M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 14
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Structures in which several polyhedra are nested in each other are sometimes called keplerates,referring to Kepler’s beautiful structure of the nested five Platonicsolids [90].There are excellent monographs on regular figures, of which wesingle out those by Coxeter and by László Fejes Tóth as especiallynoteworthy [91]. The Platonic solids have very high symmetries andone especially important common characteristic. None of the rotational symmetry axes of the regular polyhedra is unique, but each axis822 Simple and Combined Symmetries(a)(b)Figure 2-53. Two artistic representations of the regular pentagonal dodecahedron.(a) Pentagonal dodecahedron as part of the sculpture symbolizing “Industry” at theCommons in Boston (photograph by the authors); (b) Leonardo da Vinci’s dodecahedron in a book of Luca Pacioli, De Divina Proportione, published in 1509.Figure 2-54.
Johannes Kepler on Hungarian memorial stamp and his PlanetaryModel based on the regular solids [87].2.8. Polyhedra83Figure 2-55. Nesting of different regular polyhedra in the structure of W6 S8 (PEt3 )6 ,after Saito et al. [89].is associated with several equivalent axes to itself. The five regularsolids can be classified into three symmetry classes:TetrahedronCube and OctahedronDodecahedron and Icosahedron3/2·m = 3/4̃3/4·m = 6̃/4˜3/5·m = 3/10It is equivalent to describe the symmetry class of the tetrahedronas 3/2·m or 3/4̃. The skew line relating two axes means that theyare not orthogonal.
The symbol 3/2·m denotes a threefold axis, and atwofold axis which are not perpendicular and a symmetry plane whichincludes these axes. These three symmetry elements are indicated inFigure 2-50. The symmetry class 3/2·m is equivalent to a combinationof a threefold axis and a fourfold mirror-rotation axis. In both casesthe threefold axes connect one of the vertices of the tetrahedron withthe midpoint of the opposite face.
The fourfold mirror-rotation axescoincide with the twofold axes. The presence of the fourfold mirrorrotation axis is easily seen if the tetrahedron is rotated by a quarterof rotation about a twofold axis and is then reflected by a symmetryplane perpendicular to this axis. The symmetry operations chosen asbasic will then generate the remaining symmetry elements.
Thus, thetwo descriptions are equivalent.Characteristic symmetry elements of the cube are shown inFigure 2-50. Three different symmetry planes go through the centerof the cube parallel to its faces. Furthermore, six symmetry planesconnect the opposite edges and also diagonally bisect the faces.842 Simple and Combined SymmetriesThe fourfold rotation axes connect the midpoints of opposite faces.The sixfold mirror-rotation axes coincide with threefold rotationaxes. They connect opposite vertices and are located along the bodydiagonals. The symbol 6̃/4 does not directly indicate the symmetryplanes connecting the midpoints of opposite edges, the twofold rotation axes, or the center of symmetry.
These latter elements are generated by the others. The presence of a center of symmetry is well seenby the fact that each face and edge of the cube has its parallel counterpart. The tetrahedron, on the other hand, has no center of symmetry.The octahedron is in the same symmetry class as the cube.The antiparallel character of the octahedron faces is especiallyconspicuous. As seen in Figure 2-50, its fourfold symmetry axes gothrough the vertices, the threefold axes go through the face midpoints,and the twofold axes go through the edge midpoints.The pentagonal dodecahedron and the icosahedron are in the samesymmetry class.
The fivefold, threefold and twofold rotation axesintersect the midpoints of faces, the vertices and the midpoints ofedges of the dodecahedron, respectively (Figure 2-50). On the otherhand, the corresponding axes intersect the vertices and the midpointsof faces and edges of the icosahedron (Figure 2-50).Consequently, the five regular polyhedra exhibit a dual relationshipas regards their faces and vertex figures. The tetrahedron is self-dual(Table 2-3).It is an intriguing question as to why there are only five regularpolyhedra? Keeping in mind the definition that a regular polyhedronhas equal and regular polygons as its faces and all of its vertices arealike, the explanation is rather simple.
