# Five-fold symmetry in crystalline quasicrystal lattices

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To demonstrate that crystallographic methods can be applied to
index and interpret diffraction patterns from well-ordered
quasicrystals that display non-crystallographic 5-fold symmetry, we
have characterized the properties of a series of periodic
two-dimensional lattices built from pentagons, called Fibonacci
pentilings, which resemble aperiodic Penrose tilings. The computed
diffraction patterns from periodic pentilings with moderate size unit
cells show decagonal symmetry and are virtually indistinguishable from
that of the infinite aperiodic pentiling. We identify the vertices and
centers of the pentagons forming the pentiling with the positions of
transition metal atoms projected on the plane perpendicular to the
decagonal axis of quasicrystals whose structure is related to
crystalline η phase alloys. The characteristic length scale of the
pentiling lattices, evident from the Patterson (autocorrelation)
function, is ∼τ^{2} times the pentagon edge length,
where τ is the golden ratio. Within this distance there are a finite
number of local atomic motifs whose structure can be
crystallographically refined against the experimentally measured
diffraction data.

Five-fold symmetry has been associated with magic and mysticism
since ancient times. Kepler, in his *Mysterium
Cosmigraphicum*, published 400 years ago, described how he
ingeniously found the symmetry of the five Platonic polyhedra in the
structure of the solar system. Book II of his *Harmonices
Mundi* (1), on the congruence of harmonic figures, is a pinnacle in
the history of geometry, combining imaginative mathematical mysticism
with profound insights into the symmetry of polyhedra and polygonal
tilings of the plane. Kepler’s exploration of orderly arrangements of
plane pentagons has been viewed (2) as an anticipation of Penrose’s
aperiodic tilings (3), which have served as models for the geometry of
quasicrystal structures.

Quasicrystallography has developed into an elaborate discipline since
1984 when Shechtman *et al.* (4) first reported crystal-like
diffraction patterns with forbidden icosahedral symmetry from
aluminum–manganese alloys, and Levine and Steinhardt (5) coined the
name quasicrystals for the class of quasiperiodic structures.
Exposition of the results of many experimental studies on these novel
alloys, and of the efforts of physicists to model their properties are
presented in the book *Quasicrystals: A Primer*, by Janot (6);
the mathematical concepts involved in the construction of aperiodic
lattices are described in *Quasicrystals and Geometry*, by
Senechal (2).

In their endeavors, quasicrystallographers have used a variety of mathematically sophisticated but physically unrealistic models to analyze aperiodic lattices with icosahedral or decagonal symmetry. Quasicrystal structures have been represented as projections into two- or three-dimensional space from periodic models in five- or six-dimensional space. For example, such procedures have been applied by Steurer and his colleagues to calculate five-dimensional Fourier maps from three-dimensional x-ray diffraction patterns of decagonal-phase aluminum-transition metal alloy quasicrystals (7–9). Projections from these physically abstract five-dimensional constructs produce real space maps, which show correlations with the crystallographically determined atomic arrangements in related periodically ordered alloys (10–12). The success of this five-dimensional quasicrystallographic analysis suggests that, because the diffraction data is only observable in three-dimensional reciprocal space, more conventional crystallographic analysis might be applied to refine real space models of the atomic arrangements in these quasicrystals.

Quasicrystals are, by definition, aperiodic lattices. The diffraction pattern from one portion of such a lattice is indistinguishable from that of another portion. A representative portion of a quasicrystal lattice can be chosen as a large unit cell of a perfectly periodic lattice, which would yield the same diffraction pattern as the aperiodic lattice. A great variety of such periodic lattices can be constructed by selecting different portions of the aperiodic lattice as the unit cell. The fact that such lattices exist suggests that one member of this class might be transformed into any other member by localized displacive rearrangements of the constituent atoms.

Our surmise is that quasicrystals with icosahedral or decagonal symmetry may be modeled by periodic packing arrangements of icosahedra or pentagons in moderate-size unit cells that can be locally rearranged, conserving key bonding relations, to generate aperiodic lattices. In this paper, we focus on regular arrangements of pentagons in the plane, applying the same sort of packing rules as used by Dürer (13), Kepler (1), and Penrose (3) in their explorations of pentagonal tilings. The designs of these regular pentagonal tilings are related to the arrangement of transition metal atoms projected on the plane perpendicular to the axes of local 5-fold symmetry in the alloys with aluminum of the crystallographically regular η phase (10–12) and the decagonal quasicrystals (7, 8).

## Graphics Methods

To visualize the regular arrangements of pentagons (pentilings), their relation to crystal structures, quasicrystal diffraction patterns and Patterson functions, special purpose graphics routines were developed. All images were created and rendered using unique code in the postscript language (14). Pentilings were created using recursive routines, and coordinates needed for Fourier analyses were generated from the postscript code using the Aladdin Ghostscript interpreter. Once in Protein Data Bank (PDB) format, the coordinates were used with the ccp4 package (15) to calculate electron density maps, structure factors, and Patterson maps. The maps were converted to grayscale images and then embedded in postscript documents. The construction of twinned lattice images and montages used the postscript clipping and superposition capabilities.

