INFINITE TILINGS

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Alan H. Schoen

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C. INFINITE TILINGS BY RHOMBS

On Feb. 13, 2013, Alan Mackay emailed me this diffraction pattern.
He generated the pattern by illuminating (with coherent laser light) the
central core, shown below, of an infinite quasi-recursive tiling by rhombs.



C1. Penrose tilings

Below are photos of some of the people who have contributed
to our understanding of Penrose tilings and quasicrystals. The
names below the photos lead to websites that explain some of
  their contributions to this subject. For more information, check
the links below Fig. C1.2.



Nicolaas de Bruijn
9 July 1918 17 February 2012



Donald L. D. Caspar



John Conway, the author, and Chaim Goodman-Strauss
at Gathering 4 Gardner 8 in Atlanta
March 28, 2008



Martin Gardner
10 October 1914 5 May 2010



Branko Grünbaum



Alan Mackay



Roger Penrose



Marjorie Senechal



Dan Shechtman
Awarded 2011 chemistry Nobel prize for his discovery of quasicrystals



Paul Steinhardt
photo by Tony Rinaldo



Walther Steurer




  As almost everyone with an interest in science knows by now (Fall, 2011), the 2011 Nobel prize in chemistry was
  awarded to Dan Shechtman for his 1982 discovery of quasicrystals. I will say very little here about quasicrystals.  
Instead, I will descend one level to discuss a few topics in the related field of two-dimensional tilings.                  
You may find this topic pretty tame compared to three-dimensional quasicrystals, but it was pioneering work on  
2D tilings — especially by Roger Penrose — and also by Mackay, de Bruijn, Conway, Steinhardt, and others that
paved the way to our understanding of quasicrystals. Here is a New Scientist web article about the awarding       
of the Nobel prize to Shechtman, but you can easily discover for yourself many more webpages devoted to the    
     subject of quasicrystals.
Wikipedia's webpage on quasicrystals is one of the most comprehensive of these webpages.

I'm betting that you cannot fail to enjoy this Shechtman interview video and the article that accompanies it.          

In 1977, Martin Gardner introduced Roger Penrose's quasiperiodic tilings in his legendary monthly column,       
Mathematical Recreations, in Scientific American magazine. There he explained that one is free to choose either
the so-called kite and dart polygons or a pair of thin and thick rhombs ("golden" rhombs) as prototiles             
for these tilings. Figs. C1.1 and C1.2, just below, show these tiles with markings that conform to Penrose's          
'matching rules'. We'll call a tiling by replicas of the prototiles of either Fig. C1.1 or C1.2 matched if the tiling    
is edge-to-edge and the markings on the tiles are continuous across every edge. Penrose proved that                    

(a) there is no periodic matched tiling of the plane by the prototiles of either Fig. C1.1 or C1.2, and
                                  (b) there is an uncountable infinity of aperiodic matched tilings by these prototiles. Such tilings are called quasiperiodic.


Fig. C1.1


Fig. C1.2

   The periodicity of a tiling is easily demonstrated by confirming that the tiling is composed of copies of
a finite region replicated by translation in two independent directions in the plane. Quasiperiodicity,
by contrast, is not identified simply by inspecting the tiling visually. It is a subtle kind of long-range  
order that is best defined in terms of the symmetry of the diffraction pattern of the tiling.                    
You will find a detailed discussion of quasiperiodicity in the Penrose tiling entry in Wikipedia.          

Three indispensable books for the study of Penrose tilings, quasicrystals, and related topics are:         

        Tilings and Patterns (W. H. Freeman and Company, 1987), by Grünbaum and Shephard        
        Quasicrystals and geometry (Cambridge University Press, 1995), by Marjorie Senechal, and
    Crystallography of Quasicrystals (Springer, 2009), by Walther Steurer and Sofia Deloudi.

              There are several different methods of generating Penrose tilings, although they are equivalent (cf. de Bruijn).
One of these is Penrose's remarkable inflation/deflation mechanism (so named by John Conway).     

     Chapter 7 of Martin Gardner's The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and
        Problems
, W. W. Norton & Co. (2001), contains a description of inflation/deflation and how it leads to a
        proof that the number of different Penrose tilings is uncountable. Two marvelous tools for learning about
         inflation/deflation are Stephen Collins's interactive program, Bob - Penrose Tiling Generator and Explorer
  and David Austin's java applet, 'The Inflation Hierarchy', in his Penrose Tiles Talk Across Miles.       



Website information devoted to Penrose tilings, quasicrystals, and related matters includes:

(It's wise to install the latest version of Java before you try to run applets.)




Penrose is said to have remarked that before 1984 almost nobody but Alan Mackay took seriously his notion that his quasiperiodic tilings might have a physical counterpart in 3-dimensional space.


Vladimir Shevchenko (left) and Alan L. Mackay (right)

The astonishing discovery of quasicrystals in 1984 by Dan Shechtman and his co-workers changed all that. Besides the extensive experiments on the growing of solid quasicrystals from the molten state, there has been much theoretical research — especially by Steinhardt and his collaborators, by Michael S. Longuet-Higgins, and by others too numerous to mention here — aimed at explaining the growth of quasicrystals. Fragments excerpted from the book, In Our Own Image: Personal Symmetry in Discovery, introduced by Istvan and Magdolna Hargittai, describe some of the turbulent events that accompanied the discovery of quasicrystals, including Alan Mackay's independent prediction of their existence.

One might almost say now (in 2010) that the long-sought single prototile ('einstein') for two-dimensional quasi-periodic patterns has at last been found, and it is the regular decagon. But of course it is inaccurate to call this decagon a prototile, since it is the overlapping unit of a covering rather than a tiling. An article by Petra Gummelt, inspired by an idea originally suggested by Sergei E. Burkov, provides some background. Steinhardt and Hyeong-Chai Jeong have developed Gummelt's model significantly, demonstrating that Penrose coverings by decagons with matching overlaps are isomorphic to Penrose tilings by rhombs with matching edges. Their analysis is supported by HAADF electron microscopy images of decagonal AlNiCo obtained by Koh Saitoh et al. The agreement between the predictions of the overlapping decagons model and the experimental results is striking.

In a landmark article in 1981, Nicolaas G. de Bruijn described an algebraic theory of Penrose tilings.


Left to right:
David Klarner, George Polya, and Nicolaas de Bruijn
Stanford University, May 1973

He introduced the concept of pentagrids, which are composed of five superimposed grids of parallel lines unit distance apart. Below is an example of a pentagrid.


Fig. C1.3
An example of a de Bruijn pentagrid

Every pentagrid is identified by a set of five shifts γi (i = 0,1,2,3,4) — radial displacements of the five grids from the origin at the center. De Bruijn proved that the tiling by rhombs that is dual to a given pentagrid is a Penrose tiling — i.e., satisfies Penrose's matching rules — if and only if Γ, the sum of the five shifts, is equal to an integer. Among Penrose tilings for which the five shifts are equal, the one for which Γ = 1 is called SUN and the one for which Γ = 2 is called STAR. They are shown below.

A sequence of six pentagrids
If you toggle up and down using the Page Up and Page Down keys,
you will observe that
any pair a and b of these six pentagrids for which γa + γb = 1
are related by inversion in the origin at the center
and are therefore equivalent.
Hence it's unnecessary to consider shifts > 1/2.

Since SUN and STAR are the only two equal-shift tilings for which Γ = 1 and γi ≤ 1/2, they are the only two Penrose tilings with d5 symmetry.

The remarkable CARTWHEEL tiling, for which all five shifts are zero, has only d1 symmetry. Each of its ten infinite triangular sectors, which — except for the Conway 'worms' that line their sides — are congruent, contains an alternating sequence of successively larger central regions of SUN and STAR. The ratio of the distances from the origin to the centers of any two consecutive such regions is found to be equal to the golden ratio φ (≅ 1.618). CARTWHEEL is shown below.

In a 1978 AMS abstract, I conjectured that both SUN and STAR have a recursive structure that allows them to be generated from a small central core without regard for tile-matching rules. By comparing tilings, I verified that the conjecture is correct at least up to the fourth stage of recursion. The abstract describes the recursion for kite/dart tilings, not tilings by rhombs. Here I consider the tilings by rhombs. A brief summary of the recursion conjecture follows.



RECURSION CONJECTURE

The SUN and STAR Penrose tilings arise from a small central core
via a recursive sequence of
(a) radially outward reflections in the enclosing necklaces
(pentagonal rings of Conway mirror worms)
of successively larger central regions,
followed by
(b) lateral reflections of the images produced in (a)
to fill the empty triangular gaps left by (a).



Fig. C1.4
Penrose STAR
three stages
d5
pdf version: Toggle back and forth between images,
using Page Up and Page Down keys,
to see how each triangular gap in stage three
(and in all subsequent stages, according to the Recursion Conjecture)
is filled by an image of the nearby triangular tiled region, reflected in an adjacent mirror edge (red).


Fig. C1.5
Penrose SUN
three stages
d5
pdf version: Toggle back and forth between images,
using Page Up and Page Down keys,
to see how each triangular gap in stage three
(and in all subsequent stages, according to the Recursion Conjecture)
is filled by an image of the nearby triangular tiled region, reflected in an adjacent mirror edge (red).


Here I will assume the truth of the Recursion Conjecture.

At each stage of recursion, the area in the interior of the pentagonal boundary mirror of the SUN or STAR increases by the factor

f52 = (1 + 2 cos π /5)2 ≅ 6.85.

