Systematic investigations at Tübingen on long-range aperiodic order started out from point symmetry forbidden in a periodic lattice and employ the dual cell geometry of a lattice of dimension n > 3.

With my research group at Tübingen since 1982 I systematically explored the projection method from an n-dimensional lattice Λ.

A first central idea was to fix the irrational section in Bohr’s theory by the
selection of a non-crystallographic point group. Therefore the n-dimensional
lattices were chosen such that they were compatible with a representation of a
non-crystallographic point group G. The linear subspace E_{∥}^{m} for the quasicrystal,
the irrational section according to Bohr, was then uniquely determined by
decomposing the representation of the point group into its irreducible parts. A
subspace E_{∥}^{m} with irreducible but non-crystallographic representation D(G) then
carries the construction of the quasicrystal structure. To characterize this
structure we employed the geometry of the lattice Λ and its projection to
E^{m}.

A second central idea of our analysis was the notion of duality in this geometry.
We considered the geometry of high-dimensional Voronoi- or Wigner-Seitz cells
of E^{m}, defined by all points closer to a fixed lattice point than to any other one.
The vertices of these Voronoi cells, called the holes of the lattice, form the centers
of Delone cells dual to the Voronoi cells. For the concept of holes in lattices
we refer to [8]. We studied the hierarchy of m-dimensional boundaries,
0 ≤ m ≤ n, abbreviated as m-boundaries, of both Voronoi and Delone cells
associated with a lattice. We found that these boundaries admit a general
notion of duality. Any m-boundary X of a Voronoi cell can be described
as the intersection of the Voronoi cells for a finite set of lattice points.
This finite set of lattice points has a convex hull of dimension (n-m). We
showed that this convex hull is a dual (n-m)-boundary denoted as X^{*}, from
a Delone cell. This duality relation we took as a basis for the study of
quasiperiodic tilings. For m=0, the 0-boundaries of Voronoi cells are the holes
of the lattice. The duals to these 0-boundaries are precisely the Delone
cells.

Example 2.3: Duality in the hexagonal lattice A_{2}. We illustrate duality in
Fig. 5 for the root lattice A_{2}. Here we use the standard terminology of root
lattices as given for example in [8].

Fig.5. Duality in the lattice A_{2}. Upper part: The root lattice A_{2} has
hexagonal Voronoi or Wigner-Seitz cells. centered at lattice points (black squares).
Note that adjacent vertices (black and white circles) of the hexagonal
Voronoi cell are inequivalent under lattice translations. Lower part: The
dual is a tiling by two types of triangular Delone cells, centered at the
vertices of the hexagonal Voronoi cells. Duality extends to all boundaries
of complementary dimension: Any Voronoi cell is dual to a vertex of a
Delone cell, any edge of a Voronoi cell dual to an orthogonal edge of a
Delone cell, and any vertex of a Voronoi cell is dual to a Delone cell.

A particular case of duality arises for hypercubic lattices Z^{n} in E^{n}. The Voronoi
cells are hypercubes, and the holes form a second hypercubic lattice, shifted
from the vertices to the centers of the hypercubes. Duality still work,
but there is no geometric distinction between Voronoi and Delone cells
and tilings based on them. This particular case appears already in the
projection of the Fibonacci tiling from the square lattice Z^{2} E^{2}, Fig.
6.

Fig. 6. The Fibonacci tiling. (a) The square lattice and its dual: The
square lattice Z^{2} (lattice points black squares) in the plane E^{2}. Its dual Delone
cells are squares located at the holes (white squares). The irrational sections E_{∥}^{1}
are lines x_{∥} of slope τ. (b) Periodic tiling adapted to the irrational section:
We devise a new periodic tiling by two squares (A,B) of edge length
(τa,a). They are constructed locally by projecting dual pairs of edges from
Voronoi and Delone cells X,X^{*} (broken and dotted lines indicated on
the right) parallel or perpendicular to the section x_{∥} and forming the
product polytopes. (c) Quasiperiodic tiling on the irrational section:
Any horizontal line parallel to x_{∥} is tiled by the vertical edges of the two
squares (A,B) in a Fibonacci tiling with two tiles of length (τa,a). (d)
Tile windows: The perpendicular vertical projections of the dual edges
provide individual windows for the tiles. (e) Compatibility: A general
periodic function f^{p} on E^{2} can be defined on the two squares (A,B) since
these together form a fundamental domain. The values of f^{p} restricted
to a horizontal line yield a general quasiperiodic function. If moreover
the values of f^{p} on (A,B) are chosen independent of the perpendicular
coordinate x_{⊥}, its quasiperiodic restriction to a parallel line becomes
compatible with the Fibonacci tiling. Then the values of f^{p} on the two
Fibonacci tiles fix a compatible quasiperiodic function and so for them form a
fundamental domain for the tiling. (f) Covering: The parallel projections
D_{∥} of the Delone cells, black horizontal sections, yield a covering of the
Fibonacci tiling. Any such projection comprises a fundamental domain for the
tiling.

