 ABSTRACT
 DISCUSSION
 Graphics methods.
 Pentilings.
 Crystalline Phase Alloys.
 Kepler's Pentagon Tiling.
 Pentangulation.
 Fibonacci Pentilings.
 Reordering Pentilings.
 Diffraction Patterns and Patterson Functions.
 Quasicrystal Crystallography.
 Decagonal Quasicrystal Structures.
 Conclusion.
 FIGURES
 FIG. 1. Dürer's pentilings.
 FIG. 2. Projected electron density map of FeAl_{3} phase alloy and superimposed P_{0} = 2
pentile lattice.
 FIG. 3. Kepler's tiling with pentagons and decagons and a P = 34 pentiling generated
from Kepler's matrix.
 FIG. 4. Iterative pentangulation of the P_{0} = 2 pentiling.
 FIG. 5. Periodic pentilings with more than two pentagons per unit cell.
 FIG. 6. Superposition of Kepler's and Penrose's pentagon tilings.
 FIG. 7. Diffraction patterns of regular pentilings.
 FIG. 8. Patterson maps of regular pentilings.
 FIG. 9. Embellished P_{8} = 89 Patterson map from Fig. 8.
 FIG. 10. Indexing of the P_{8} = 89 diffraction pattern.
 Table 1. Fibonacci Pentilings.
 REFERENCES
Proc. Natl. Acad. Sci. USA
Vol. 93, pp. 1427114278, December 1996
Colloquium Paper
This paper was presented
at a colloquium entitled "Symmetries Throughout the Sciences,"
organized by Ernest Hendley, held May 11 and 12, 1996, at the
National Academy of Sciences in Irvine, CA.
Fivefold symmetry in crystalline
quasicrystal lattices
DONALD L. D. CASPAR AND ERIC FONTANO
Institute of Molecular Biophysics,
Florida State University, Tallahassee FL 323064380
ABSTRACT To demonstrate that
crystallographic methods can be applied to index and interpret
diffraction patterns from wellordered quasicrystals that display
noncrystallographic fivefold symmetry, we have characterized
the properties of a series of periodic twodimensional 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
phase alloys. The characteristic length scale of the
pentiling lattices, evident from the Patterson (autocorrelation)
function, is ~ ^{2} times the pentagon edge length,
where 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.
Fivefold 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 which combines 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 crystallike
diffraction patterns with forbidden icosahedral symmetry from
aluminummanganese 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); and the mathematical concepts involved in the construction
of aperiodic lattices are described in Quasicrystals and geometry,
by Senechal (2).
In their endeavors, quasicrystallographers have utilized 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 threedimensional space from periodic models in five
or sixdimensional space. For example, such procedures have been
applied by Steurer and his colleagues to calculate fivedimensional
Fourier maps from threedimensional Xray diffraction patterns
of decagonal phase aluminumtransition metal alloy quasicrystals
(7,8,9). Projections from these physically abstract fivedimensional
constructs produce real space maps which show correlations with
the crystallographicallydetermined atomic arrangements in related
periodically ordered alloys (10,11,12). The success of this fivedimensional
quasicrystallographic analysis suggests that, because the diffraction
data is only observable in threedimensional 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 that 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 moderatesize unit cells that can be locally rearranged,
conserving key bonding relations, to generate aperiodic lattices.
In this paper, we focus consideration 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 fivefold symmetry in the alloys with aluminum of the crystallographically
regular phase (10,11,12) 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 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 utilized 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 edgetoedge 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 that was 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
Fig. 24) is shown in Fig. 1a. 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. 1a) 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} = E, long axis b_{0}
= ^{2}E and included angle = 108°,
where is the golden mean ( = ^{2}  1 = ^{1} + 1 = ½(5+1) = 2 cos 36° = (2 sin l8°)^{1} = 1.618034...). The unit cell can also be
represented by the 36° lozenge of edge length ^{2}E.
Because the ratio of the edge length of the unit cell lozenge
to that of the gap is ^{2}, the fraction of the
unit cell area occupied by pentagons, defined as the packing density,
is _{0} = (1  ^{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. 1b).
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 fivecoordinated 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 that
they called pentamer tubes. In the 72 pentamer icosahedral virus
capsids of this oncogenic family (17), the 12 pentavalent pentamers
make edgetoedge contacts with their neighbors, as in the pentagonal
dodecahedron; but the 60 hexavalent pentamers utilize, in addition