Since 2 cos π/5 = (1+√ 5)/2 = φ, the golden ratio (≅ 1.618), the total tiled area in successive steps is proportional to

1+0φ
2+3φ
13+21φ
  89+144φ
...

Let S1 = {1, 2, 13, 89,...} and S2 = {0, 3, 21, 144,...}.
Since the Fibonacci sequence is
{.., 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...}, where
    F1 = 1,
    F2 = 0,
    F3 = 1,
    F4 = 1,
    F5 = 2,
    F6 = 3,
etc., then
    S1 = {F1, F5, F9, F13, ...} and
    S2 = {F2, F6, F10, F14, ....}.



De Bruijn's landmark paper is entitled "Algebraic theory of Penrose's non-periodic tilings of the plane", Nederl. Akad. Wetensch. Proceedings Ser. A 84 (Indagationes Math. 43) (1981) 38-66. It was reprinted in The Physics of Quasicrystals, ed. P. J. Steinhardt and S. Ostlund, World Scientific Publ. Comp., Singapore (1987), pp. 673-700. For a concise summary of parts of the paper, see the extraordinarily elegant Laura Effinger-Dean's undergraduate honors thesis.

The duality between a vertex (point of intersection) in a pentagrid and a thin or thick rhomb in the corresponding Penrose tiling is illustrated in the representative pentagrid shown in Fig. C1.3. For any pair of intersecting grid lines, the smaller angle of intersection is either 36º or 72º. The sides of a rhomb dual to a vertex are orthogonal to the two grid lines that intersect at the vertex.


Fig. C1.6
In this representative pentagrid, the two gray rhombs
illustrate the duality between rhombs and points of intersection.
The amplitude of each vector in the central shift star for this pentagrid is equal to 1/4.



CARTWHEEL (Γ = 0)

Here is the pentagrid for CARTWHEEL.




Next — two differently shaded images of each of the d5-symmetric Penrose tilings, SUN and STAR. In the second image of each pair, concentric pentagonal rings of Conway mirror-worms are distinguished by color. The role of these mirrors in the conjectured recursive structure is explained below.

SUN (de Bruijn shift = 2/10). (TWO images: Scroll down for the second image.)
STAR (de Bruijn shift = 4/10). (TWO images: Scroll down for the second image.)
Here are the pentagrids for SUN and STAR.
pentagrid for SUN
pentagrid for STAR



Below are images of

Generalized Penrose tilings are de Bruijn tilings with equal grid shifts for which Γ is a half-integer. Each of the first two has d5 symmetry, and the third has d10 symmetry. They are not recursively structured, and they are not governed by Penrose-like matching rules. A smaller portion of the third one, which has d10 symmetry, appears as Fig. 4.7 on p. 44 in Laura Effinger-Dean's undergraduate honors thesis.

de Bruijn (1/10) (de Bruijn shift = 1/10)
de Bruijn (3/10) (de Bruijn shift = 3/10)
de Bruijn (5/10) (de Bruijn shift = 5/10)

Here are the pentagrids for these three tilings. Note the emergence of d10 symmetry (if you ignore the colors of the lines!) in the third pentagrid.

de Bruijn pentagrids for the three generalized Penrose tilings
(Toggle with Page Down and Page Up keys)




C2. d7-symmetric generalized Penrose tilings from de Bruijn's heptagrids


Here are seven of the eight tilings of this generalized class.
(I haven't taken the time to construct the eighth one — de Bruijn (0/14) — which is a rough analog of the Penrose CARTWHEEL.)

de Bruijn (1/14)-1
de Bruijn (2/14)-1
de Bruijn (2/14)-2
de Bruijn (2/14)-3
de Bruijn (3/14)-1
de Bruijn (4/14)-1
de Bruijn (4/14)-2
de Bruijn (5/14)-1
de Bruijn (6/14)-1
de Bruijn (6/14)-2
de Bruijn (7/14)-1





C3. RPn tilings (recursive rhombic tilings of dn symmetry)





The seven liberal arts,
in the Hortus deliciarum of Herrad von Landsberg
(from Wikipedia)

  • C3.0 Introduction

    I call any pseudo-Penrose tiling for n ≥ 3 an RPn tiling if it
        (a) contains every shape of rhomb in SRIn (cf. § B1), viz., n ⁄ 2 shapes for even n and (n − 1) ⁄ 2 shapes for odd n;
        (b) is recursive in the restricted sense described below in § C3.1; and
        (c) has dihedral symmetry dn.
    An RPn tiling is a tiling for even n that satisfies both (b) and (c), but not (a), because it omits one or more of the shapes of rhomb in SRIn.

    Here's an example of a three-stage RP7 tiling:

  • Fig. C3.0.0
    Three-stage RP7 tiling (2,2,1)P
    pdf version (the colors are better!)

    cf. the blunt tiling (2,2,1)B,
    which — although it has the same core as (2,2,1)P —
    is forced to have a somewhat more fragmented structure.

    A number of additional examples of RP7 tilings are in § C3.12.
    Examples of RPn tilings for n = 3, 4, 5, 6, 8, and 9 are in § C3.13, and § C3.14 shows examples of RPn tilings for n = 4, 6, 8, 10, 12, 14, and 16.

    A particular subset of the RPn tilings in § C3.14 allows tilings constructed by a common algorithm. It has the following properties:
    The prototiles of the tiling, which exists for every n > 5, are the two smallest rhombs in SRInρ1 and ρ2. Aside from a central star-like core comprised of 2n specimens of ρ1, the tiling consists of 2n central triangular sectors that are related by rotation through the angle π ⁄ n. Each sector is the union of translated images of a single pattern unit (fundamental domain) composed of two specimens of ρ1 and one specimen of ρ2.

    It is somewhat misleading to apply the term 'recursive' to RPn tilings for n > 7 and RPn tilings for n > 6, because — as Fig. C3.0.1 demonstrates up to n = 12 and Fig. C3.0.2 demonstrates up to n = 42 — with increasing n, the fraction of the tiling inherited from the previous stage that is unaltered in the current stage decreases significantly because of increasing overlap (cf. § C3.1).



    Fig. C3.0.1
    The geometrical basis of recursion in RPn and RPn tilings (n ≥ 3)
    (For details, cf. § C3.1)

    For example: in the RPn tiling for n = 14 in Fig. C3.14.14, only 42 of the 273 non-mirror tiles (~ .15) generated in the second stage are reflected images of tiles in the first stage, and a comparable fraction of the non-mirror tiles in the third stage are reflected images of tiles in the second stage. In this case, it is impossible to increase the first of these two 'transfer fractions', but the second transfer fraction could be increased somewhat by modifying the tiling. (Cf. § C3.1 for an explanation of these terms.)

    The ultimate justification for calling RPn and RPn tilings recursive is that the necklaces that decorate the mirrors of every stage after the first are generated recursively.

    Although the principal topic of § C3 is RP7 tilings, most of the discussion in § C3.1 to C3.11 is applicable also to RPn tilings for odd n ≠ 7.

    RP7 tilings were an afterthought of a 1978 conjecture (cf. § C1) suggesting that the d5-symmetric Penrose SUN and STAR tilings are structured in strictly recursive fashion. RP7 tilings are designed to mimic this putative recursive structure, at least partially, but unlike Penrose tilings, they are not quasiperiodic and cannot be generated by de Bruijn's algorithm (cf. § C2).

    Fig. C3.0.3 shows the three prototile rhombs of RP7 tilings, each with sides of unit length. Rhomb j (j = 1, 2, 3) has upper and lower face angles 2 ⁄ 7.
    For all n ≥ 3, the ⌊(n − 1) ⁄ 2⌋ prototile rhombs are defined in similar fashion: rhomb j (j = 1, 2, ⌊(n − 1) ⁄ 2⌋) has upper and lower face angles 2n. For even n, it is convenient to augment the inventory of prototiles by defining the 'zero rhomb' — a single line segment of unit length. Although it is merely the intersection of two adjacent rhombs and has zero area, it is useful in describing the composition of symmetrical radial strings of rhombs in RPn tilings for even n (cf. § C3.13).

    Fig. C3.0.3
    The three prototile rhombs for n = 7


    Fig. C3.0.4
    The regular 14-gon is the smallest convex region
    for which there exists a centro-symmetric tiling by the three RP7 prototiles




  • C3.1 The recursive structure of RP7 tilings
  • RP7 tilings are created by iterating the reflection of a finite d7-symmetric tiled region, beginning with a core tiling, in the edges of its heptagonal boundary mirror. Each stage of iteration produces a heptagonal annulus that encloses — and thereby enlarges — the tiling of the previous stage. The mirrors M1, M2, M3 for the first three stages are shown in the schematic diagram of Fig. C3.1.1. The structure suggests a two-dimensional onion whose concentric layers increase exponentially in thickness from the center outward.

    In the Penrose SUN and STAR, the analogous reflection of the pentagonal region generated up to a given stage leaves five empty triangular gaps, which are magically filled by lateral reflection of the newly generated tiled regions adjacent to the gaps (cf. Figs. C1.4 and C1.5). In the case of RP7 tilings there is no such magic. Instead, as illustrated below in Fig. C3.1.1, in addition to gaps (green), overlaps (red) are created in each new heptagonal annulus. Each gap-overlap pair must be replaced by an orderly ad hoc arrangement of tiles called a wedge, in order to connect the adjacent tiled regions (gray) seamlessly. The portion (gray) of the reflected tiling that survives without replacement of tiles is called the remainder.