Take first the simplest regularpolygon, the equilateral triangle, as the face for a polyhedron. Atleast three of them need to join at a vertex to make a solid. This isthe basis for the tetrahedron. When there are four and five of them,the octahedron and icosahedron are obtained, respectively. However,when we try to have six equilateral triangles to join at a commonvertex, they will lie flat, yielding a regular hexagon. Obviously,larger numbers are out, too.
Take now the next regular polygon, thesquare. Three of them at a vertex will yield the cube. Four squares,however, will lie in a plane; thus, with the square, there is only onekind of regular polyhedron. The next regular polygon is the regularpentagon. Joining three of them will eventually lead to the regulardodecahedron. Four of them cannot fit. It is impossible to build a2.8. Polyhedra85Figure 2-56. The four regular star polyhedra [93].
From the left, the small stellateddodecahedron; great dodecahedron; great stellated dodecahedron; and the greaticosahedron. Used by permission from Oxford University Press.polyhedron with three regular hexagons in the vertex since they willlie flat. Here, we reached the limit.If the definition of regular polyhedra is not restricted to convexfigures, their number rises from five to nine [92]. The additional fourare depicted in Figure 2-56; they are called by the common name ofregular star polyhedra.
One of them, viz., the great stellated dodecahedron, is illustrated by the decoration at the top of the Sacristy of St.Peter’s Basilica in Vatican City in Figure 2-57.The sphere deserves special mention. It is one of the simplestpossible figures and, accordingly, one with high and complicatedFigure 2-57. Great stellated dodecahedron as decoration at the top of the Sacristyof St. Peter’s Basilica, Vatican City (photograph by the authors).862 Simple and Combined Symmetriessymmetry. It has an infinite number of rotation axes of infiniteorder. All of them coincide with body diagonals going through themidpoint of the sphere.
The midpoint, which is also a singular point,is the center of symmetry of the sphere. The following symmetryelements may be chosen as basic ones: two infinite-order rotation axeswhich are not perpendicular plus one symmetry plane. Therefore, thesymmetry class of the sphere is ∞/∞·m. Concerning the symmetry ofthe sphere George Kepes quotes Copernicus [94]:.
. . the spherical is the form of all forms mostperfect, having need of no articulation; andthe spherical is the form of greatest volumetriccapacity, best able to contain and circumscribe allelse; and all the separated parts of the world—I mean the sun, the moon, and the stars—areobserved to have spherical form; and all things tendto limit themselves under this form—as appears indrops of water and other liquids whenever of themselves they tend to limit themselves. So no one maydoubt that the spherical is the form of the world, thedivine body.An artistic representation of a sphere is shown in Figure 2-58.Figure 2-58.
Artistic representation of a sphere in front of the World Trade Centerin New York City, which was also destroyed in the terror attack on September 11,2001 (photograph by the authors).No.Namea1Truncated tetrahedron2Truncated cubea3Truncated octahedrona4Cuboctahedronb5Truncated cuboctahedron6Rhombicuboctahedron7Snub cube8Truncated dodecahedrona9Icosidodecahedronb10Truncated icosahedrona11Truncated icosidodecahedron12Rhombicosidodecahedron13Snub dodecahedronaTruncated regular polyhedron.bQuasiregular polyhedron.2.8. PolyhedraTable 2-4. The Thirteen Semiregular PolyhedraNumber ofNumber of rotation axesFacesVerticesEdges2-fold3-fold4-fold5-fold814141426263832323262629212242412482424603060120606018363624724860906090180120150366666615151515151544444441010101010100333333000000000000066666687882 Simple and Combined SymmetriesFigure 2-59.
The 7 special semi-regular polyhedra. First two rows: the so-calledtruncated regular polyhedra; Third row: the quasi-regular polyhedra.In addition to the regular polyhedra, there are various familiesof polyhedra with diminishing degrees of regularity. The so-calledsemi-regular or Archimedean polyhedra are similar to the Platonicpolyhedra in that all their faces are regular and all their vertices arecongruent. However, the polygons of their faces are not all of the samekind.
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