to an edgetoedge contact, one overlapped corner and two pointtopoint
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.
FIG. 1: Dürer's pentilings. (a) The
repeating unit of the P_{0} = 2 pentiling consists
of two pentagons (orange) and one lozenge gap arranged with
c2mm plane group symmetry. The primitive unit cell is
marked by solid lines or dashed lines in the alternate lozenge
shape. The pair of 36° wedge regions, comprising a fifth of the
pentile lattice, was used to construct the pentagonally twinned
lattice (b) by applying fivefold rotation symmetry about
the central pentagon. The twinning of the P_{0}
= 2 pentiling does not alter the local coordination pattern marked
by dotted lines. The ten boundary twinning lines mark rows of
pentagons that are shared by neighboring P_{0}
= 2 lattice domains oriented at 36° to each other. The orange
pentagons simulate Dürer's twinning pattern (13).
Crystalline
Phase Alloys. The arrangement of transition metal
atoms in the crystalline, 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 ± .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 twodimensional 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 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 , whereas with iron it is 0.44%
smaller. There are some small but nontrivial 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
rearrangements of some of the aluminum atoms must be involved
in this meshing.
In the Fe and Co 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 threedimensional arrangement of the transition metal atoms
in the pentagonal prisms of the 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 twodimensionally projected phase
lattice which would allow discrete disordering to form a quasiperiodic
lattice? It is evident that localized displacive rearrangements
can not be made in the regular P_{0} = 2 pentiling.
FIG. 2: Projected electron density map of FeAl_{3} phase alloy and superimposed P_{0} = 2
pentile lattice. The density map was constructed from the atomic
coordinates (10), applying a temperature factor of 5 Å^{2}
to all atoms and projecting the density on the plane perpendicular
to the monoclinic b axis. The twinning was modeled by
rotating a portion of the map by 36° and rejoining as illustrated
in Fig. 1b. The regular P_{0} = 2 pentiling,
marked by the dashed lines, closely fits the map with the iron
atoms located at the vertices and centers of the pentagons of
mean edge length E = 4.78 Å. The common set of pentagons
at the twinning boundary is marked by solid lines, with portions
of the pattern connecting pentagon centers on either side of the
boundary indicated by fine dashed lines.
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
Fig. Aa of his Harmonices Mundi Book II is shown in a computer
graphics facsimile in Fig. 3a. Senechal (2) considered
this a nonrepeating pattern and concluded that Penrose's first
pentagonal tiling family (3) (cf. Fig. 6) is essentially
a completion of Kepler's Fig. 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 fivecornered 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. 3a. These hexagon units have exactly the
same shape and packing arrangement as those formed by connecting
the pentagon centers in Fig. 1b 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. 3b.
Each decagonal cavity can be fitted with three pentagons and
each pair of fused pentagons with six. This completed tiling
has pentile number P = 34. The three pentagons in each
decagon can be placed in 10 different orientations conserving
the number of possible edgetoedge contacts. There are 52 ways
in which the six pentagons can be fitted into the fused decagon
pair conserving edgetoedge contacts. (There are 12 more ways
that these pentagons can be fitted without overlap which sacrifice
an edgetoedge contact, but these are excluded by our local packing
rule for optimizing contacts.) Therefore, there are 5,200 ways
in which 12 pentagons can be added to Kepler's 22 pentagon matrix
to conserve the number of contacts and packing density (There
are, however, only 1,314 combinations with nonidentical autocorrelation
functions, counting updown 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
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.
FIG. 3: (a) Kepler's tiling with pentagons
and decagons (1) and (b) a P = 34 pentiling generated
from Kepler's matrix. The pattern formed by connecting the pentagrams
in (a) can be continued indefinitely, as in Fig. 1b,
to construct a perfect pentagonal twin of the elongated hexagonal
units comprised of 22 pentagons, two decagons, two fused decagons
and two pentagrams. By arranging Kepler's matrix unit with translational
symmetry and filling the decagonal spaces with twelve pentagons,
a regular P = 34 pentiling is formed in (b). Flipping
some pentagons at random will produce an aperiodic tiling.
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; and 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_{n1}, with F_{0}
= 0, F_{1} = 1 and