    In designing a wedge, it is inevitably found necessary to replace some tiles that lie outside the the red and green regions of Fig. C3.1.1. After the second recursion stage, wedge design is challenging, because of the large number of rhombs involved.

    I assume without proof that it is possible to construct a wedge at every stage of recursion.


    Fig. C3.1.1
    Schematic diagram of the first three stages of a RP7 tiling.
    M1, M2, and M3 are the boundary mirrors of these three stages.





    Fig. C3.1.2 illustrates the difference between odd and even n in the expansion of the kth central polygonal region.

    Fig. C3.1.2
    odd n                                                             even n
    Rk is the circumradius and rk is the inradius of the kth central polygonal region.

    The linear expansion ratio fn = Rk+1/Rk = rk+1/rk
    for the concentric polygonal regions of consecutive stages
    is equal to
    1 + 2 cos π/n for odd n ≥ 3
    and
    3 for even n.

    For n = 7,
    f7 ≅ 2.8019

    Although I conjecture that RP7 tilings can in principle be extended indefinitely from the center outward, in practice they are restricted to only a few stages, because the time required to design a wedge eventually becomes prohibitive. At each stage, the tiled area increases by the factor g7 = f72 ≅ 7.85. (Note that g72 ≅ 62,  g73 ≅ 484,  g74 ≅ 3799,  and g75 ≅ 29,825.)





  • C3.2 The structural components of RP7 tilings

    At the center of every RP7 tiling (e.g., pointed tiling (1,1,0)P) lies a d7-symmetric core enclosed by a heptagonal necklace composed of seven beadstrings (Conway worms), each of which is a string of one or more beads (cf. Figs. C3.3.1 and C3.3.2). A bead is a d2-symmetric polygon, not necessarily convex, tiled by prototile rhombs. If every bead — with the possible exception of a bead at the center of the beadstring (central bead) — is a single rhomb, the beadstring is called minimal. (Central beads are discussed in § C3.7.) The regular heptagon whose edges bisect the beadstrings is called a mirror (e.g., the green heptagons in pointed tiling (1,1,0)P). In many of the examples shown below, beadstrings are distinguished by color from other components of the tiling.

    Beads related by reflection in a radial line of reflection of the tiling are called conjugate. (Every central bead is self-conjugate.)

    The boundary curve of every beadstring (the 'beadstring polygon') necessarily has d2 symmetry. It is symmetrical by reflection in both
        (a) the mirror edge that bisects it, and
        (b) the perpendicular bisector of that mirror edge.

    Although the tiling of a non-central bead is not required to be symmetrical, so long as it is related by reflection to the tiling of its conjugates, it is always possible for the tiling of a bead to be symmetrical by reflection in a line perpendicular to the mirror edge associated with the bead (d1 symmetry). Only in central beads is the tiling forced — by the d7 symmetry of the tiling — to have d1 symmetry. Central beads are found only in beadstrings whose signatures (cf. § C3.4) have at least one odd component (cf. § C3.7). If the signature has exactly one odd component, the central bead may be either simple, i.e., tiled by a single rhomb, or compound, i.e., tiled by more than one rhomb (cf. § C3.7). If the signature has more than one odd component, the central bead must be compound.

    As an example, consider the beadstrings of the three necklaces in the three-stage blunt tiling (0,0,2)B-2. Although there is no central bead in the beadstrings of the innermost necklace, the beadstrings of each of the next two necklaces have a central bead. The signatures (cf § 3.4) of these first three beadstrings are (0,0,2)B, (1,2,4)B, and (4,7,10)B, respectively.

    In some of the tiling examples in § C3.12, every beadstring is minimal, i.e., contains at most one compound bead. In other examples in § C3.12 (e.g., Fig. C3.12.1), some beadstrings contain compound beads that could be replaced by two or more simple beads.

    If the boundary of every bead were not symmetrical by reflection in the mirror edge e associated with it, the tiling on opposite sides of the beadstring would not be related by reflection in e. But it is not necessary for the boundary of every individual bead to have d2 symmetry, so long as every beadstring is symmetrical by reflection in e.




  • C3.3 Pointed tilings vs. blunt tilings

    I consider here principally two types of RPn and RPn tilings — pointed and blunt.

    In pointed RP7 tilings (cf. Fig. C3.3.1), beadstrings are terminated at each end by a bead vertex (call it a vertex of type V) that coincides with a vertex of the heptagonal mirror (red). Adjacent beadstrings are joined at type V bead vertices. In some cases, depending on the shape of the terminal beads of the beadstring, they may also be joined along a common edge.

    In blunt RP7 tilings (cf. Fig. C3.3.2), a corner rhomb that is congruent to rhomb 1 (cf. Fig. C3.0.3) is inserted at each end of every beadstring. The pair of corner rhombs lengthens the mirror edge associated with the beadstring by tan π/n (cf. Fig. C3.3.3).



    Fig. C3.3.1
    The core (green), mirror (red heptagon), and necklace (closed chain of seven 6-rhomb convex beadstrings)
    for the pointed tiling (1,1,0)P





    Fig. C3.3.2
    The core (orange and blue), mirror (red heptagon), and necklace (closed chain of seven 3-rhomb convex beadstrings)
    for the blunt tiling (0,0,1)B
    A corner rhomb (blue) is inserted at each end of every blunt beadstring.



    Fig. C3.3.3
    The corner rhomb ρ0 incident at each end of every blunt beadstring
    (cf. Figs. C3.0.3 and C3.3.2)

    If a tiling is pointed, its signature, beads, beadstrings, and mirrors are also called pointed (with a similar convention for blunt tilings). The vertices of pointed mirrors coincide with bead vertices, but each vertex of a blunt mirror lies in the interior of a corner rhomb.

    As shown in Fig. C3.3.2, blunt beadstrings are not incident at mirror vertices. Every blunt beadstring is terminated by a bead edge (call it an edge of type E) that is orthogonally bisected by the associated mirror edge. Adjacent blunt beadstrings share a vertex (call it a vertex of type V*) that does not coincide with a vertex of the heptagonal mirror. In some blunt tilings, depending on the shape of the bead at the ends of the beadstring, adjacent beadstrings may also share a common edge. Every corner rhomb shares an edge and a vertex with each of its two adjacent beadstrings. A corner rhomb is regarded as part of the necklace but not as part of a beadstring.

    It is clear from Fig. C1.5 that the Penrose SUN can be classified as a blunt tiling. The Penrose STAR (cf. Fig. C1.4), which is neither blunt nor pointed, could perhaps be called overlapping blunt.




  • C3.4 Beadstring signatures and the linear expansion matrix

    A RP7 tiling grows recursively by outward expansion from its core via successive reflections in the edges of concentric heptagonal mirrors of exponentially increasing size. At each reflection, the entire tiling in the interior of the outermost mirror is reflected, except for the rhombs in the outermost necklace.

    The composition of each beadstring in the outermost necklace at the kth stage of recursion is specified by the kth signature

    σk = (σk(1), σk(2), σk(3)),

    where σk(1), σk(2), and σk(3) are non-negative integers. σ1 is called the initial signature. If σ1 = (0,0,0), the tiling is called the null tiling.

    For both pointed and blunt beadstrings, the three components of the signature σk are defined as follows:

    In either the left or right half of a [horizontally oriented] beadstring of the kth necklace,
    σk(j) is the number of boundary edges with projected length sin /7 (j=1,2,3)
    on the mirror edge that bisects the beadstring
    (cf. Fig. C3.4.1).



    Fig. C3.4.1
    The three prototile rhombs for n = 7

    In a beadstring composed of single-rhomb beads,
    rhomb ρj is oriented so that it contributes 2sin /7 to the total length of a mirror edge.

    In either the left or right half of each horizontal beadstring in Fig. C3.3.1, for example, one of the two contributing boundary edges has projected length sin π /7 and the other has projected length sin 2π /7. Hence the signature is (1,1,0)P.

    Similarly, in Fig. C3.3.2, in either half of each horizontal blunt beadstring, the single contributing boundary edge has projected length sin 3π /7, and the signature is therefore (0,0,1)B.

    It is convenient to assign signatures to beads according to the same convention as for beadstrings.

    Now define the vector

    S = (sin π /7, sin 2π /7, sin 3π /7).

    In pointed tilings, (λk)pointed — half the length of an edge of the kth mirror — is equal to the scalar product of σk and S:

    (λk)pointed = σk S,

    In blunt tilings, (λk)blunt — half the length of an edge of the kth mirror — is equal to the scalar product of σk and S plus the term (1/2) tan π/7 ( 0.2403) contributed by one corner rhomb (cf. Fig. C3.5.2).

    (λk)blunt = σkS + (1/2) tan π/7.

    Fig. C3.4.2
    The contribution (1/2) tan π/7 of a corner rhomb to
    the half-edge length (λk)blunt of a blunt mirror (cf. Fig. C3.3.2)



    A simple trigonometric identity (previously unknown?) makes it possible to assimilate the corner rhomb into the expression for (λk)blunt, so that both pointed and blunt tilings can be treated in a unified way:

    (1/2) tan π/7 = sin π/7 + sin 2π/7 − sin 3π/7

    = (1,1,− 1) S
    = δ S,

    where
    δ = (1,1,− 1).

