M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 61
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It has a simple rod shapewith a regular helical array of protein molecules and there is a singlestranded ribonucleic acid molecule embedded in this protein coat. Themodel of TMV is shown in Figure 8-19 in two representations. AaronKlug called attention to an important difference between ordinarypolymers and biological macromolecules: “The key to biologicalspecificity is a set of weak interactions. A polymer chemist could startbuilding the model in the middle or at any other point.” However, forbuilding a model for a biological macromolecule, it is “important tofind the special sequence for initiating nucleation” [26].Whereas helical symmetry is characterized by a constant amount oftranslation accompanied by a constant amount of rotation, in spiralsymmetry the amounts of translation and rotation change graduallyand regularly.
D’Arcy Thompson capitalized this word in his description, “. . .a Spiral is a curve which, starting from a point of origin,Figure 8-19. Models of the tobacco mosaic virus (TMV) structure; Left: AaronKlug with the model at the Laboratory of Molecular Biology, Cambridge, UK(photograph by the authors); Right: graphical representation of the model (courtesyof Aaron Klug, Cambridge, UK).3928 Space-Group Symmetriescontinually diminishes in curvature as it recedes from that point; or,in other words, whose radius of curvature continually increases. .
.”[27]. An important difference between helices and rods is that whilehelices can only form around rods, spirals may form along a rod orin a plane and the scattered leaf arrangement and the sunflower seedplate may serve as their respective examples. An artistic double spiralis seen in Figure 8-20 as detail of a sculpture from the garden of aresearch institute where structures of biological macromolecules areinvestigated.Interesting chemical examples of spirals occur in systems withchemical oscillations. Oscillating reactions are often called Belousov–Zhabotinsky reactions. Boris Belousov communicated his first observation in an obscure Russian medical publication [28] in the 1950s andit was followed by Anatol Zhabotinsky’s first systematic studies [29]in the 1960s.
Although the chemical community was slow in catchingup and many viewed the first reports on oscillating reactions withscepticism, research on nonlinear chemical phenomena has greatlyexpanded by now along with research of nonlinear phenomena inFigure 8-20. Artistic double spiral. Detail of a sculpture in the garden of theWeizmann Institute, Rehovot, Israel (The Inner Light by Gidon Graetz; photographby the authors).8.4. Rods, Spirals, and Similarity Symmetry393(a)(b)(d)(c)(e)Figure 8-21.
(a) Spiral ring pattern in a reacting Belousov-Zhabotinsky system[30]; (b) Tombstone in the Jewish cemetery, Prague (photograph by the authors);(c) Galaxy, courtesy of Bruce Elmegreen, Yorktown Heights, New York [31]; (d)Nautilus (photograph by and courtesy of Lloyd Kahn, Bolinas, California); (e) Fernfrom the Big Island, Hawaii (photograph by the authors).other fields. Figure 8-21 illustrates the spiral structure in a Belousov–Zhabotinsky reaction. The two spirals shown make a heterochiral pair,paralleled by one on a tombstone in Figure 8-21. Spirals abound innature.
Some examples are shown also in Figure 8-21.A gradual and regular change in size may appear by itself, that is,without being part of a spiral. A regular change in size characterizes,for example, homologous series, such as the alkanes, Cn H2n+2 ,3948 Space-Group Symmetries. . . C4 H10 , C5 H12 , C6 H14 , C7 H16 , . . .with the increment of a methylene group, CH2 . Examples from outsidechemistry are shown in Figure 8-22, where, again, it is up to ourimagination to extend the series to infinity. All the spirals above andthe phenomenon of phyllotaxis as well as the homologous series, theseries of railway wheels, and the family of mountain goats can beconsidered as examples of similarity symmetry [32].
The mathematical concept of similarity holds one of the keys to understanding theprocesses of growth in the natural world [33]. Similarity symmetry isa good example of the convenient extension of the symmetry concept.In addition to the definitions offered in the Introduction, here, wesuggest another one: A pattern is symmetrical if there is a simple ruleFigure 8-22. Examples of similarity symmetry (photographs by the authors). Top:railway wheels, Foundry Museum, Budapest; Bottom: mountain goats, BudapestZoo.8.5. Two-Dimensional Space Groups395Figure 8-23.