## Pentilings

We define a pentiling as an arrangement of regular pentagons in
the plane in which each pentagon makes edge-to-edge contact with two,
three, four, or five neighbors, thereby sharing vertices in such a way
that no gaps large enough to contain another pentagon are left in the
array. A periodic pentiling is a regular lattice with *P*
pentagons in the unit cell. *P* is called the pentile number.

The simplest and most compact periodic pentiling is the first tile
pattern formed by pentagons described by Dürer (13) in *A
Manual of Measurements of Lines, Areas, and Solids by Means of Compass
and Ruler*, which he published in 1525. This tile pattern (which
was illustrated by Dürer in his figure 24) is shown in Fig.
1*a*. The pentile number for this
lattice is *P*_{0} = 2, the zero subscript indicating
that this is the fundamental member of its class.

The repeating motif of the *P*_{0} = 2 pentiling
(Fig. 1*a*) consists of the two regular pentagons of edge
length *E* and the 36° lozenge gap, also with edge length
*E*. The crystallographically defined unit cell is the
parallelogram with short axis **a**_{0} =
τ*E*, long axis **b**_{0} =
τ^{2}*E*, and included angle γ = 108°, where
τ is the golden mean (τ = τ^{2} − 1 =
τ^{−1} + 1 = ½(+1) = 2 Cos 36° =
(2 Sin 18°)^{−1} = 1.618034… ). The unit cell can also
be represented by the 36° lozenge of edge length
τ^{2}*E*. Because the ratio of the edge length of
the unit cell lozenge to that of the gap is τ^{2}, the
fraction of the unit cell area occupied by pentagons, defined as the
packing density, is ρ_{0} = (1 − τ^{−4}) =
0.854102. Each pentagon is joined to three neighbors, thus the
coordination number is *C*_{0} = 3.

Dürer (13) demonstrated that the *P*_{0} = 2
pentiling can be perfectly pentagonally twinned (Fig. 1*b*).
He wrote: “you can combine pentagons in the following manner: First
draw a pentagon and place pentagons of the same size on each side. Then
place … pentagons on their sides … . This will result in the
formation of five narrow lozenges between them. Then add pentagons in
the angles which will have formed, so that these will touch the narrow
lozenges with their corners. You can continue in this manner as long as
you desire”. The twinning interrupts the regular translational
symmetry and produces one five-coordinated pentagon, but does not alter
any of the other local contact relations.

Dürer’s *P*_{0} = 2 pentiling appears in
various guises in the structure of matter. For example, this pattern
was found by Kiselev and Klug (16) in the cylindrical surface lattices
formed by pentamers of papovavirus coat proteins, which they called
pentamer tubes. In the 72 pentamer icosahedral virus capsids of this
oncogenic family (17), the 12 pentavalent pentamers make edge-to-edge
contacts with their neighbors, as in the pentagonal dodecahedron; but
the 60 hexavalent pentamers use, in addition to an edge-to-edge
contact, one overlapped corner and two point-to-point contacts. In
various polymorphic aggregates of polyoma virus pentamers (18),
different combinations of these adaptable contacts occur. Thus, the
polymorphic packing of these virus pentamers is too complex to be
analyzed using the simple pentiling notions that are appropriate for
decagonal quasicrystals.

## Crystalline η Phase Alloys

The arrangement of transition metal atoms in the crystalline, η
phase alloys with aluminum can be modeled in two-dimensional projection
by Dürer’s *P*_{0} = 2 pentiling. Fig.
2 illustrates the projected atomic structure of
the FeAl_{3} intermetallic compound determined by Black (10).
All the crystals he examined were twinned. One of the twinning
arrangements he inferred from his atomic model (11) is illustrated at
the right side of Fig. 2. The regular pentiling does not perfectly fit
the map of Black’s projected unit cell because the ratio of the long
to short axes from his measurements is 1.6108 ± 0.0005 rather
than the expected golden ratio 1.6180, and his included angle is 17′
short of the expected 108°. These discrepancies are so small that
they are hardly discernible in the frame of this figure, which contains
about 18 of the 7.745 × 12.476 Å two-dimensional unit cells. (In
three dimensions, the lattice is **c** face centered, which
doubles the 7.745 Å **a** axis for the crystallographic unit
cell.)