    Hence

    (λk)blunt = (σk + δ)S
    = τkS,
    where
    τk = σk + δ.
    τk is called the kth expanded signature.


    The expansion of the tangent in sines for n = 7 led me to conjecture that for all odd n ≥ 3,


    where

    For n = 3, 5, 7, 9, 11, …, this expansion takes the forms

    (1/2) tan π/3   =     sin π/3
    (1/2) tan π/5   =  sin π/5   +  sin 2π/5
    (1/2) tan π/7   =     sin π/7   +  sin 2π/7      sin 3π/7
    (1/2) tan π/9   =  sin π/9    sin 2π/9    +  sin 3π/9      sin 4π/9
    (1/2) tan π/11 =     sin π/11 +  sin 2π/11  −  sin 3π/11    sin 4π/11  +  sin 5π/11
    (1/2) tan π/13 = −  sin π/13 +  sin 2π/13  +  sin 3π/13    sin 4π/13    sin 5π/13 +  sin 6π/13
           …

    The pattern of signs in this set of equations is

    +
    − +
    + + −
    − + + −
    + + − − +
    − + + − − +
    + + − − + + −
    − + + − − + + −

           …

    Bob Fitzgerald promptly proved my conjecture that the tangent expansion in sines is valid for odd n ≥ 3. Here's Bob's proof.

    Although it's not relevant to the present discussion, I also conjectured that for odd j,

    (Bob's proof requires only a trivial addition to prove this generalization.)


    The recursive relation between the signatures σk and σk+1 of consecutive beadstrings is expressed by the symmetric tri-diagonal linear expansion matrix E3 shown in Fig. C3.4.3. For odd n, let m = (n − 1)/2. E3 is the n = 7 version of the m x m linear expansion matrix Em that is applicable to all RPn tilings for odd n. All of the non-zero elements eij of Em are equal to 1, except for emm = 2.


    Fig. C3.4.3
    The linear expansion matrix E3 for n = 7

    For pointed tilings,
    σk+1T = E3 σkT,

    and for blunt tilings,

    σk+1T = E3 (σk + δ)TδT

    = E3 τkT δT.

    Here is proof that for linear recursion via the matrix Em — with m = (n − 1)/2 — between signatures of consecutive pointed beadstrings for odd n, the ratio of the edgelengths of the associated mirrors is equal to the linear expansion ratio fn (cf. § C3.1).

    And here is proof that for linear recursion between expanded signatures of consecutive n = 7 blunt beadstrings via the matrix E3, the ratio of the edgelengths of the associated mirrors is also equal to the linear expansion ratio fn (cf. § C3.1).




  • C3.5 Rhomb population totals in consecutive generations

    Let

    ρk = (ρk(1), ρk(2), ρk(3)),

    where ρk(j) = the number of specimens of rhomb j (j = 1,2,3) in the interior of the mirror Mk of generation k (cf. Fig. C3.1.1),

    and let Ak = the area in the interior of Mk.

    Here's my proof that if

    ρk+1T = E32 ρkT,
    where
    E32 = E3 E3
                 
    which is equal to
    or
      ,
    Fig C3.5.1

    then
    Ak+1/Ak = gn (cf. § C3.1).




  • C3.6 Allowed initial signatures

    It might appear that the appropriate first step in the construction of a RP7 tiling would be to choose a core. As it happens, however, only one-third of all possible initial signatures allow the construction of recursively related necklaces. Consequently, a more prudent course is to choose the initial signature first, then construct the first necklace, and finally tile the core.

    An initial signature σ1 = (σ1(1), σ1(2), σ1(3)) is called allowed if and only if

                  the beadstring with signature σ2 = (σ2(1), σ2(2), σ3(3)) contains two replicas of the beadstring with signature σ1 — one at either end.

    Hence σ2(1) ≥ 2σ1(1), σ2(2) ≥ 2σ1(2), and σ2(3) ≥ 2σ1(3).
                 
    An initial signature that is not allowed is called forbidden.

    I have proved that

                  for pointed tilings, the initial signature σ1 = (σ1(1), σ1(2), σ1(3)) is allowed if and only if

    σ1(1) + σ1(3) ≥ σ1(2) ≥ σ1(1),

                  and for blunt tilings, the initial signature σ1 = (σ1(1), σ1(2), σ1(3)) is allowed if and only if

    σ1(1) + σ1(3) ≥ σ1(2) ≥ σ1(1) − 1.

    For both pointed and blunt tilings, it is readily proved that if a given signature σk is allowed, its successor σk+1 is allowed. Hence the signature of every descendant of an allowed initial signature is allowed.

    No example is known of an initial signature — whether allowed or forbidden — for which no core can be designed, but it has not been proved that such signatures do not exist.

    I have proved that for pointed tilings, the number of allowed initial signatures for which 0 ≤ σ1(i) ≤ n (i=1,2,3) is

    N(n) = (n+1)(n+2)(2n+3)/6.

    Since the total number of pointed signatures for which 0 ≤ σ1(i) ≤ n (i=1,2,3) is (n+1)3, it follows that only one-third of all possible initial pointed signatures are allowed. A similar result holds for initial blunt signatures.




  • C3.7 Beadstring signature parity

    The parity Π of a beadstring signature σk = (σk(1), σk(2), σk(3)) is defined as

    Π(σk)= (σk(1), σk(2), σk(3)) (mod 2).

    Signature parity determines whether or not a beadstring incorporates a central bead (cf. § C3.2) and — if it does — whether that bead is simple or compound.

    Π(σk) has eight possible values. Parity (0,0,0) is called even; the seven other parities are called odd.

    (0,0,0)
    (1,0,0)
    (0,1,0)
    (0,0,1)
    (1,1,0)
    (1,0,1)
    (0,1,1)
    (1,1,1)

    The illustrated catalog of initial signatures shows paired examples of (a) pointed and (b) blunt beadstrings for each of the eight parity classes.

                  In beadstrings with even parity (0,0,0), there is no central bead.

                  In beadstrings with odd parity (1,0,0), (0,1,0), or (0,0,1), there is a central bead, and a simple bead will suffice.

                  In beadstrings with odd parity (1,1,0), (1,0,1), (0,1,1), or (1,1,1), there is a central bead, and it must be compound.

    The catalog of initial signatures includes cores, beadstrings, and mirrors for every initial signature σ1 — pointed or blunt, allowed or forbidden — for which 0 ≤ σ1[i] ≤ 2 (i = 1,2,3). Each signature σ1 is followed by the signature of its daughter σ2. If σ1 is forbidden, σ2 is marked with an asterisk. Inside the necklace defined by each initial signature is an example of a tiled core. The order in which the rhombs are placed in each beadstring and also the arrangement of rhombs in each core are uniquely determined only for the smallest beadstrings.

    In each of the catalog illustrations from Fig. 1 to Fig. 26, the figure labelled "a" shows a pointed beadstring, and the figure labelled "b" shows a blunt beadstring, but the cores for the two figures are identical. For the signature (2,2,2) in catalog Fig. 27, beadstrings and cores for each of the six possible bead sequences are shown for both pointed and blunt tilings.

    Fig. C3.7.1 shows four pointed compound beads of minimum area. Every blunt bead, whether simple or compound, can be constructed simply by attaching a string of parallel rhombs to the boundary of the bottom half of the corresponding pointed bead.


    Fig. C3.7.1
    The smallest possible pointed compound beads for beadstrings of odd parity

    For pointed signatures

                  (a) if the initial signature has even parity (0,0,0),
                        all subsequent signatures have parity (0,0,0).

                  (b) if the initial signature has odd parity,
                        subsequent signatures cycle repeatedly through the seven odd parities
                        in counter-clockwise order as listed in Fig. C3.7.2a.


    Fig. C3.7.2a
    One period of the [counter-clockwise] odd-parity sequence for pointed beadstrings

    For blunt signatures,

                  (a) if the initial signature has odd parity (1,1,1),
                        all subsequent signatures have parity (1,1,1).

                  (b) if the initial signature has parity ≠ (1,1,1),
                        subsequent signatures cycle repeatedly through the seven parities
                        listed in Fig. C3.7.2b, in counter-clockwise order.


    Fig. C3.7.2b
    One period of the [counter-clockwise] seven-term parity sequence for blunt beadstrings




  • C3.8 Uniqueness of beadstring signatures

    I conjectured as a result of numerical experiments that iff n is prime, the sum

    c1 sin π/n + c2 sin 2π/n + … + c(n-1)/2 sin [(n-1)/2]π/n,

    where c1, c1, … c(n-1)/2 are rational numbers, is equal to zero iff
    c1 = c2 = … = c(n-1)/2 = 0.

    If true, this conjecture would imply that for prime n,

    no two pointed beadstrings with different signatures have the same length
    and
    no two blunt beadstrings with different signatures have the same length.

    I was unable to prove the conjecture, but Bob Fitzgerald made light work of this one too! Here's Bob's proof.

    The conclusion, then, is that if n is prime, for a given initial signature, the composition of beadstrings that are recursively related via the expansion matrix E3 (cf. § C3.4) is uniquely determined. But because of the peculiar coincidence that not only the edge length but also the area of each polygonal mirror is expressed as a linear combination of sines with rational coefficients, it follows from Bob's proof that the populations of the three rhombs contained inside each mirror are also unique.




