It is a well-known fact that fivefold rotational symmetry is incompatible with crystalline lattice periodicity. Suppose that a Bravais lattice would contain a regular pentagon (Figure l), then it is possible to construct a smaller pentagon that also belongs to the lattice, as can be seen in Figure 1. This implies that a periodic lattice with fivefold symmetry can not satisfy the Delaunay condition. Thus, it came as a big surprise when Shechtman et al. (1) discovered quasicrystals (QC) (2) that exhibited diffraction patterns with five-fold (icosahedral) symmetry and sharp Bragg peaks characteristic of long-range order. Later, QC with eight-fold, tenfold and twelve-fold 2D symmetry were reported. The origin of this apparent paradox lies in a confusion between the concepts of long-range order and periodicity. Quasiperiodic lattices or tilings, such as illustrated in Figure 2 (for eightfold 2D symmetry), are not periodic but do give rise to diffraction patterns with sharp Bragg peaks due to their underlying long-range order. Mathematically, this fact has been known since the beginning of this century due to the work of Harald Bohr (3). In the example of Figure 2, one can recognize many local tile configurations that look as the drawing of a cube in perspective. This observation leaves us actually with a clue as to how such tilings are generated, viz. by a method of cut and projection in a higher dimensional space (4) as illustrated for a one-dimensional quasicrystal, the Fibonacci lattice, in Figure 3a. The physical space, here of dimension d = 1, is embedded in a higher dimensional (d = 2) superspace that contains a hypercubic periodic lattice. The “atoms” of this hyperlattice are the beads and are called the atomic surfaces. Where these atomic surfaces intersect the physical space (also called parallel space), one has an atomic position. If the slope of the parallel space with respect to the basis of the hyperlattice is rational, this procedure will give rise to a periodic one-dimensional lattice. On the other hand, if the slope is irrational, we will obtain a non-periodic structure that is completely ordered, due to the deterministic construction that gave rise to it. This quasiperiodic lattice is characterized by a non-periodic sequence of long (L) and short (C) first neighbor distances. Real 3D QC mostly have icosahedral symmetry and are described by embedding physical space in a 6D superspace. The orthogonal complement of physical space in this superspace is also called perpendicular space. The atomic surfaces here are complicated 3D objects. Several alloys are known now which have resolution limited sharp Bragg peaks, e.g. AlFeCu, AlPdMn. Of the latter centimeters, large single grain samples can be obtained. The mathematical construction of quasicrystals is analogous to the construction of the modulated incommensurate structures, as illustrated in Figure 3. There is, however, an important difference. It has been shown by Levitov (5) that the symmetry requirements make it, in general, impossible to have continuous atomic surfaces in quasicrystals. By shifting the physical space parallel to itself in Figure 3b, a dynamical mode is generated, which is called a phason. Due to the intrinsic discontinuity of the atomic surfaces in QC, the analogous shift gives rise to atomic jumps, which by analogy have been dubbed phasons as well, although this terminology is unfortunate. In the toy model of Figure 2, phasons correspond to jumps inducing a change of perspective in the drawing of the cube, as illustrated on the right.