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All pattern classes based on a square lattice were systematically applied to the cube in a similar manner, utilising an area equal to the whole unit cell. Further constructions involved extracting smaller areas of the unit cell for use as tiles. The area constitutes either a square or equilateral triangle and must be capable of creating the full repeating pattern when symmetries are applied. The smaller the area of the pattern used to tile the polyhedron, the larger the scale of the pattern on the solid.

As shown in Figure 16, the area of the fundamental region is one-third of the unit cell area. All-over pattern class p3m1 is based on a hexagonal lattice, and is produced through a combination of three-fold rotational centres and reflection axes. As indicated in Figure 17, the fundamental region is 25 one-sixth of the unit cell. Centres of three-fold rotation are positioned at the intersections of the reflection axes, which are present along the longest diagonal of the unit cell and alternate with glide-reflection axes in all three directions.

16 / - p6mm hex. ½ ¸ p6mm hex. 16 / - p6 hex. ½ ¸ 5- 2- 3- - - p6mm hex. ½ ¸ 5- 2- 3- ¸ ¸ p4 sq. 1 ¸ 3- 2- 4- - - p4mm sq. 1 ¸ 3- 2- 4- ¸ ¸ p4gm sq. 1 ¸ 3- 2- 4- - - p4 sq. 2 Patterning the tetrahedron Patterning of the tetrahedron is possible with certain patterns possessing two-fold and six-fold rotational symmetry. An area equivalent to half the unit cell forms the repeating unit in each case. When a p2 or c2mm pattern is applied to the tetrahedron (as shown in Figures 42 – 43 and 44 – 45 respectively), the pattern exhibits no rotation at the vertices but tetrahedral axes of two-fold rotation are maintained through the mid-point of opposite edges.

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381st Bomber Group

by George

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