  • C3.9 Orientation of beadstring rhombs

    Fig. C3.9.1
    The three prototile rhombs for n = 7
    In a beadstring composed of single-rhomb beads,
    rhomb ρj is oriented so that it contributes 2sin /7 to the total length of a mirror edge.


    Let us call a vertex of the prototile rhomb ρj (j = 1, 2, 3) odd or even according to whether the face angle of ρj at that vertex is an odd or even multiple of π ⁄ 7. If π ⁄ 7 is the unit of measurement for angles, the value of the angle is called the angle index. Two of the angle indices of each ρj are even and two are odd. As shown in Fig. C3.9.1, if a beadstring rhomb ρbead is oriented so that its odd vertices lie on a mirror edge (pointed tiling) or on a line parallel to a mirror edge (blunt tiling), its contribution to the length of the mirror edge is 2sin jπ ⁄ 7. Let us call this orientation the standard orientation of ρbead. No other orientation is possible for such a rhomb in a RP7 tiling.

    If a rhomb in a RP7 tiling is bisected by a radial line of reflection, it must be oriented so that its even vertices lie on that line. Its contribution to the length of the line is 2cos jπ ⁄ 7, and it too is described as being in standard orientation.

    In a blunt RP7 tiling, non-standard orientation for a beadstring rhomb ρbead can be ruled out by a parity argument: if the even vertices of ρbead lie either on a mirror edge or on a line parallel to a mirror edge, the odd vertices lie on a perpendicular line (cf. Fig. C3.9.1). Since it is impossible to surround an odd vertex with a tiling by rhombs that is symmetrical by reflection in the mirror edge (because the angle indices of the two sectors to be tiled (gray) are non-integer), ρbead cannot be incorporated into a bead.


    Fig. C3.9.1
    Hypothetical non-standard orientation for a beadstring rhomb ρbead in a blunt tiling
    iodd is the odd angle index of the face angle of the rhomb at its bottom vertex

    (b) Pointed RP7 tilings

    If the rhomb ρbead in Fig. C3.9.1 is a central bead, then the argument in (a) proves that its non-standard orientation makes the tiling impossible. If it is a non-central bead, the reflection symmetry of the beadstring forces every other bead in the beadstring — whether convex or non-convex, simple or compound — to be in non-standard orientation. Otherwise, the angle index of each sector incident at the vertex common to every pair of neighboring beads (gray in Fig. C3.9.2) would be non-integer.


    Fig. C3.9.2
    Hypothetical non-standard orientation for a pair of adjacent beads in a pointed tiling

    Now consider the two bead polygons incident at a vertex of the heptagonal mirror associated with a necklace. The face angle index of the mirror is 5. The only possible values for the face angle index of each of the two bead polygons incident at the vertex are 2 and 4 (6 would define a condition of overlap.)

    To be continued. (Proof still incomplete!)



    Here's an example of what happens at the center when the rhombs that are bisected by a radial line of reflection symmetry are in non-standard orientation:


    Fig. C3.9.3
    n = 7
    false tiling
    d7

    Note the trapezoidal tiles at the center (cf. the 'pseudo-compound rosette' for n = 7/2 in Fig. B1.3).

    The insertion — at the center of the tiling — of the regular heptagon, which does not admit a tiling by the three prototile rhombs,
    is forced by the non-standard orientation of the rhombs that are bisected by a radial line of reflection symmetry.
    pdf version


    Here's a similar example for n = 5:


    Fig. C3.9.4
    n = 5
    false tiling
    d5

    Note the trapezoidal tiles at the center (cf. the 'pseudo-compound rosette' for n = 5/2 in Fig. B1.3).

    The insertion — at the center of the tiling — of the regular pentagon, which does not admit a tiling by the two prototile rhombs,
    is forced by the non-standard orientation of the rhombs that are bisected by a radial line of reflection symmetry.

    pdf version



  • C3.10 Tiling wedges

    The first step in the construction of a wedge is the design of a beadstring for the necklace that will enclose the entire enlarged tiling. It is not obvious without proof that in every generation there exists a string of beads of precisely the right length to bridge the gap (base of green isosceles triangle in Fig. C3.1.1) between the two beadstring replicas inherited from the previous generation. That such a beadstring always exists (and is of unique composition) is proved in § C3.4 and § C3.8.

    Although a mathematical description of the recursive structure of beadstring composition is slightly simpler for pointed tilings than for blunt tilings (cf. § C3.4), it is no more difficult to design wedges for blunt tilings than for pointed tilings.

    In the three-generation pointed tiling (1,1,0)P, the arrangement of rhombs in the wedge just below the bottom vertex of the innermost mirror is identical to the arrangement of rhombs just beyond each vertex of the next larger mirror. Although it is not essential to design each wedge so that it incorporates a portion of the wedge of the previous stage, it does save labor to do so. The three-generation blunt tiling (2,2,1)B-1 provides an additional example.




  • C3.11 Rings of 14 stars in RP7 tilings of d7 symmetry

    The concentric regular heptagons M1, M2, and M3 in the diagram in the middle of Fig. C3.11.1 represent the three innermost mirrors of the tiling. (The colored tilings at the left and right sides of the central image should have been rotated by π/7 about their centers.)

    Fig. C3.11.1
    'Proof [almost] without words'

    The images of seven uniformly spaced stars
    produced by reflection in the mirror edges at each stage
    define a ring of fourteen stars
    with centers at the vertices of a regular 14-gon.




    C3.12 Gallery of RP7 tilings

    When you examine some of the images listed below, if you are able to freeze the display mid-course by right-clicking with the mouse, you may be able to see what overlaps look like before they are replaced by wedges. (Pointed tiling (1,1,0)P-8 is a good example. After the image is displayed, toggle between zoom up and zoom down.)


  • Fig. C3.12.1
    (0,0,1)B-1 is a three-stage blunt tiling



    Fig. C3.12.2
    (1,1,0)P-1 is a three-stage pointed tiling


                   
    Examples of both
    pointed and blunt tilings

    blunt tiling (1,0,0)B-1 (27.16 KB)
    blunt tiling (1,0,0)B-2 (27.02 KB)
    blunt tiling (0,0,1)B-1 (60.62 KB)
    blunt tiling (0,0,1)B-2 (307.88KB)
    blunt tiling (0,0,1)B-3 (38.43 KB)
    blunt tiling (0,0,1)B-4 (321.45KB)
    blunt tiling (0,0,1)B-5 (322.36KB)
    blunt tiling (0,0,1)B-6 (319.33KB)
    blunt tiling (0,0,1)B-7 (321.44KB)
    blunt tiling (0,0,1)B-8 (320.75KB)
    pointed tiling (0,0,1)P-1 ( 33.84 KB)
    pointed tiling (0,0,1)P-2 ( 56.84 KB)
    pointed tiling (0,0,1)P-3 ( 35.25 KB)
    pointed tiling (1,1,0)P-1 ( 36.15 KB)
    pointed tiling (1,1,0)P-2 ( 8.47 MB)
    pointed tiling (1,1,0)P-3 ( 36.47KB)
    pointed tiling (1,1,0)P-4 (378.34KB)
    pointed tiling (1,1,0)P-5 (366.75KB)
    pointed tiling (1,1,0)P-6 (361.01KB)
    pointed tiling (1,1,0)P-7 (332.30KB)
    pointed tiling (1,1,0)P-8 ( 8.28 MB)
    pointed tiling (1,1,0)P-9 ( 6.01 KB)
    pointed tiling (1,1,0)P-10 ( 36.15KB)
    pointed tiling (1,0,1)P-1 ( 58.12KB)
    pointed tiling (0,1,1)P-1 ( 90.39KB)
    blunt tiling (0,0,2)B-1 (140.11KB)
    blunt tiling (0,0,2)B-2 (140.11KB)
    blunt tiling (1,1,1)B-1 (197.05KB)
    blunt tiling (1,1,1)B-2 (198.26KB)
    blunt tiling (1,1,1)B-3 (209.82KB)
    blunt tiling (1,1,1)B-4 (202.73KB)
    blunt tiling (2,1,0)B-1 ( 3.31 MB)
    blunt tiling (2,1,1)B-1 (41.82 KB)
    blunt tiling (2,2,1)B-1 ( 8.54 MB)
    pointed tiling (2,2,1)P-1 (36.48KB)
    pointed tiling (2,2,1)P-2 (328.01KB)
    blunt tiling (2,2,2)B-1 (106.75KB)
    blunt tiling (2,2,2)B-2 (108.11KB)

    blunt tiling [variant]B-1 (108.55KB)
    blunt tiling [variant]B-2 (116.11KB)
    blunt tiling [variant]B-3 (111.20KB)
    blunt tiling [variant]B-4 (109.00KB)




    C3.13 Gallery of RPn tilings (n ≠ 7) (cf. § C3.12 for RP7 tilings)

    Recall (from § 3.4) that the components σk(j) (j = 1, 2, …, ⌊ n ⁄ 2 ⌋) of the signature σk of the kth stage of the tiling are defined as follows:
                  In either the left or right half of a [horizontally oriented] beadstring of the kth necklace,
                  σk(j) is the number of boundary edges with projected length sin /n on the mirror edge that bisects the beadstring.

    Accompanying each of the figures below is a list of signatures for the first few stages of the tiling.

    The linear expansion ratio fn is equal to three for even n tilings and to 1 + 2 cos πn for odd n tilings (cf. Fig C3.1.2).