The dark electron density peaks in Fig. 2 correspond to
projections of pairs of iron atoms along the direction of the
8.083 Å monoclinic axis. It is evident that the iron atoms, in
projection, are located very nearly at the vertices and centers of the
regular pentagons of the *P*_{0} = 2 pentiling. The
arrangement of the lighter aluminum atoms surrounding the iron atoms,
as seen in the projected electron density map, is more complicated, but
these details will not concern us in relating possible arrangement of
the transition metal atoms to various pentilings.

Cobalt and aluminum form an η phase alloy (12) with structure and
twinning very similar to that of FeAl_{3}, with which it forms
solid solutions. The projected unit cell dimensions of 7.592 ×
12.340 Å with included angle 107°54′ are close to those of the iron
alloy. For the cobalt alloy, the axial ratio is 0.46% greater than
τ, whereas with iron it is 0.44% smaller. There are some small but
non-trivial differences in the number and arrangement of the aluminum
atoms in these two crystals, which suggest some adaptability in the
aluminum coordination that maintains the pentagonal arrangement of the
transition metal atoms in different lattices and different
environments. For example, along the twinning line marked in Fig. 2,
the iron atom arrangement is common to the differently oriented lattice
domains, but small localized rearrangements of some of the aluminum
atoms must be involved in this meshing.

In the iron and cobalt η phase alloys, the transition metal atoms are
arranged at the corners of pentagons of mean edge lengths 4.78 and 4.70
Å in flat planes separated by 4.04 and 4.06 Å, respectively, in the
direction of the monoclinic axes of double these separations. The
transition metal atoms on the pentagon axes are located about 0.25 Å
above or below the midplane of the pair of pentagons and are spaced
alternately closer together and further apart along the axes of the
pentagonal prisms. In each column the shorter axial separation is
opposite the longer separation in the three neighboring prisms. Some of
the details of the three-dimensional arrangement of the transition
metal atoms in the pentagonal prisms of the η phase crystals will be
relevant for considering possible quasicrystalline arrangements. But
what we are concerned with at this point is the question: Are there
periodic pentilings that can conserve the local pentagonal arrangement
of the two-dimensionally projected η phase lattice, which would allow
discrete disordering to form a quasiperiodic lattice? It is evident
that localized displacive rearrangements of the pentagons cannot be
made in the regular *P*_{0} = 2 pentiling.

## Kepler’s Pentagon Tiling

Consideration of Kepler’s orderly arrangements of pentagons (1)
led to insight into ways in which periodic pentilings may be
aperiodically disordered. Kepler’s tiling with pentagons and decagons
from figure Aa of his *Harmonices Mundi* Book II is shown in a
computer graphics facsimile in Fig.
3*a*. Senechal (2) considered this a
nonrepeating pattern and concluded that Penrose’s first pentagonal
tiling family (3) (see Fig. 6) is essentially a completion of Kepler’s
figure Aa. In his description of the procedure he used to combine
pentagons and decagons in this figure, Kepler commented: “If you
really wish to continue the pattern, certain irregularities must be
admitted, two decagons must be combined … . So as it progresses
this five-cornered pattern continually introduces something new”.
However, examination of his pattern shows that a basic repeating unit
can be outlined by connecting the star centers to form the elongated
hexagons marked in Fig. 3*a*. These hexagon units have exactly
the same shape and packing arrangement as those formed by connecting
the pentagon centers in Fig. 1*b* of Dürer’s perfect
pentagonal twinning of the *P*_{0} = 2 pentiling.
Adding pentagons to extend the regular pattern of Kepler’s tiling, we
can say with Dürer: “You can continue in this manner as long
as you desire”.

Kepler’s repeating unit can be assembled into a lattice with perfectly
regular translational symmetry, as shown in Fig. 3*b*. Each
decagonal cavity can be fitted with three pentagons and each pair of
fused decagons with six. This completed tiling has pentile number
*P* = 34. The 3 pentagons in each decagon can be placed
in 10 different orientations conserving the number of possible
edge-to-edge contacts. There are 52 ways in which the 6 pentagons can
be fitted into the fused decagon pair conserving edge-to-edge contacts.
(There are 12 more ways that these pentagons can be fitted without
overlap that sacrifice an edge-to-edge contact, but these are excluded
by our local packing rule for optimizing contacts.) Therefore, there
are 5200 ways in which 12 pentagons can be added to Kepler’s 22
pentagon matrices to conserve the number of contacts and packing
density. (There are, however, only 1314 combinations with non-identical
autocorrelation functions, counting up-down and enantiomorphic pairs
only once.) The pentagons forming Kepler’s matrix can also be
conservatively reoriented. For example, the five pentagons bordering
each pentagram can be flipped in 15 different combinations. Thus, there
are an extremely large number of isomers of the *P* = 34
pentiling, and a larger number of ways in which any of these regular
lattices can be discretely disordered, conserving the number of
contacts and packing density.