Example 2.4: Scaling in the lattice Z^{2}: For the Fibonacci projection we
describe the scaling property of the lattice Z^{2}, compare [33]. First of all we write
down in terms of two unit length orthogonal column vectors a basis matrix
B = (b^{1},b^{2}),

| (7) |

These two basis vectors connect a lattice point to two next neighbours (black
squares) in Fig. 6. The upper and lower row of B describe the projections of the
basis vectors (b^{1},b^{2}) along the directions (x_{
∥},x_{⊥}) in Fig. 6. Now it can be checked
that the basis matrix B obeys the equation

| (8) |

The right-hand matrix multiplication describes a linear transformation with
determinant Δ = -1 of the basis vectors with integer coefficients, that is a
symmetry transformation of the lattice Z^{2}. On the left-hand side, this
multiplication is expressed in the form of a scaling of the basis vectors with
scaling factors (τ,-τ^{-1}) in the parallel and the perpendicular direction
respectively. So the scaling in parallel space by τ can be interpreted as
part of a transform of the lattice into itself. When this scaling is applied
to the two original tiles of length (a,τa), it scales them into two tiles
of length (τa,τ^{2}a = (τ + 1)a). The scaled tiles can be decomposed into
the original tiles. By repeating this scaling one can generate a Fibonacci
string of arbitrary length. In any step of this generation, the frequency
of occurence of each of the two tiles is given by successive /Fibonacci
numbers.

We return to the general projection from a lattice. We found that, when m
becomes the dimension of the physical or parallel space E_{∥}, this linear subspace is
tiled by the parallel projections of m-boundaries from Voronoi cells. The
perpendicular projections to the complementary subspace E_{⊥}^{n-m} plays an
important role in the dual projection technique: it determines the so called
windows of the objects. The role of a window in E^{n} for the tiling on E^{m} is
defined by the rule: Whenever the irrational section E^{m} within E^{n} hits a window
for a tile in E^{n}, the tile will appear in the tiling. Specifically, if the tiling
consists of parallel projections of m-boundaries say from Voronoi cells,
the windows for these tiles are the perpendicular projections of the dual
(n - m)-boundaries. The windows for the vertices of the canonical tilings
( , Λ), (^{*}, Λ) projected from Voronoi or Delone cells, are the perpendicular
projections of the set of Delone cells and of Voronoi cells of Λ respectively.
Examples of vertex windows for the icosahedral tilings are shown in Figs. 10,
11.

Choosing as an alternative the projections of m-boundaries of Delone cells, E_{∥} is
again tiled by these. For given lattice and projection, the dual Voronoi and Delone
boundaries provide two alternative canonical tilings which we denote as
( , Λ), (^{*}, Λ) respectively. The general scheme was developed in the doctoral
thesis of M. Schlottmann and published in [26].

Fig. 7. Penrose rhombus tiling . The planar Penrose tiling ( ,A_{4}) is formed by
two types of rhombs. They can be constructed as projections of 2-boundaries from
the Voronoi cells of the root lattice A_{4} in E^{4}.

In [1] (1990), I studied with M Baake, M Schlottmann and D Zeidler the root
lattice A_{4} in 4-dimensional space. We showed that the projection of 2D Voronoi
boundaries yields the Penrose rhombus tiling Fig. 7, and the dual projection of
Delone boundaries yields the Tübingen triangle tiling Fig 8. These tilings were
used to model decagonal quasicrystals [29], [30] and were related to the Burkov
model.