    A necessary condition for the existence of blunt RPn tilings is that tan π/n can be expressed as a linear combination, with integer coefficients, of sines of integer multiples of π/n (cf. § C3.4). This requirement is met by odd n but not by even n. Hence there are no blunt RPn tilings for even n.

    For the 'crystallographic integers' n = 2, 3, 4, and 6, RPn tilings are found to have translation symmetry. The RP3 tiling in Fig. C3.13.1 immediately below has translation symmetry if the coloring is ignored. In this tiling (and also in some of the other figures), mirror edges are shown as faint line segments superimposed on necklace beadstrings.

    For those illustrations in which mirror edges are not drawn and necklaces are not distinguished by color, necklaces may be recognized by working toward the interior from the outermost necklace (which is easy to recognize because it is incident at the pattern boundary).

    In most of the schematic diagrams like the one at the right in Fig. C3.13.1, gaps are green and overlaps are red (cf. § C3.1) .


    Fig. C3.13.1
    RP3
    n = 3
    f3 = 2
    σ1 = (2)P
    σ2 = (4)P
    σ3 = (8)P
    σ4 = (16)P
    four stages
    p3m1
    pdf version





    Fig. C3.13.2
    RP4
    n = 4
    f4 = 3
    σ1 = (1,1)P
    σ1 = (3,3)P
    σ1 = (9,9)P
    three stages
    p4m
    pdf version




    Fig. C3.13.3
    RP5
    n = 5
    f5 = 1 + φ (≅ 2.618)
    σ1 = (1,2)P
    σ2 = (3,5)P
    σ3 = (8,13)P
    three stages
    d5
    pdf version



    Fig. C3.13.4
    RP5
    n = 5
    f5 = 1 + φ (≅ 2.618)
    σ1 = (1,2)B
    σ2 = (3,5)B
    σ3 = (8,13)B
    three stages
    d5
    pdf version





    Fig. C3.13.5
    RP6
    n = 6
    f6 = 3
    σ1 = (1,3,1)P
    σ2 = (3,9,3)P
    σ3 = (9,27,9)P
    three stages
    p6m
    pdf version





    Fig. C3.13.6
    RP6
    n = 6
    f6 = 3
    σ1 = (2,3,2)P
    σ2 = (6,9,6)P
    σ3 = (18,27,18)P
    three stages
    p6m
    pdf version




    Fig. C3.13.7
    RP8
    n = 8
    f8 = 3
    σ1 = (0,2,1,1))P
    σ2 = (0,6,3,3))P
    σ3 = (0,18,9,9))P
    two stages
    p8m
    pdf version




    Fig. C3.13.8
    RP9
    n = 9
    f9 ≅ 2.879
    Applying the expansion matrix E4, with δ9 = (-1,1,1,-1) (cf. § C3.4), to the intial signature
    σ1 = (0,0,1,2)B
    to obtain subsequent signatures yields
    σ2 = (1,1,3,5)B
    and
    σ3 = (3,5,9,13)B,
    but the second signature here is
    σ2 = (2,2,3,4)B,
    not (1,1,3,5)B.
    Because of the identity
    sinπ ⁄ 9 + sin2π ⁄ 9 = sin4π ⁄ 9,
    these two different values for σ2 define beadstrings of the same length.
    (There is a similar degeneracy for every odd n that is composite.)

    two stages
    d9
    pdf version


  • C3.14 Gallery of RPn tilings
  • A necessary condition for the existence of blunt RPn tilings is that tan π/n can be expressed as a linear combination, with integer coefficients, of sines of integer multiples of π/n (cf. § C3.4). This requirement is met by odd n but not by even n. Hence there are no blunt RPn tilings for even n. But if n is equal to twice an odd integer and the prototiles of the tiling are restricted to the rhombs of SRIn ⁄ 2, then a blunt tiling is possible.

    If n is equal to twice an odd integer, the rhombs that are bisected by radial lines of reflection to corners of the tiling are oriented oppositely to those that are bisected by radial lines of reflection to edge midpoints of the tiling (cf. Figs. C3.14.2, C3.14.3, C3.14.10, and C3.14.14).


    Fig. C3.14.1
    RP4
    n = 4
    Has one prototile — ρ2— of the two — ρ1, ρ2 — in SRI4
    f2(2) = 3
    σ1 = (0,1)P
    σ2 = (0,3)P
    σ3 = (0,9)P
    three stages
    p4m
    pdf version
    The [degenerate] beadstrings and their associated mirror edges are identical in this example.




    Fig. C3.14.2
    RP6
    n = 6
    Has one prototile — ρ2 — of the three — ρ1, ρ2, ρ3 — in SRI6
    f2(3) = 3
    σ1 = (0,1,0)P
    σ2 = (0,3,0)P
    σ3 = (0,9,0)P
    three stages
    p6m
    pdf version





    Fig. C3.14.3
    RP6
    n = 6
    Has one prototile — ρ2 — of the three — ρ1, ρ2, ρ3 — in SRI6
    f6 = 3
    σ1 = (0,1,0)P
    σ2 = (0,3,0)P
    σ3 = (0,9,0)P
    three stages
    p6m
    pdf version



    Fig. C3.14.4
    RP8
    n = 8
    Has two prototiles — ρ2, ρ4 — of the four — ρ1, ρ2, ρ3, ρ4 — in SRI8
    f2(4) = 3
    σ1 = (0,1,0,0)P
    σ2 = (0,3,0,0)P
    σ3 = (0,9,0,0)P
    three stages
    d8
    (1,0)P-1




    Fig. C3.14.5
    RP8
    n = 8
    Has two prototiles — ρ2, ρ4 — of the four — ρ1, ρ2, ρ3, ρ4 — in SRI8
    f2(4) = 3
    σ1 = (0,1,0,0)P
    σ2 = (0,3,0,0)P
    σ3 = (0,9,0,0)P
    three stages
    d8
    (1,0)P-2



    Fig. C3.14.6
    RP8
    n = 8
    Has two prototiles — ρ2, ρ4 — of the four — ρ1, ρ2, ρ3, ρ4 — in SRI8
    f2(4) = 3
    σ1 = (0,1,0,0)P
    σ2 = (0,3,0,0)P
    σ3 = (0,9,0,0)P
    three stages
    d8
    (1,0)P-3




    Fig. C3.14.7
    RP8
    n = 8
    Has two prototiles — ρ2, ρ4 — of the four — ρ1, ρ2, ρ3, ρ4 — in SRI8
    f2(4) = 3
    σ1 = (0,1,0,0)P
    σ2 = (0,3,0,0)P
    σ3 = (0,9,0,0)P
    three stages
    d8
    pdf version




    Fig. C3.14.8
    RP8
    n = 8
    Has two prototiles — ρ2, ρ4 — of the four — ρ1, ρ2, ρ3, ρ4 — in SRI8
    f2(4) = 3
    σ1 = (0,0,1,0)P
    σ2 = (0,0,3,0)P
    σ3 = (0,0,9,0)P
    three stages
    d8
    pdf version

    This tiling bears a slight resemblance to the 1977-1982 'Ammann-Beenker tiling'
    illustrated in the Tiling Encyclopedia of Dirk Frettlöh and Edmund Harriss.
    That tiling, unlike this one, is the dual of a de Bruijn multigrid.





    Fig. C3.14.9
    RP10
    n = 10
    Has two prototiles — ρ2, ρ4 — of the five — ρ1, ρ2, ρ3, ρ4, ρ5 — in SRI10
    f2(4) = 3
    σ1 = (0,1,0,0,0)P
    σ2 = (0,3,0,0,0)P
    σ3 = (0,9,0,0,0)P
    three stages
    d10
    pdf version





    Fig. C3.14.10
    RP10
    n = 10
    Has two prototiles — ρ2, ρ4 — of the five — ρ1, ρ2, ρ3, ρ4, ρ5 — in SRI10
    f2(5) = 3
    σ1 = (0,0,1,0,0)P
    σ2 = (2,0,1,0,1)P
    σ3 = (4,0,5,0,2)P
    three stages
    d10
    pdf version



    Fig. C3.14.11
    RP12
    n = 12
    Has two prototiles — ρ2, ρ4 — of the six — ρ1, ρ2, ρ3, ρ4, ρ5, ρ6 — in SRI12
    f2(6) = 3
    σ1 = (0,1,0,0,0,0)P
    σ2 = (0,3,0,0,0,0)P
    σ3 = (0,9,0,0,0,0)P
    three stages
    d12
    pdf version





    Fig. C3.14.12
    RP12
    n = 12
    Has three prototiles — ρ2, ρ4, ρ6 — of the six — ρ1, ρ2, ρ3, ρ4, ρ5, ρ6 — in SRI12
    f2(6) = 3
    σ1 = (0,0,1,0,0,0)P
    σ2 = (0,0,3,0,0,0)P
    σ3 = (0,0,9,0,0,0)P
    three stages
    d12
    pdf version


    Fig. C3.14.13
    RP14
    n = 14
    Has two prototiles — ρ2, ρ4 — of the seven — ρ1, ρ2, ρ3, ρ4, ρ5, ρ6, ρ7 — in SRI14
    f2(7) = 3
    σ1 = (0,1,0,0,0,0,0)P
    σ2 = (0,3,0,0,0,0,0)P
    σ3 = (0,9,0,0,0,0,0)P
    three stages
    d14
    pdf version