It is evident that the transition metal atoms in the η phase alloys
could be arrayed in pentagonal columns corresponding to the pentagons
of any of the isomeric or discretely disordered *P* = 34
pentilings, conserving very similar local packing arrangements. Our
surmise is that all the isomers and discretely disordered versions of
the *P* = 34 pentiling just described will have very
similar autocorrelation functions and, therefore, similar diffraction
patterns. We will compare diffraction patterns and Patterson
(autocorrelation) functions of different pentilings after more
systematic analysis of their designs and packing properties.

## Pentangulation

How can the possible periodic pentilings with *P* >
2 be systematically enumerated? An obvious strategy is that used by
Penrose (3) to generate aperiodic tilings with pentagons: starting with
a pentagonal array of pentagons, each pentagon was subdivided into six
smaller pentagons and the gaps in the array between the larger
pentagons were filled with the smaller ones; this process was iterated.
Geometers call this process substitution tiling (2). In our application
of this substitution tiling strategy to enumerate periodic pentilings,
we call the process *pentangulation* by analogy with the
*triangulation* process used to enumerate possible icosahedral
surface lattice designs in the quasiequivalence theory of icosahedral
virus construction (19).

As illustrated in Fig. 4, pentangulation of
*P*_{0} = 2 generates a pentiling with
*P*_{4} = 13 (the significance of the subscript will
be described shortly); pentangulation of *P*_{4} = 13
generates *P*_{8} = 89; and pentangulation of
*P*_{8} = 89 generates *P*_{12} =
610. The series 2, 13, 89, 610, 4181, … are the fourth order
Fibonacci numbers starting with 2. As a reminder, the Fibonacci numbers
are defined: *F*_{n+1} = *F*_{n} +
*F*_{n−1}, with *F*_{0} = 0,
*F*_{1} = 1, and *F*_{−n} =
(−1)^{n−1 }*F*_{n}; and starting from
*F*_{0}, the series begins 0, 1, 1, 2, 3, 5, 8, 13,
21, 34, 55, 89, … . Thus, the series of pentangulations starting
with *P*_{0} = 2 have pentile numbers
*F*_{3}, *F*_{3+4},
*F*_{3+8}, *F*_{3+12}, … .
The *p*th pentangulation of *P*_{0}
(designated by the superscript *p*^{★}) is:
*P*_{0 }^{p★}=
*P*_{4p} =
*F*_{3+4p}. In general,
*P*_{i}^{★} = 7 *P*_{i}
− *P*_{i}^{−★}, where
*P*^{− ★} is the inverse pentangulation.

At each pentangulation, the ratio of the edge length of the smaller to
the larger pentagons is τ^{−2}; or if we consider the
pentagon size to remain constant on pentangulation, the edge length of
the unit cell will increase by τ^{2} and its area by
τ^{4}. Thus, the fraction of the area occupied by pentagons
after the pth pentangulation is ρ_{4p} =
*P*_{4p}α/τ^{4p}A_{0}, where
α is the unit pentagon area and *A*_{0} is the area
of the *P*_{0} = 2 unit cell. Since
α/*A*_{0} = ½ (1 −
τ^{4}), ρ_{4p} =
(*P*_{4p}/2τ^{4p})(1 −
τ^{−4}). For *P*_{4} = 13,
ρ_{4} = 0.8099767; for *P*_{8} = 89,
ρ_{8} = 0.8090374; and for *P*_{12} = 610,
ρ_{12} = 0.8090174. It is evident that the pentagon packing
density on successive pentangulations starting with
*P*_{0} = 2 converges to ρ_{∞} =
τ/2 = 0.8090170.

## Fibonacci Pentilings

The fact that the pentangulations of *P*_{0} = 2
generate a fourth order sequence of Fibonacci pentile numbers, and the
*P* = 34 pentiling generated from Kepler’s net
represents another Fibonacci pentile number, suggests that the sequence
of pentilings with *P*_{n} =
*F*_{n+3} may be particularly relevant for
characterizing the geometry of decagonal quasicrystals. Nevertheless,
it is possible to produce periodic pentilings with any number of
pentagons ≥ 2 in the unit cell.

Fig. 5 illustrates regular pentilings for
*P* = 3, 4, 5, and 8. Two versions of *P* =
4 are shown to demonstrate that this is not a very interesting pentile
number. The *P* = 4a pentiling has exactly the same
pentagon packing density as *P*_{0} = 2, and can be
generated from it by multiple twinning along the oblique rows using
Dürer’s scheme shown in Fig. 1*b*. The alternate
*P* = 4b arrangement has a considerably lower pentagon
packing density than any of the other illustrated pentilings. The
packing density in any periodic pentiling can be calculated by dividing
the area of the pentagons in the unit cell by the total pentagon and
gap areas. There are three different size gaps in a regular pentiling
that are allowed by our local packing rules: a lozenge L, trigram
(Penrose’s “paper boat”) T, and pentagram (star) or
allopentagram (as in the *P* = 4b pentiling) S. The
relative areas of these gaps, taking the pentagon area α = 1,
are L = 2 (3τ + 1)^{−1}, T = (τ + 3)(3τ +
1)^{−1}, and S = 2τ^{−1}. Thus,
ρ(*P* = 4b) = 2τ^{−2} = 0.763932.
Successive pentangulation of *P* = 4b gives packing
densities that converge to τ/2. (The first pentangulation generates
*P* = 29 with ρ = 0.808057 and the second generates
*P* = 199 with ρ = 0.808997.)

