**Figure 2**
The geometric principle can be encoded in a tiling. The figure demonstrates the relation of extended symmetry groups with tilings. For simplicity, the principle is demonstrated for the two-dimensional case of tenfold symmetry; the same principle has been applied to icosahedral symmetry in three dimensions and has resulted in the library of point arrays used here (Keef & Twarock, 2009; Wardman, 2012). (*a*) A decagon, a geometric representation of tenfold symmetry, superimposed on a Penrose-type tiling such that its corners coincide with vertices of the tiling. (*b*) Addition of a translation to the rotational symmetries of the decagon (*i.e.* an affine extension of tenfold rotational symmetry) results in translated copies of the decagon (shown in blue) with corners also coinciding with vertices of the tiling. (*c*) Subsequent rotations about the tenfold axis at the centre of the original decagon result in ten copies of the translated decagon. (*d*) Since corner points of the decagon are geometric representations of the (rotational) symmetries, the (artificial) decagonal edges are faded away. An iterative process of translation and rotation leads to the addition of further points. For example, in the second iteration step the red points in (*e*) are added and more vertices of the tiling are covered. |