    Fig. C3.14.14
    RP14 tiling
    n = 14
    Has three prototiles — ρ2, ρ4, ρ6 — of the seven — ρ1, ρ2, ρ3, ρ4, ρ5, ρ6, ρ7 — in SRI14
    f2(7) = 3
    σ1 = (0,0,0,0,0,1,0)P
    σ2 = (0,0,0,0,0,3,0)P
    σ3 = (0,0,0,0,0,9,0)P
    three stages
    d14
    pdf version
    pdf version of modified tiling






    Fig. C3.14.15
    RP16
    n = 16
    Has two prototiles — ρ2, ρ4 — of the eight — ρ1, ρ2, ρ3, ρ4, ρ5, ρ6, ρ7, ρ8 — in SRI16
    f2(8) = 3
    σ1 = (0,1,0,0,0,0,0,0)P
    σ2 = (0,3,0,0,0,0,0,0)P
    σ3 = (0,9,0,0,0,0,0,0)P
    three stages
    d16
    pdf version




  • C3.15 Some history
  • In his legendary 1977 essay, Martin Gardner introduced Penrose tilings in his Scientific American column, Mathematical Recreations. SUN and STAR show the central regions of the two d5-symmetric Penrose tilings. (These images were derived here as duals of de Bruijn pentagrids.) In 1981, Nicolaas G. de Bruijn proved that the dual of every regular pentagrid that satisfies a simple restriction on the translational shifts γj of the five grids of the pentagrid is identical to a Penrose tiling created by arranging rhombs constrained by Penrose's matching rules.

    In 1977, after studying Gardner's article, I persuaded a few students to join me in the study of Penrose tilings. I ordered steel-rule dies for cutting kites, darts, and long and short bow-ties from color-printed cardstock, and we soon amassed a large supply of die-cut tiles. I decided to focus on the three special Penrose tilings: SUN (d5 symmetry), STAR (d5 symmetry), and CARTWHEEL (Γ = 0)

    When I examined the image of the CARTWHEEL tiling, I noticed that each of its ten triangular sectors contains an infinite alternating sequence of successively larger replicas of the central portion of the SUN or the STAR. I also observed, but did not prove, that for any two consecutive replicas in this sequence, the distance from the center of the tiling to the center of the replica increases by a multiplicative factor equal to the golden ratio.

    Looking at the growth outward from the center of the SUN and STAR, I noticed an apparently recursive structure, which is described below. I tried unsuccessfully to prove that this recursion continues beyond its first few iterations. In the April 1978 Notices of the American Mathematical Society I published a preliminary report on this topic, although I failed to state that I had not actually proved anything!

    In late 1978, I welcomed two surprise visitors at my home: Hank Saxe and Cynthia Patterson, ceramic artists extraordinaire from Taos, New Mexico. Hank had been my student and close friend at CalArts several years earlier. I insisted on holding my two guests (and their golden retriever) hostage until they agreed to consider designing and making Penrose patterns from ceramic tiles. I knew that Hank was expert in the production of colored ceramic glazes, and I was convinced that together Hank and Cynthia would produce spectacular works of mathematical art. They didn't disappoint me. They quickly obtained permission from Roger Penrose to make ceramic versions of his tilings, and you can see samples of their work on their website .

    In 1979, I encouraged my students to make a 24' x 24' CARTWHEEL tiling by gluing colored paper rhombs (2 cm. edgelength) onto thirty-six 4' x 4' sheets of masonite. We planned to use the panels for a traveling Penrose exhibit at Illinois high schools. When the project was still unfinished at the end of the semester, we stored the panels in our department building. Unfortunately, the panels — and other property stored in the building — mysteriously disappeared during the end-of-semester break, and work on the project was never resumed.

    In those days, I hadn't yet heard of Robert Ammann's work on aperiodic tilings. It wasn't until 1981 that Nicolaas de Bruijn published his pair of ground-breaking monographs explaining the connection between multigrids and Penrose tilings. (At a Cincinnati AMS meeting soon afterward, David Klarner invited me to spend the evening with him and his friend de Bruijn, who graciously gave me copies of his multigrid monographs.) Now for the first time it became possible to construct Penrose tilings without having to proceed step-by-step, following matching rules and hoping you wouldn't end up at an impasse and have to backtrack. Until de Bruijn's breakthrough, matching rules were the principal device for forcing aperiodicity. I also benefited from reading the early articles by the physicist Paul Steinhardt and his students. This was still several years before Shechtman and his colleagues at NIST astonished the world by discovering three-dimensional quasicrystals, confirming the earlier expectations of Roger Penrose and Alan Mackay (and perhaps nobody else!).

    I naively wondered whether one could find a set of matching rules for tilings by rhombs for n = 7 that would also force aperiodicity, i.e., allow only non-periodic tilings. I tried everything I could think of, but nothing worked. It was only later that I learned that such matching rules had been proved impossible for n = 7. I was intrigued by tilings by the three rhombs of SRI7, but I recognized that tilings by those rhombs — or any other non-Penrose rhombs — lack the special charm of authentic Penrose tilings based on the magic number five.



    Detour into rhombic rosettes, ROMBIX, and other things

    One day in December 1979 I suddenly decided to invent a new tiling puzzle, by somehow marrying the idea of a polyomino (the brainchild of Sol Golomb) with the idea of tiling a regular 2n-gon by a set of n(n − 1)/2 rhombs, as described by Donald Coxeter. I decided to experiment by constructing a twin, in every possible edge-to-edge configuration, from every possible pair of rhombs in Coxeter's set of n(n − 1)/2 rhombs. When I tested this idea for every n ≤ 10, I was startled to discover that so long as one adds exactly one specimen of each single rhomb ('keystone') to the collection of twins, the combined area of all the pieces is precisely equal to that of the regular 2n-gon. I quickly proved that this holds for all integer n ≥ 3.

    I found that I was able to tile the interior of the regular 2n-gon with some arrangement of the pieces of every set of order n ≤ 12. Of course it took a lot longer for the larger values of n. At first I called the puzzle 'CYCLOTOME', but later I shortened the name to 'ROMBIX'.

    Several years and a couple of patents later, injection-molded ROMBIX sets composed of the sixteen pieces for n = 8 were manufactured in China and marketed in the U.S. The manufacturer was unwilling to spend any money to advertise them, however, and production stopped after a couple of years. But in 1991 Kate Jones of KADON made a laser-cut acrylic version of ROMBIX that is still sold today.



    Back to RP7 tilings

    Immediately below is a 2005 photo of a paper RP7 tiling I started to paste in 1981. It's the same tiling as (0,0,1)B-1, (0,0,1)B-2, (0,0,1)B-3, and (0,0,1)B-4. After a couple of weeks of cutting and pasting, it was still barely half finished. But I became bored with it and left it in this unfinished state for the next twenty-four years.



    Fig. C3.17.1
    A paste-up of (0,0,1)B-3, left unfinished in 1981


    In 2005, my interest in (0,0,1)B-3 was revived by a phone call from Tom Rodgers, the Atlanta impresario who founded — and continues to host — the now binennial Gatherings for Gardner. Tom told me that he had seen ceramic versions of d7 rhombic tilings on the Saxe-Patterson website and wanted to use such a design as a logo for G4G7. After Tom's call, I decided to take up cutting and pasting again. Here's what the tiling looks like now:


    Fig. C3.17.2
    The completed (0,0,1)B-3 paste-up

    When I examined the unfinished panel, it suggested some mathematical questions that I probably hadn't thought much about in 1981 but that now seemed to demand answers. I've since found answers for most — but not all — of these questions. Many of the questions and answers are discussed in § C3.1-C3.8.

    The most awkward of these questions is how to prove that as the pattern grows radially outward, it can retain the d7 symmetry of the central nucleus (core) at every stage of recursion, i.e., that a proper wedge exists at every stage (cf.§ C3.1 and § C3.10). I still can't answer this question, but at least it can now be said (cf.§ C3.4 and § C3.8) that for both pointed and blunt tilings,

           (a) the d7-symmetric necklaces that decorate the heptagonal boundary mirrors can be constructed at every stage of recursion, and
           (b) for prime n the composition of every necklace is unique.

    In practice it hardly matters whether one can prove that a tiling can be extended symmetrically outward forever, since the tiling expands so rapidly at each stage that it is unrealistic to consider physical tilings (or even mere computer images!) that exceed four or five stages.The precise number of stages depends, of course, on whether you're planning to cover just the floor of a room or an area the size of Rhode Island.

    RP7 tilings are in a very rough sense self-similar centro-symmetric tilings of the Euclidean plane by rhombs. The three prototile rhombs of RP7 tilings are unmarked, and there are no matching rules. The two prototile rhombs of Penrose tilings, by contrast, are marked in such a way as to prevent periodic tilings and allow only aperiodic tilings, when Penrose's matching rules are imposed on the marked tiles. Although there is a superficial connection between RP7 tilings and the two most symmetrical examples of Penrose tilings, the d5-symmetric SUN and STAR, none of the subtle features of Penrose tilings are found in RP7 tilings. These features include:

    (a) quasiperiodicity, which implies Conway's town theorem ('local isomorphism'),
    (b) projection from a five-dimensional cubic honeycomb,
    (c) a basis for mathematical modeling of the 3-dimensional quasicrystals discovered by Dan Shechtman and his collaborators in 1984,
    (d) the dual relationship between the rhombs of a tiling and the vertices of a de Bruijn pentagrid,
    (e) derivation by the
    inflation of a finite tiling,
    (f) matching rules for kite/dart tilings and tilings by rhombs,
    (g) ubiquitous role of the golden ratio and Fibonacci sequences,
    etc.