For the Fibonacci pentilings, as shown in Table
1, the pentagon packing density converges more
closely with increasing pentile number to τ/2 than for any other
pentiling not in this class. The closer the packing density is to
τ/2, the more exactly the periodic pentiling represents the ideal
of an infinite aperiodic array. All possible Fibonacci pentilings can
be derived by pentangulation starting from the
*P*_{0} = 2 already described, and the
*P*_{1} = 3, *P*_{2} = 5, and
*P*_{3} = 8 pentilings illustrated in Fig. 5.
Furthermore, these pentilings can all be twinned with themselves, or
twinned with each other in various combinations, as illustrated in Fig.
5 for a particular combination of *P*_{2} = 5 with
*P*_{3} = 8.

It is evident from Figs. 1 and 5 and the rule for pentangulation that
the unit cells for Fibonacci pentile numbers of even index
(*P*_{2m} = 2, 5, 13, 34 … ) are 108°
parallelograms with edge lengths **a** =
τ^{m+1}*E* and **b** =
τ^{m+2}*E*, where *E* is the pentagon edge
length; for the odd index Fibonacci pentile numbers
(*P*_{2m+1} = 3, 8, 21, 55, … ) the unit cells
are 108° rhombs with edge length **a** =
**b** = τ^{m+2}*E*. Thus, the area of
any of these unit cells with pentile number *P*_{n} =
*F*_{n+3} is *A*_{n} =
τ^{n}*A*_{0}, and the pentagon packing
density *ρ _{n} =
P*

_{n}α/

*A*

_{n}= [1/2]

*P*

_{n}τ

^{−n}(1 − τ

^{−4}). From the definition of

*P*

_{n}and the relations for powers of τ: τ

^{n}=

*F*

_{n}τ +

*F*

_{n−1}and τ

^{−n}= (−1)

^{n−1}(

*F*

_{n}τ − F

_{n+1}), it can be shown that ρ

_{n}= ½ τ [1 + (−1)

^{n}τ

^{−(2n + 6)}]. The fractional pentagon density difference (ρ

_{n}− ρ

_{∞})/ρ

_{∞}starts at +0.0557281 for

*P*

_{0}and oscillates between negative and positive values of rapidly diminishing magnitude for successive odd and even index pentile numbers, as listed in Table 1.

Other parameters listed in Table 1 that are important for
characterizing the Fibonacci pentile packings are the mean coordination
*C*_{n}, which defines the average number of
edge-to-edge contacts per pentagon and *V*_{n}, which
defines the number of pentagon vertices contained within the unit cell.
The sum *P*_{n} + *V*_{n} =
2*P*_{n+1} corresponds to the total number of
sites per unit cell (in projection) that could be occupied by
transition metal atoms (M) with the geometry of the η phase
illustrated in Fig. 2. The product
ρ_{n}(*P*_{n} +
*V*_{n})/*P*_{n} ☰
ρ_{n}(M) is a measure of the density of M atom sites as a
function of the order of the pentile number. From the numbers in Table
1 it can be seen that although the density of pentagon centers in the
*P*_{0} = 2 pentiling (corresponding to the geometry
of the crystalline η phase alloys) is 5.57% greater than that of the
limiting aperiodic pentiling, the density of total M sites (pentagon
vertices + centers) is 2.13% smaller for the *P*_{0}
= 2 pentiling than for the aperiodic limit. Thus, less ordered packing
arrangements of the pentagonal columns of transition metal atoms can
lead to slightly denser M atom packing while maintaining the same
nearest neighbor separations. Such a density increase would be
accommodated by a reduction in aluminum content.

It is evident from Table 1 that it is not necessary to go to a very large pentile number to reach a pentagon packing density that would be experimentally indistinguishable from the infinite limit of τ/2. Furthermore, the occurrence of only a small number of local motifs in these large period pentilings suggests that ordered lattices with moderate size periods may provide adequate models for less ordered states.

