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By Mazurov V.D.
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Additional resources for A Characterization of Alternating Groups II
The cube also exhibits a total of nine planes of reﬂection, illustrated in Figure 33. A net for the cube is shown in Figure 34. 5], as seen in Figure 35. The octahedron is composed of eight equilateral triangular faces, twelve edges and six vertices. 1]. The octahedron displays six axes of two-fold rotation passing through the midpoint of opposite edges, four axes of three-fold rotation connecting the centre of opposite faces and three axes of four-fold rotation joining opposite vertices. In addition to rotational symmetry, the octahedron exhibits nine planes of reﬂection.
A total of nine points of two-fold rotation are evident: at the centre of the unit, at each of the unit corners and the mid-points of the unit sides. The fundamental region occupies half the area of the unit cell, as shown in Figure 11 which illustrates the construction of class p2 on a parallelogram lattice. Figure 11: Example of a p2 all-over pattern 21 All-over pattern class p2mm is based upon either a rectangular or a square lattice, exhibiting two alternating axes of horizontal reﬂection and two alternating axes of vertical reﬂection.
In addition to rotational symmetry, the tetrahedron possesses six planes of reﬂection passing through axes of two-fold rotation and the edges of the tetrahedron. As noted previously, the tetrahedron is its own dual polyhedron and therefore connecting the centres of the faces of a tetrahedron forms another tetrahedron. The symmetry characteristics of the tetrahedron are illustrated in Figure 31 and the relevant net for the tetrahedron is shown in Figure 32. 3 The cube The regular hexahedron, more commonly known as the cube, consists of six square faces that meet at right angles, any of which may be regarded as the base.