F_{n} = (1)^{n1}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
pth 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 ^{2}; or if we consider
the pentagon size to remain constant on pentangulation, the edge
length of the unit cell will increase by ^{2}
and its area by ^{4}. Thus, the fraction of the
area occupied by pentagons after the pth pentangulation
is _{4p} = P_{4p}/^{4p}A_{0}
where is the unit pentagon area and A_{0}
is the area of the P_{0} = 2 unit cell. Since
/A_{0} =
½(1  ^{4}),
_{4p}
= (P_{4p}/2^{4p})(1
 ^{4}). For P_{4} = 13, _{4}
= 0.8099767; for P_{8} = 89, _{8}
= 0.8090374; and for P_{12} = 610, _{12}
= 0.8090174. It is evident that the pentagon packing density
on successive pentangulations starting with P_{0}
= 2 converges to = /2 = 0.8090170.
FIG. 4: Iterative pentangulation of the P_{0}
= 2 pentiling. In the basic pentangulation process (a),
six small pentagons are fit into each of the two larger pentagons
of the P_{0} = 2 pentiling and a single small pentagon
is fit into the lozenge gap to generate the P_{4}
= 13 pentiling. The repeating unit of P_{4} =
13 comprises 13 pentagon centers, 29 pentagon vertices, one pentagram,
one trigram and three lozenge gaps. Because five small pentagons
fit in the pentagram gap and three fit in the trigram gap, the
next stage of pentangulation (b) generates P_{8}
= 6 × 13 + 5 × 1 + 3 × 1 + 1 × 3 = 89 pentagons. Continuing this
process (c), pentangulation of P_{8} = 89
generates P_{12} = 610.
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. 1b. 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 = 1, are
L = 2(3 + 1)^{1}, T = (
+ 3)(3 + 1)^{1} and S = 2^{1}.
Thus, (P = 4b) = 2^{2} = 0.763932.
Successive pentangulation of P = 4b gives packing densities
that converge to /2.
(The first pentangulation generates
P = 29 with = 0.808057 and the second generates
P = 199 with = 0.808997.)
For the Fibonacci pentilings we enumerate in Table 1, the pentagon
packing density converges more closely with increasing pentile
number to /2 than for any other pentiling not in this
class. The closer the packing density is to /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 Fig. 1 and Fig. 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 = ^{m+1}E and
b = ^{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 = ^{m+2}E.
Thus the area of any of these unit cells with pentile number
P_{n} = F_{n+3} is A_{n}
= ^{n}A_{0}, and the pentagon packing
density _{n} = P_{n}/A_{n}
= ½P_{n}^{n}(1  ^{4}).
From the definition of P_{n} and the relations
for the powers of : ^{n} = F_{n}
+ F_{n1} and ^{n} = (1)^{n1}(F_{n}  F_{n+1}), it can be demonstrated that
_{n} = ½[1 + (1)^{n}^{(2n
+ 6)}]. The fractional pentagon density difference (_{n} 
)/ 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 edgetoedge contacts per
pentagon; and V_{n}, the number of pentagon vertices
contained within the unit cell. The sum P_{n}
+ V_{n }= 2P_{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 phase illustrated in Fig. 2. The product _{n}(P_{n}
+ V_{n})/P_{n} _{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 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 /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.
FIG. 5: Periodic pentilings with more than two pentagons
per unit cell. Pentagons in a repeating unit are orange and unit
cells are marked by solid lines. The regular Fibonacci pentilings
P_{0} = 2, P_{1} = 3, P_{2}
= 5 and P_{3} = 8 are the first four members of
the series listed in Table 1 whose pentagon packing densities
are approximately /2. As explained in the text, the two
versions of P = 4 illustrated have packing densities that
differ significantly from that of the closest Fibonacci pentilings
(P_{1} = 3 and P_{2} = 5). Different
pentiling lattices can be twinned with themselves or each other
in various combinations. P = 4a corresponds to periodic
twinning of P_{0} = 2. The illustrated twinning
of P_{2} = 5 and P_{3} = 8, with
the common boundary marked, exemplifies one of the many ways in
which periodic pentilings can be combined to generate less regular
pentilings.
Table 1. Fibonacci Pentilings
Pentile Number P_{n} = F_{n+3}
number of pentagons in unit cell. (F_{n+3} is
sequence of Fibonacci numbers, starting with 2.) Pentagon vertex
number V_{n} = P_{n} + 2P_{n1}
number of pentagon vertices in unit cell. Mean number of M
atom sites per pentagon (P_{n} + V_{n})/P_{n}
= 2P_{n+1}/P_{n}; asymptotic value
= 2 = 3.236068. Mean pentagon coordination C_{n}
= (P_{n+1} + P_{n4})/P_{n};
asymptotic value = ^{2} + ^{4}
= 2.763932. Pentagon packing density _{n} = ½P_{n}^{n}(1  ^{4}) = ½[1 + (1)^{n}^{(2n
+ 6)}];
asymptotic value = /2 = 0.809017. M atom site
density _{n}(P_{n} + V_{n})/P_{n}
_{n}(M) = ^{2}[1 + (1)^{n+1}^{(2n+8)}];
asymptotic value = ^{2} = 2.618034. Values for
(_{n}(M)  (M))/(M)
= (1)^{n+1}^{(2n+8)} are not listed
because this is equal to (_{n+1}  )/
which is listed for the succeeding pentile number.
n
 P_{n}
 V_{n}
 (P_{n} + V_{n})/P_{n}
 C_{n}
 (_{n}  )/