## Reordering Pentilings

The larger the pentile number, the larger the number of
isomers and the greater the possibilities for introducing discrete
disorder by flipping pentagons without altering the mean coordination
or packing density (see Fig. 3*b*). Such local displacive
rearrangements can also transform one ordered pentiling into another
with comparable packing density. Fig. 6
illustrates the superposition of Kepler’s pentagonally-twinned matrix
of pentagons (from Fig. 3*a*) on the first “aperiodic”
tiling presented by Penrose (3). Penrose’s tiling can be represented
as the second pentangulation of Dürer’s (13) pentagonally
twinned *P*_{0} = 2 pentiling to generate the 5-fold
twin of the *P*_{8} = 89 pentiling. The superposition
in Fig. 6 requires 11 pentagon flips within the frame. There are many
other ways in which Kepler’s tiling, or regularly periodic versions of
it, can be superimposed on Penrose’s tiling, which require different
numbers of pentagon flips. These pentagon flips should correspond to
energetically equivalent local packing arrangements.

A pentagon flip in atomic terms corresponds to interchanging a pentagon
center and vertex in one orientation for a vertex and center in the
other. The displacement in the plane of the pentagon for the two sites
is *E* Tan 18°, which, if *E* = 4.7 Å,
involves a lateral movement of 1.5 Å. In the η phase alloys (10,
12), the axial metal atoms sit ≈1.7 Å above or below the pentagonal
plane. Interchange of axial and vertex positions would involve coupled
movements of ≈2.3 Å. The activation energy for such coordinated
movements in columns of atoms might be very high, but could be
facilitated by lattice defects designated as phasons in
quasicrystallography (6). Even if such flips are rare in the locally
well-ordered condensed state, during crystallization columns of atoms
would have to choose between pentagon axial and vertex sites. Thus,
under conditions favoring a pentagon packing density ρ ≅ τ/2,
any of the local atomic arrangements corresponding to
moderate-to-larger size periodic pentilings would be equally probable.
As we will show, these lattices have virtually indistinguishable
diffraction patterns.

## Diffraction Patterns and Patterson Functions

“Atomic” models based on several of our Fibonacci pentilings
were constructed by placing atoms at the centers and vertices of the
pentagons. This construction for *P*_{0} = 2
corresponds to omitting the aluminum atoms from the projected map of
Black’s (10) η phase FeAl_{3} structure in Fig. 2. A
montage of the modeled diffraction patterns and Patterson functions for
*P*_{0} = 2, *P*_{4} = 13,
*P*_{6} = 34, and *P*_{8} = 89 are
shown in Figs. 7 and 8.
The arrangement of the pentagons in the *P*_{0} = 2
pentiling has orthogonal mirror (*mm*) symmetry, and
arrangements were chosen for the even order pentilings with a line of
mirror symmetry perpendicular to the **a** axis (i.e., along the
long axis of the 36° lozenge unit cell). Thus, the Fourier transforms
have *mm *symmetry and the single quadrant displayed for each
model in Fig. 7 represents all the diffraction data. The Patterson
functions (Fig. 8) were calculated from the squared structure factors
illustrated in Fig. 7. All unit cells have the same golden ratio shape
because the Fibonacci pentile numbers all have even order indices. The
cell dimensions for *P*_{0} = 2 are **a**
= τ*E* = 7.60 Å and **b** =
τ^{2}*E* = 12.30 Å; for
*P*_{4} = 13, *P*_{6} = 34, and
*P*_{8} = 89 these dimensions are scaled up by
τ^{2}, τ^{3}, and τ^{4}, respectively.

In both reciprocal and Patterson space, the patterns for
*P*_{4}, *P*_{6}, and
*P*_{8} show dominant 10-fold symmetry,
characteristic of the averaged pentagonal packing of the pentagons in
these moderate size unit cells. Small departures from perfect 10-fold
symmetry are evident in the *P*_{4} = 13 patterns,
but such undecagonal features are more difficult to detect in the
*P*_{6} = 34 and *P*_{8} = 89
patterns. For the *P*_{0} = 2 patterns, even though
the unit cell axes are at an angle of 108°, there is little
indication of 10-fold symmetry. Nevertheless, there are evident
correlations in the distribution of the short vectors of the
*P*_{0} = 2 Patterson function compared with those of
its more pentangulated relations, indicative of the common local atomic
packing relations.

It has been argued (6) that large unit cell models are inappropriate to
represent quasicrystal structures because “unit cells in crystal
models have to be so large that they would imply a physically
implausible range of interaction”. We are not proposing that any
particular crystal model represents the actual atomic arrangement of a
quasicrystal. What we have demonstrated is that a wide range of crystal
models with common local pentagonal coordination relations have nearly
indistinguishable decagonal Fourier transforms and autocorrelation
functions. *A priori*, any one of our crystal models with a
moderate size unit cell is as likely a representation of the actual
projected arrangement of the transition metal atoms in a defect-free
domain of a decagonal quasicrystal (7, 8) as any particular defect-free
aperiodic model with pentagon packing density ρ ≅ τ/2. Periodic
pentiling models with moderate size unit cells provide a rational
foundation for crystallographically refining the actual local atomic
arrangements that build up the quasiperiodic decagonal crystal
structures.