Fig. 8. Tübingen triangle tiling. (a) The planar Tübingen triangle tiling:
(^{*},A_{
4}) is formed by two types of triangles, constructed as projections of
2-boundaries from the Delone cells of the root lattice A_{4} in E^{4}. (b) Dual
projections: This tiling and the Penrose tiling shown in Fig. 7 exemplify dual
projections from the same lattice. (c) Pentagon covering: A covering of the
triangle tiling is given by two types of pentagons, projections of two types of
Delone cells from A_{4}, shaded in the Figure. The four shaded pentagons together
comprise 20 triangle tiles in all possible orientations.

The lattice A_{4} also has a scaling symmetry [1]. The two dual tilings ( ,A_{4}),
(^{*},A_{
4}) admit two inequivalent Robinson-decompositions into triangles of the
shapes appearing in Fig. 8. From the scaling symmetry there result two different
stone-inflations of these triangular tilings. The stone-inflation for the Penrose
tiling is shown in Fig. 9. This stone inflation can be used to construct
quasiperiodic wavelets. The details are given in [31].

Fig. 9. Robinson decomposition of the Penrose tiling. The Penrose tiles can be rearranged into two triangles of the same shape as in the triangle tiling Fig. 8, but with different matchings. The orientation of any triangle is marked by a black dot. The tiles admit a stone inflation: When scaled by τ = (1 + ), each tile can be decomposed into copies of unscaled preimages.

Work on icosahedral projections and tilings was extended with Z Papadopolos and
D Zeidler in [27] (1992) by analyzing the root lattice D_{6}, compare [8], also
denoted as the face-centered hypercubic lattice in 6 dimensions. This lattice is
inequivalent to the lattice Z^{6}. The lattice D_{
6} can be identified by indexing its
diffraction pattern which belongs to its reciprocal lattice, the so called
body-centered hypercubic lattice. The lattice D_{6} occurs in stable phases of
AlMnCu. We constructed the six tiles for both the Voronoi- and Delone-based
tilings ( ,D_{6}) and (^{*},D_{
6}).

Fig. 10. Icosahedral tilings from Voronoi cells of the root lattice D_{6}. Top:
The three types of Delone cells D^{a},D^{b},D^{c} of the root lattice D_{
6} in E^{6} project in
perpendicular space on two icosahedral and one dodecahedral polytope. They
form the window of the tiling ( ,D_{6}) and of its three types of vertices. The
relative frequencies of the three types of vertices correspond to the volumes of the
projected Delone cells. Bottom: The tiles are two rhombohedra and four
pyramids on a rhombus base. Their vertices marked by black circles, double and
single circles are projections of the three types (a,b,c) of holes in the lattice.

The lattice has three types a,b,c of translational inequivalent holes, that is,
vertices of its Voronoi cells, and hence three types of dual Delone cells. Under
icosahedral projection, the tiling ( ,D_{6}) has the six projections of the
3-boundaries of Voronoi cells as tiles. These are shown in Fig. 10. We showed in
[28] that the tiling ( ,D_{6}) is equivalent to a tiling constructed with methods of
scaling by Danzer [9]. This tiling in turn is related to a tiling constructed by
Levine and Steinhardt [35], [36] which in this way is shown to belong to the D_{6}
module.

Fig. 11. Icosahedral tiling from Delone cells of the root lattice D_{6}. Top:
The Voronoi cell of D_{6} has as perpendicular projection Kepler’s triacontahedron.
Its vertices are marked as in Fig. 10. It forms the window for the tiling (^{*},D_{
6})
and its vertices. Bottom: The tiles are six tetrahedra projected from
3-boundaries of the three Delone cells of D_{6}. Their edges have two sizes scaled by
τ.

The second tiling (^{*},D_{
6}) based on duality has the six projected 3-boundaries of
the Delone cells, Fig. 11, as its tiles. The window for this tiling, the projection of
the Voronoi domain of D_{6}, is again Kepler’s triacontahedron [16]. This tiling can
be related to the one by Mosseri and Sadoc [40].

For properties of root lattices we refer to [8]. A condensed description of the dual projection method is given in [33]. Up to now, all the tiling constructions for decagonal and icosahedral quasicrystals were shown by us to be related to the canonical tilings based on duality.