0  2
 4  3
 3  +5.5728 × 10^{2}

1  3
 7  3.33333
 2.66666  2.1286 × 10^{2}

2  5
 11  3.2
 2.8  +8.1306 × 10^{3}

3  8
 18  3.25
 2.75  3.1056 × 10^{3}

4  13
 29  3.23077
 2.76923  +1.1862 × 10^{3}

5  21
 47  3.23810
 2.76171  4.5310 × 10^{4}

6  34
 76  3.23529
 2.76471  +1.7307 × 10^{4}

7  55
 123  3.23636
 2.76364  6.6107 × 10^{5}

8  89
 199  3.23596
 2.76405  +2.5251 × 10^{5}

9  144
 322  3.23611
 2.76389  9.6449 × 10^{6}

10  233
 521  3.23605
 2.76395  +3.6840 × 10^{6}

11  377
 843  3.23607
 2.76393  1.4072 × 10^{6}

12  610
 1,364  3.23607
 2.76393  +5.3749 × 10^{7}

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 (cf. Fig. 3b). Such local displacive
rearrangements can also transform one ordered pentiling into another
with comparable packing density. Fig. 6 illustrates the superposition
of Kepler's pentagonallytwinned matrix of pentagons (from Fig.
3a) on the first "aperiodic" tiling presented
by Penrose (3). Penrose's tiling can be represented as the second
pentangulation of Dürer's (13) pentagonallytwinned P_{0}
= 2 pentiling to generate the fivefold 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 that 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
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 wellordered 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 /2, any of the local
atomic arrangements corresponding to moderatetolarger size periodic
pentilings would be equally probable. As we will show, these
lattices have virtually indistinguishable diffraction patterns.
FIG. 6: Superposition of Kepler's and Penrose's
pentagon tilings. Penrose's tiling (3), marked by the boldly
outlined pentagons, is a perfect fivefold twin of the P_{8}
= 89 pentiling whose unit cells are marked by the portions of
the lightly outlined 36° rhombs centered on the starred pentagon.
This facsimile of his tiling has been extended 11% to the left
of his margin to give the frame a golden aspect ratio. Kepler's
pentagonally twinned matrix of shaded pentagons, from Fig. 3a,
is centered on a pentagram in a region of Penrose's tiling that
has only local fivefold symmetry. The 11 sites within this frame
at which pentagon flips are necessary to transform from Penrose's
to Kepler's pattern can be recognized by the shaded pentagons
that do not match the bold outline. Kepler's matrix cells, outlined
with dashes, each contain an isomer of the P_{6}
= 34 unit, which can be distinguished by the orientation of the
twelve pentagons inside the decagonal spaces. Curiously, four
of these cells contain the same isomer. The density of the P_{6}
= 34 pentiling is 0.0148% greater than that of P_{8}
= 89, amounting to one more pentagon in an area ~ 20× that shown
here.
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) 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 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 = E = 7.60 Å
and b = ^{2}E = 12.30 Å; for
P_{4} = 13, P_{6} = 34 and P_{8}
= 89 these dimensions are scaled up by ^{2}, ^{3}
and ^{4} respectively.
In both reciprocal and Patterson space, the patterns for P_{4},
P_{6} and P_{8} show dominant 10fold
symmetry, characteristic of the averaged pentagonal packing of
the pentagons in these moderate size unit cells. Small departures
from perfect 10fold 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 10fold symmetry. Nevertheless, there are evident
correlations in the distribution of the short vectors of the P_{0}
= 2 Patterson function compared to 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 defectfree domain of a decagonal
quasicrystal (7,8) as any particular defectfree aperiodic model
with pentagon packing density /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.
FIG. 7: Diffraction patterns of regular pentilings.
Model electron density maps similar to Fig. 2 were calculated
by placing identical atoms with temperature factors of 5 Å^{2}
at the vertices and centers of pentagons in the P_{0}
= 2, P_{4} = 13, P_{6} = 34 and
P_{8} = 89 pentilings from Figs. 1a, 4a,
3b and 4b respectively. The pentagon edge length
E in this construction is 4.7 Å. Fast Fourier transforms
of these maps were calculated to 0.9 Å resolution and the
amplitudes were scaled by setting F_{00} to unity.
The maximum scaled amplitude in each pattern is F_{max}
~ 0.2. The quadrants, which include all independent Fourier coefficients
for each pattern, are aligned with the (0,k) axis horizontal.
The patterns from the higher order pentilings have evident tenfold
symmetry. Comparison across the boundary between the P_{6}
= 34 and P_{8} = 89 patterns demonstrates that
they are virtually indistinguishable. The structure factors,
which are discrete points in reciprocal space, are depicted as
Gaussian disks with radii proportional to the amplitudes as indicated
in the F scale at the top. The intensity range displayed
is ~ 100.
FIG. 8: Patterson maps of regular pentilings. Sectors
of the four Patterson maps are aligned with their a axes
vertical and their unit cells outlined (the b axis of P_{8}