## Quasicrystal Crystallography

For a crystallographer, a crystal is like an orderly forest that is useful for determining the average structure of the trees. The repeating unit may be a clump of trees related by noncrystallographic symmetry or constrained to grow in non-equivalent configurations. These complexities can aid the crystallographer in seeing the trees more clearly. Quasicrystallographers have, however, had difficulty seeing the trees for the forest. The aperiodic space-filling and periodic higher dimensional representations of quasicrystalline forests are mathematically elegant, but these abstractions have tended to obscure sight of the trees. It is evident that these atomic trees are locally ordered in clusters, which are arranged quasiperiodically. How can the structure of these clumps be most clearly visualized?

One way to approach the question of how the atoms are arranged in a
quasicrystal is to look at the Patterson function that epitomizes all
the information about the interatomic vectors contained in the
diffraction pattern. Fig. 9 shows an embellished
version of the *P*_{8} = 89 Patterson function
calculated from the diffraction pattern that is shown indexed in Fig.
10. An obvious feature of the pentiling
Patterson emphasized by the circles drawn about the origin and
quasiorigins is that the characteristic length scale is
∼τ^{2} times the pentagon edge length *E*. The
central portion of this Patterson, which is repeated at the
quasiequivalent origins, is indistinguishable from that of any
pentiling with comparable or larger cell dimensions or from a five-fold
average of any of these Pattersons (which would represent the
autocorrelation function of the pentiling with infinite size unit cell
that is the ultimate quasiperiodic lattice). Thus, the Patterson
function solution at the length scale ∼τ^{2}*E*
for a suitable crystalline representation of the decagonal diffraction
data is a solution for any crystalline or quasicrystalline
representation of this data. This data can be analyzed
crystallographically.

Quasicrystallographers have concluded that the diffraction maxima in
decagonal patterns, such as in Fig. 10, cannot be assigned rational,
two-dimensional Miller indices because the Bragg spacings along lattice
lines correspond to incommensurate periodicities. There are, however,
no incommensurate periodicities in the world of experimental distance
measurements. For example, the diffraction pattern from a helix with an
irrational screw axis would be indexed by an experimentalist as the
Fourier-Bessel transform of a helix with **u** units in
**t** turns. The exact values of **u** and **t**
depend on the accuracy with which the ratio of the layer-line spacings
corresponding to the helix pitch and unit axial translation can be
measured. The more precise this measurement, the larger the calculated
repeat distance. Indexing a large helix repeat distance does not imply
an implausible range of interaction, but does indicate the skill of the
fiber diffractionist in making precision measurements. The same
considerations can be applied to indexing experimentally recorded
quasicrystal diffraction patterns.

A set of selection rules for indexing the diffraction maxima in the
decagonal patterns could be formulated as for helical diffraction, but
all that is needed is the choice of pentile model and the rule for
indexing the lowest resolution set of strong reflections. Odd and even
order pentile numbers correspond to fat and thin golden ratio rhombic
unit cells as already noted, and either will represent the decagonal
quasicrystal symmetry. For the even order (*P*_{2m})
Fibonacci pentile diffraction patterns illustrated in Fig. 7, with unit
cell dimensions **a** = τ^{m+1}*E* and
**b** = τ**a**, the index of the first strong
(*h*,0) reflection along the **a*** axis is
*h* = *P*_{m} (=
*F*_{m+3}); for the first strong (0,*k*)
reflection along the **b*** axis, *k* =
*P*_{m+1} (= *F*_{m+4}); and for
the first (*h*,*k*) reflection at ≈36° between
the (*h*,0) and (0,*k*), *h* =
*P*_{m−1}, *k* =
*P*_{m}. These three reflections have very nearly the
same Bragg spacing. For the *P*_{8} = 89 diffraction
pattern in Fig. 10 for which m = 4, the ratio of the spacings
**d***_{0,21} : **d***_{8,13} :
**d***_{13,0} is 1 : 0.99949 : 1.00164. For higher
resolution decagonally related reflections or for higher order
pentiling lattices, these ratios approach much closer to 1 (e.g.,
**d***_{34,0}/**d***_{0,55} = 1.00024
and **d***_{0,89}/**d***_{55,0} =
1.00009). An experimental diffraction pattern can be represented as the
5-fold average of a pentile model pattern that best fits the measured
width of the Bragg reflections.

A curious feature of quasicrystal diffraction patterns, associated with what is called phason disorder, is a sharpening of the diffraction peaks with increasing angle (20). Such resolution-dependent peak sharpening for decagonal quasicrystals can be accounted for by pentagonal averaging of diffraction from pentile lattice domains with relatively small pentile number. For a multiply twinned mosaic of such domains (see Fig. 5), whose densities vary above or below the asymptotic value for higher order pentilings (Table 1), the effect on the diffraction pattern would be to enhance small angle diffuse scatter and average decagonally related Bragg peaks with slightly different spacing. This peak broadening would progressively diminish for higher resolution reflections.

