetryThe contour of the simple and powerful oriental symbol yin yang(Figure 2-12a) has twofold rotational symmetry in that a half rotationabout the axis perpendicular to the midpoint of the drawing bringsback the original figure. This rotation axis is a symmetry axis. TheTaiwanese stamp with two fish (shown in Figure 2-12b) is reminiscent of yin yang.The order of a rotation symmetry axis tells us how many timesthe original figure reoccurs during a complete rotation.
The elementalangle is the smallest angle of rotation by which the original342 Simple and Combined Symmetries(a)(b)Figure 2-12. Contour of yin yang and two fish on a Taiwanese stamp.figure is reproduced. Thus, for twofold rotational symmetry, theorder of the rotation axis is two and the elemental angle is 180◦ .The corresponding numbers for threefold, fourfold, etc., rotational◦symmetries are three and 120◦ , four and 90 , etc., respectively. Theorder of rotation axes (n) may be 1, 2, 3, . . .
up to infinity, ∞, thusit may be any integer. The order 1 means that a complete rotation isneeded to bring back the original figure, thus there is a total absenceof symmetry which means asymmetry. A one-fold rotation axis is anidentity operator. The other extreme is the infinite order; the circlehas such symmetry. This means that any, even infinitesimally smallrotation leads to congruency.Figures 2-13–2-15 illustrate rotational symmetries in flowers,rotating parts of machinery, and hubcaps. Seldom does exclusively rotational symmetry have functional importance in flowers.In contrast, the motion of rotating parts in machinery is reinforcedby having only rotational symmetry and no symmetry planes.
ThereFigure 2-13. Hawaiian flowers with only rotational symmetry (photographs by theauthors).2.2. Rotational Symmetry35Figure 2-14. Rotating parts of machinery (photographs by the authors).are hubcaps with rotational symmetry only, but just as well can ahubcap with multiple symmetry planes fulfill its function—servingas protection and decoration. It is only our perception that mightfavor a hubcap with rotational symmetry over a hubcap with highersymmetry.
The perception is that rotation favors motion whereassymmetry planes stop motion. This is why we suggest that recycling companies, banks, and transportation companies often chooselogos with rotational symmetry only. A small sampler of examples ispresented in Figure 2-16. Finally, a curious appearance of rotationalonly symmetries is found in sculptures of two, three or even moreinterweaving fish and dolphins (Figure 2-17).Figure 2-15. Hubcaps with only rotational symmetry (photographs by the authors).362 Simple and Combined Symmetries(a)(b)(c)Figure 2-16.
Logos: (a) Recycling; (b) Banking; (c) Transportation (photographsby the authors).(a)(b)(c)Figure 2-17. Sculptures of interweaving fish and dolphins: (a) Twofold in Washington, DC; (b) Threefold in Prague; (c) Fourfold in Linz, Austria (photographs bythe authors).2.3. Combined Symmetries372.3.
Combined SymmetriesThe symmetry plane and the rotation axis are symmetry elements.If a figure has a symmetry element, it is symmetrical. If it has nosymmetry element, it is asymmetrical. Even an asymmetrical figurehas a one-fold rotation axis; or, actually, an infinite number of onefold rotation axes.The application of a symmetry element is a symmetry operation andthe symmetry elements are the symmetry operators. The consequenceof a symmetry operation is a symmetry transformation. Strict definitions refer to geometrical symmetry, and will serve us as guidelinesonly.
They will be followed qualitatively in our discussion of primarilynon-geometric symmetries, according to the ideas of the Introduction.So far symmetries with either a symmetry plane or a rotationaxis have been discussed. These symmetry elements may also becombined. The simplest case occurs when the symmetry planesinclude a rotation axis.2.3.1. A Rotation Axis with Intersecting Symmetry PlanesA dot between n and m in the label n·m indicates that the axis is inthe plane.
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