= 89 extends beyond the image frame). The circle has radius 52.4
Å = ^{5}E, which is equal to the
a axis of P_{8} = 89 and the b axis
of P_{6} = 34, and is marked every 36° to emphasize
the tenfold symmetry.
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 nonequivalent 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 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 that 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 ~ ^{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 fivefold 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 ~ ^{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, can not be assigned
rational, twodimensional 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 having u units in t turns.
The exact values of u and t depend on the accuracy
with which the ratio of the layerline 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 = ^{m+1}E
and b = 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_{m1}, 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 fivefold 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). Pronounced broadening of lowresolution
peaks in decagonal patterns can be accounted for by pentagonal
averaging of diffraction from lattice domains with relatively
small pentile numbers. For a multiply twinned mosaic of such
domains (cf. 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.
FIG. 9: Embellished P_{8} = 89 Patterson
map from Fig. 8. Unit cell axes are marked and the a axis
is vertical. The map has mirror symmetry parallel and perpendicular
to the a axis at the origin and at a/2 marked with
a cross. The centers of symmetry at the left and right margins
are also marked. Circles of radius E and ^{2}E
are drawn about the crystallographic origins. Circles of radius
E are drawn about the quasiequivalent origins related
by the local noncrystallographic tenfold axes and the crystallographic
centers of symmetry. Circles of radius ^{2}E
can also be drawn around the quasiequivalent origins which fit
like that about the crystallographic origin.
FIG. 10: Indexing of the P_{8} =
89 diffraction pattern. The quadrant in Fig. 7 is enlarged and
the most intense spots are labeled with their h,k
indices. The thick lines, which are parallel to the crystal axes
and are spaced at intervals of consecutive Fibonacci numbers,
pass through rows of lattice points including the most intense
spots. The thin lines, at angles of ±36° to the (0,k)
direction, are noncrystallographic but nearly pass through centers
of the intense spots related by the noncrystallographic tenfold
symmetry.
Decagonal Quasicrystal Structures.
The twodimensional pentile lattices appear to provide reasonable
trial models for the projected arrangement of the transition metal
atoms in the plane perpendicular to the fivefold 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 fivedimensional 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}^{m+1}E
cos 18°. The asymptotic value of (P_{m})^{1}^{m+1}
= ^{1} + ^{3} = 1  ^{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 fivedimensional
Fouriers (7,8) and the mean edge length of 4.70 Å for the
phase Al_{13}Co_{4} alloy (12).
There are many fascinating aspects to the decagonal quasicrystal
diffraction patterns obtained by Steurer and his 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 longtomoderate 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 longrange lateral correlations
in the updown displacements of the axial metal atoms that are
regularly staggered between adjacent columns in the 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 fivedimensional Fourier analyses
of the decagonal quasicrystals (7,8,9) 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 ~ ^{2}E
12 Å, which corresponds, for example, to the vertextovertex
span of a pentagram or the centertocenter separation of a pair
of pentagons attached to the nonadjacent 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 twocoordinated. The metal atoms at the vertices of a fivecoordinated
pentagon would be expected to be equivalently related; but in
all other pentagon environments, these atoms will be quasiequivalently
related, as in the crystalline phase structures (10,11,12),
which means that the five edgelength distances need not be exactly
equal.
Experimental diffraction data for the vectors between the transition
metal atoms can be obtained by multiwavelength anomalous dispersion
measurements. This 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 atommetal atom vectors,
these vector distances could be refined within the framework of
any reasonable size threedimensional 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 fivedimensional quasiclothes, and we
surmise that similar sixdimensional garments will prove to be
unnecessary in the icosahedral quasicrystalline domain.
We thank Bin Yu for assistance with the crystallographic
calculations. This work has been supported by a U. S. public
health service research grant from the National Institutes of
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