## Decagonal Quasicrystal Structures

The two-dimensional pentile lattices appear to provide reasonable
trial models for the projected arrangement of the transition metal
atoms in the plane perpendicular to the 5-fold axes of decagonal
quasicrystals such as Al_{65}Cu_{20}Co_{15}
(7) and Al_{70}Ni_{15}Co_{15} (8). The Bragg
spacings for the first equatorial decagonal set of strong
reflections—which have been given five-dimensional indices
(10000)—are 3.765 and 3.794 Å, respectively, for these two
quasicrystals. The indexing of a pentile lattice diffraction pattern
(Fig. 10) indicates that the mean spacing of this set of strong
reflections is the limiting value of
(*P*_{m})^{−1}τ^{m+1}*E*
Cos 18°. The asymptotic value of
(*P*_{m})^{−1}τ^{m+1} =
τ^{−1} + τ^{−3} = 1 − τ^{−4}
= 0.854102. Thus, the mean pentagon edge lengths corresponding to the
Bragg spacings of 3.765 and 3.794 Å are *E* = 4.635 and
4.671 Å, respectively. These values are in good accord with the
density maps projected from the calculated five-dimensional Fourier (7,
8) and the mean edge length of 4.70 Å for the η phase
Al_{13}Co_{4} alloy (12).

There are many fascinating aspects to the decagonal quasicrystal diffraction patterns obtained by Steurer and colleagues (7, 8). The satellite reflections around the strong Bragg maxima are suggestive of ghost spectra (21) due to microdomain structure or directional quasiperiodic density fluctuations on a long distance scale. The pronounced small angle diffuse scatter is characteristic of uncorrelated density fluctuations on a long-to-moderate distance scale. The strongly modulated diffuse diffraction on the odd order 8.08-Å layer planes perpendicular to the pentagonal prism axes is related to absence of long-range lateral correlations in the up-down displacements of the axial metal atoms that are regularly staggered between adjacent columns in the η phase alloys (10, 12). Analysis of this diffuse scatter, applying methods developed in protein crystallography (22), could provide a measure of the decay distance for the lateral correlations in the axially staggered displacements and the magnitude of the mean square fluctuations in the lateral positions of the variable atoms.

From the density maps derived by five-dimensional Fourier
analyses of the decagonal quasicrystals (7–9) it should be possible to
construct real space models, which could be refined against the
diffraction data using conventional crystallographic least squares
methods. The characteristic distance scale evident from the Patterson
function (Fig. 9) indicates that the distinctive structural motifs need
only be modeled to distances ∼τ^{2}*E* ≅ 12 Å,
which corresponds, for example, to the vertex-to-vertex span of a
pentagram or the center-to-center separation of a pair of pentagons
attached to the non-adjacent sides of a central pentagon. Within this
distance, there are a finite number of motifs, all or most of which
will be represented in any reasonable size periodic pentile lattice.
Individual pentagons can be five-, four-, three-, or two-coordinated.
The metal atoms at the vertices of a five-coordinated pentagon would be
expected to be equivalently related; but in all other pentagon
environments, these atoms will be quasiequivalently related, as in the
crystalline η phase structures (10–12), which means that the five
edge-length distances need not be exactly equal.

Experimental diffraction data for the vectors between the transition
metal atoms can be obtained by multi-wavelength anomalous dispersion
measurements. These data would also identify possible correlations in
the positions of the two types of transition metals in the stable
decagonal quasicrystals. Having established the number of independent
variables for these metal atom–metal atom vectors, these vector
distances could be refined within the framework of any reasonable size
three-dimensional periodic pentile lattice. The number of parameters
required to refine all the possible aluminum atom configurations in the
distinguishable environments is likely to exceed the 253 independent
structure factor terms that have been measured to 0.5-Å resolution for
the Al_{70}Ni_{15}Co_{15} decagonal
quasicrystal (8). In this case, stereochemical and energetic restraints
could be applied to refine the structure as in macromolecular
crystallography (23, 24).

## Conclusion

We have demonstrated that in the decagonal quasicrystalline realm the Emperor need not wear five-dimensionally quilted quasiclothes, and we surmise that similar six-dimensional garments will prove to be unnecessary in the icosahedral quasicrystalline domain.

## Acknowledgments

We thank Bin Yu for assistance with the crystallographic calculations. This work has been supported by U.S. Public Health Service Research Grant CA47439-08 from the National Institutes of Health’s National Cancer Institute.

## Footnotes

- Copyright © 1996, The National Academy of Sciences of the USA

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