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Chapter 1
A First Familiarization
with Geometric Shapes:
The Dense Packings
They have eyes, but don’t see…
The Koran (Surah 7, Verse 179)
Why such a chapter in which neither crystallography nor chemistry appear? The notions presented here, which have the origins in both philosophy and mathematics, first concern the macroscopic Universe, but have also found a marvellous resonance in the nano-world of atomic arrangements as soon as the developments in physics at the beginning of 20th century paved way for this relation. It is therefore through these strange resonances that we shall progressively tackle crystal chemistry.
1.1The Archimedian Polyhedra
The five Platonic polyhedra reflected the regularity and the identity of their faces. A question then arose: can other polyhedra present different kinds of faces, all of them remaining regular? More than a century after the death of Plato, Archimedes of Syracuse (–287/–212), a disciple of Euclid (–330?/–260?), provided the solution by introducing the concept of truncation, first applied to the Platonic polyhedra, and leading to the so-called Archimedian polyhedra, with different regular faces having edges of same length.
Figure 1.1 shows the progressive truncations of the vertices of the cube. They generate successively three Archimedian polyhedra: (i) the truncated cube with six regular octogonal faces and eight equilateral triangles, obtained when the truncation arises at one-third of the length of the edge; (ii) when done at the middle of each edge, it leads to the cuboctahedron, with six square faces and eight equilateral triangles; when it reaches two-thirds of the edge, the truncated octahedon appears (six square and eight regular hexagonal faces). Finally, a truncation corresponding to a full edge gives the octahedron, another Platonic polyhedron with its eight triangular faces.
Fig. 1.1. Construction of three Archimedian polyhedra (noted A) [truncated cube, cuboctahedron and truncated octahedron], and of the Platonic octahedron (P), by progressive truncations of the vertices of a cube.
The octahedron is the dual form of the cube. Duality represents the reciprocity between two polyhedra. It can be illustrated when considering the center of a polyhedron and the lines perpendicular to each face containing this center. Joining together the intersections of these lines with the faces of the original polyhedron provides the shape of the dual polyhedron (Fig. 1.2 for the duality cube-polyhedron).
Fig. 1.2. Geometric construction of the octahedron, the dual form of the cube.
Fig. 1.3. Duality relations and characteristics of the five Platonic polyhedra.
Using this rule, it is easy to show that the dodecahedron is the dual form of the icosahedron and vice-versa. On the contrary, the dual form of the tetrahedron is the tetrahedron itself (Fig. 1.3).
From Fig. 1.3, it is noteworthy that, for two dual forms, (i) the number of faces in one form becomes the number of vertices of the other and vice-versa and (ii) the number of edges remain invariant for the two forms. Whatever the polyhedra, the Descartes–Euler relation between the number F of faces, the number E of edges and the number V of vertices is always verified.
The above rules of truncation applied to the five Platonic polyhedra lead to the 13 Archimedian polyhedra (Fig. 1.4).
All the announced polyhedra are not represented here. Two are missing: the snub cube and the snub dodecahedron. Before truncation, a rotation of the faces is necessary. Figure 1.5 explains their construction, first in the case of the cube. After the surface burst of the cube (which is equivalent to an intermediary rhombicuboctahedron), the initial surfaces of the cube are rotated by 16°. In this case, the squares of the intermediary rhombicuboctahedron are transformed into two triangles with a common edge, and the snub cube is formed.
In Fig. 1.4, under the names of each polyhedron, a succession of numbers appears in red. They correspond to the Schläfli notation (Schläfli: Swiss mathematician (1814–1895)) which allows a reduced definition of these polyhedra, beside their complicated names. In a general way, the first numeral represents the type of face of each polyhedron (3 for triangle, 4 for square, 5 for pentagon, 6 for hexagon, etc…); the second gives the number of each type of face joining at a vertex. For instance, in the notation of a tetrahedron: [3,3], the first 3 means triangle and the second that three triangles meet at a vertex. For the truncated tetrahedron (noted 31 · 62 or (simpler) 3 · 62), 3 and 6 indicate that the faces are both triangular and hexagonal, and that there is one triangle and two hexagons (superscript numerals) joining at a vertex. The notation of the dual forms of the Archimedian polyhedra, which begins with the letter V, gives the number of edges intersecting at each type of vertex and the number of times when each type is associated to a face. For example, the Schläfli symbol of the rhombic dodecahedron (rhombic from the Greek rhombus (lozenge)), which is the dual form of the cuboctahedron, is V(3.4)2. This means that 2 vertices are formed by the intersection of 3 edges, and that 2 other vertices correspond to that of 4 edges (Fig. 1.6).
Fig. 1.4. Construction and names of the Archimedian polyhedra obtained by truncation of Platonic polyhedra.
Fig. 1.5. Construction of the snub cube and of the snub dodecahedron by burst and rotation (the sign ≡ means here equivalent to).
Appendix 1 at the end of the book provides, for each polyhedron, its various geometrical characteristics and also their two-dimensional development for the readers who have difficulties to see in three dimensions and need three-dimensional models which are easy to build with paper.
Fig. 1.6. The rhombic dodecahedron, dual form of the cuboctahedron. It can be generated from a cube by adding on each of its faces a hemi-octahedron whose edges are identical to those of the cube.
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