Fractals decorating a book cover [34]. (Computer graphics by andcourtesy of Clifford A. Pickover, Yorktown Heights, New York).to generate it. According to this inclusive definition, for example,Mandelbrot’s fractals fall naturally into the realm of symmetry. Anexample is shown in Figure 8-23.8.5. Two-Dimensional Space GroupsThere are altogether 17 symmetry classes of one-sided planarnetworks. Figure 8-24 illustrates them in a way analogous to the sevensymmetry classes of the one-sided bands (Figs. 8-5 and 8-6). Themost important symmetry elements and the coordinate notations of thesymmetry classes are also given.
The first letter (p or c) in this notation refers to translation. The next three positions carry informationon the presence of various symmetry elements, where m denotes asymmetry plane, g a glide-reflection plane, and 2, 3, 4, or 6 denotesa rotation axis.
The number 1, or a blank, indicates the absence ofa symmetry element. The representations of the symmetry classesin Figs. 8-5 and 8-24 were inspired by the illustrations inside thecovers of Buerger’s Elementary Crystallography [35]. Along with thepurely geometrical configurations, Figure 8-24 presents 17 Hungarian3968 Space-Group SymmetriesFigure 8-24. (Continued on next page)8.5. Two-Dimensional Space Groups397Figure 8-24. The 17 symmetry classes of one-sided planar networks with the mostimportant symmetry elements and the notations of the classes indicated.
Along withthe geometrical configurations, Hungarian needlework patterns are presented forillustration. A brief description of the origin of these patterns is given here [36]:p1 and p4 Patterns of indigo dyed decorations on textiles for clothing. Sellye,Baranya county, 1899.p2 Indigo dyed decoration with palmette motif for curtains. Currently verypopular pattern.p3, p6, p6mm, p3m1, and p31m Decorations with characteristic bird motifs frompeasant vests.
Northern Hungary.pm Decoration with tulip motif for table-cloth. Cross-stitched needlework. Fromthe turn of the last century.pmm2 Bed-sheet border decoration with pomegranate motif. Northwest Hungary,19th century.p4mm Pillow-slip decoration with stars. Cross-stitched needlework. Transylvania,19th century.cm Pillow-slip decoration with peacock tail motif. Cross-stitched needlework.Much used throughout Hungary around the turn of the last century.cmm2 Bed-sheet border decoration with cockscomb motif. Cross-stitched needlework.
Somogy county, 19th century.pg From a pattern-book of indigo dyed decoration. Pápa, Veszprém county, 1856.pgg2 Children’s bag decoration. Transylvania, turn of the last century.pmg2 Pillow-slip decoration with scrolling stem motif. Much used throughoutHungary around the turn of the last century.p4gm Blouse-arm embroidery. Bács-Kiskun county, 19th century.3988 Space-Group Symmetriesneedlework patterns. A scheme for establishing the symmetry class ofone-sided two-dimensional space groups is given in Figure 8-25.The lattice of the planar networks with two-dimensional spacegroups is defined by two noncollinear translations. Such a lattice isshown in Figure 8-26a.
Given a particular lattice, the question is,which pair of translations should be selected to describe it? An infinitenumber of choices exists for each translation because a line joiningany two lattice points is a translation of the lattice. Figure 8-26b showsa plane lattice and some of the possible choices for translation pairsto describe it. A primitive cell is defined by choices of translationpairs such as t1 and t2 or t3 and t4 . Only one lattice point is associatedwith each primitive cell. This is understood if each lattice point inFigure 8-26 is considered to belong to four adjacent cells, or only onefourth of each point to belong to any one cell.
As each cell containsfour corners, all this adds up to one whole point. Alternatively, bydisplacing any one primitive cell, each primitive cell will contain onlyone lattice point. On the other hand, a multiple cell contains one ormore lattice points in addition to the one shared at the corners. Thetranslation pair t5 and t6 , for instance, defı̀nes a double cell.
A cell iscalled a unit cell if the entire lattice can be derived from it by translations. Thus, a unit cell may be either primitive or multiple. The unitcell is chosen usually to represent best the symmetry of the lattice.The translations selected as the edges of the plane unit cell are a andb, and for a space lattice, a, b, and c.
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