  • C3.18 Stereoscopic view of (1,0,0)B stepped pyramid



    Fig. C3.18.1
    cross-eyed stereoscopic view of the (1,0,0)B tiling
    transformed into a stepped pyramid


    C4. Rhombic wallpaper (periodic tilings derived from a variant form of de Bruijn multigrids)

    • C4.0   Introduction
      •   The trivial examples derived from regular and uniform tilings
      •   How densely can rosettes be embedded?
      •   Adaptation of the Gessel and de Bruijn algorithms
      •   The six rosettes for n = 5
      •   Single-lattice and poly-lattice tilings
      •   Lattice stars
      •   The classes and types of RW tilings
      •   Superdense, dense, and sparse tilings
      •   Examples of straight row tiling lattice stars 4.5
    • C4.1   Straight row tilings, aligned: even n
    • C4.2   Straight row tilings, aligned: odd n
    • C4.3   Straight row tilings, staggered: even n
    • C4.4   Straight row tilings, staggered: odd n
    • C4.5   Zig-zag row tilings: odd n
    • C4.6   Square lattice tilings: even n
    • C4.7   Square lattice tilings: odd n
    • C4.8   Rectangular lattice tilings: even n
    • C4.9   Hexagonal lattice tilings: even n
    • C4.10 Hexagonal lattice tilings: odd n
    • C4.11 Additional examples of RWn
    • C4.12 How to construct a RWn
    • C4.13 Origins of RWn




    C4.0 Introduction

    Using the n(n − 1)/2 rhombs of SRIn as prototiles, it's easy to construct examples of periodic tilings in which there are no embedded rosettes (aside from the rosette composed of a single square when n is even). Here is an example of such a tiling for n = 7:


    Fig. C4.0.1
    Periodic tiling for n = 7 in which
    there are no embedded rosettes

    I will ignore all such tilings, and I'll call periodic tilings by the rhombs of SRIn in which rosettes of order n are embedded

    rhombic wallpaper (or RW) of order n.

    For n = 2, 3, 4, and 6, there exist RW tilings that I call the trivial examples. They are based on

        two of the three regular tilings:
              4.4.4.4, tiled by squares,
              6.6.6, tiled by hexagons,
    and on
        two of the eleven uniform tilings:
              4.8.8, tiled by squares and octagons and
              4.6.12, tiled by squares, hexagons, and 12-gons.


    Figs. C4.0.1 a and b

    
    Figs. C4.0.2 a and b


    Figs. C4.0.3 a and b


    Figs. C4.0.4 a and b
    Four regular and uniform tesselations (left) and the four trivial examples of RW tilings based on them (right)

    Now consider the following admittedly frivolous question:

    How densely can non-overlapping rosettes of order n be embedded in a RW tiling of order n?

    This question is related to a slightly more restricted one:

    Imagine an infinite orchard of identical trees arranged on a square lattice. Let ρtree = the tree density (number of trees per unit area). A straight cable is stretched from each tree to 2n other trees, which are called its 'connected neighbors'. Each cable is incident at no tree other than the two at its ends. The arrangement of the 2n connected neighbors ('connected-neighbor configuration') is identical, up to rotation about a vertical axis, for all trees. The number of distinct crossed pairs of cables ('nodes') at each point at which m cables cross is equal to m(m − 1)/2. Let ρnode = the node density (total number of crossed cable pairs per unit area of the rhombic tiling dual to the set of nodes), and call the ratio ρnode/ρtree the tree-normalized node density.

    For given n, which periodic orchard and connected-neighbor configuration have the smallest tree-normalized node density?

    This orchard model is a paraphrase of the 'star grid' scheme for generating RW tilings that is described below.

    In the first three of the four examples shown above (Figs. C4.0.1b-C4.0.3b), the rosettes of order n have maximal density, i.e., they occupy the largest possible fraction of the tiling area among all RW tilings of the same order. But the density of the tiling for n = 6 in Fig. C4.0.4b is almost seven percent less than that of the kagome tiling in Fig. C4.0.5, which is conjectured to have maximal density.


    Fig. C4.0.5
    density = (8√ 3 − 12)/3 ≅ .618802

    For small n, there are a few empirical rules — described below — that predict which RW tiling has maximal density. For every n>2, there is an uncountable infinity of RW tilings in which the density of the rosettes of order n is less than maximal. I describe below a systematic procedure, adapted from Gessel's algorithm for rosettes and from de Bruijn's multigrid algorithm for aperiodic tilings, for generating RW tilings of both maximal and sub-maximal density. For n ≠ 2, the density can be made arbitrarily close to any value less than or equal to the maximal density, by using the splitting and augmenting operation illustrated in § 4.8.

    * * * * *
    (For a concise description of de Bruijn's multigrid algorithm, see pp. 33-44 of Laura Effinger-Dean's undergraduate honors thesis.
    The original article by de Bruijn is entitled "Algebraic theory of Penrose's non-periodic tilings of the plane", Nederl. Akad. Wetensch. Proceedings Ser. A 84 (Indagationes Math. 43) (1981) 38-66. It was reprinted in The Physics of Quasicrystals, ed. P.J. Steinhardt and S. Ostlund, World Scientific Publ. Comp., Singapore (1987), pp. 673-700.)

    * * * * *

    In any aperiodic tiling for n ≥ 3 that is the dual of a de Bruijn multigrid, one cannot help noticing the not necessarily symmetrical rosettes of order n that are embedded here and there. (Here is an example for n = 7.) The occurrence of rosettes is statistically inevitable in these aperiodic tilings. A rosette is the dual of the k(k − 1)/2 points of intersection of any set of k lines in which each line intersects every other at a unique point. In every de Bruijn pentagrid, arrangements of five lines that satisy this condition occur infinitely often, since Conway's town theorem guarantees that the portion P of the tiling dual to every such arrangement is no farther from its nearest replica than slightly more than twice the diameter of P. Here is an example of one of the six five-line arrangements in a pentagrid:


    Fig. C4.0.6
    Ten points in a de Bruijn pentagrid that are dual to a rosette of order 5

    These six line arrangements define the following six rosettes:


    Fig. C4.0.7
    The six rosettes for n = 5

    For n > 5, without analogs of Conway's n = 5 town theorem we don't have an upper bound on the distance between a rosette that is embedded in a de Bruijn aperiodic tiling and its nearest replica. (Incidentally, for n > 5, the number of ways a rosette of order n can be tiled by the rhombs of SRIn is unknown. But I have found that for n = 6, this number is at least 49.)

    The scheme I have devised to explore the rosette density problem for n ≥ 3 can be used to create an uncountable infinity of RW tilings containing embedded rosettes of order n. It generates a 'star grid' from one or more 'lattice stars'. Every intersection of two lines of the star grid is the dual of a rhomb in a RW tiling.

    I'll first describe the scheme for single-lattice tilings, in which a rosette of order n is centered at every lattice point of the tiling. There are no other such rosettes in the lattice fundamental domain. Then I will describe the version for poly-lattice tilings, in which a rosette of order n is centered at each of m points (m>1) in the lattice fundamental domain. In this case, the rosettes are centered at the lattice points of m congruent sub-lattices of the tiling. (The kagome tiling in Fig. C4.0.5 is an example of a poly-lattice tiling with m = 2.)

    Scheme for single-lattice tilings

      (i) Choose a lattice L.
      (ii) Construct a lattice star composed of 2n distinct rays (line segments), each of which extends from a common root-lattice point P0 to a terminal lattice point Pk (k = 1, 2, ..., 2n). The 2n terminal lattice points are all distinct.
      (iii) Generate a star grid by translating the lattice star to each lattice point of L. The rhombs of the tiling are dual to the vertices of the star grid.

    Scheme for poly-lattice tilings (m congruent sub-lattices)

      (i) Choose a lattice L.
      (ii) Construct m lattice stars—one for each of m sub-lattices Lk (k = 1, 2, ..., m) congruent to L, each composed of 2n distinct rays. In each lattice star, every ray extends from a common root-lattice point P0 to a terminal lattice point Pk (k = 1, 2, ..., 2n), which may belong either to the same sub-lattice or to a different sub-lattice. The 2n terminal lattice points are all distinct.
      (iii) Generate a star grid by translating the lattice star for each sub-lattice Lk to every other point of Lk. The rhombs of the tiling are dual to the vertices of the star grid.


    For the unit square lattice (integer lattice), we define the magnitude of a lattice star — using the conventions of taxicab geometry — as the sum of the 'manhattan lengths' of the 2n rays of the star. By this measure, the lattice star in Fig. C4.0.8, for example, has magnitude 44.


    Fig. C4.0.8
    Star magnitude = 44

    For RW tilings for small n on either square or rectangular lattices, rosette density and star magnitude are found to be inversely correlated. Examples are shown in § 4.8. When two different lattice stars for the same n have the same magnitude, the rosette density is usually found to be larger in the tiling for which the tree-normalized node density in the star grid is smaller. (If the tree-normalized node density were replaced by a weighted count of nodes, each weight being the area of the rhomb dual to each node, then the tree-normalized node density would be exactly inversely correlated.)




    It is convenient to identify RW tilings of order n by class and type. These terms are defined below. Tilings are ranked according to density as follows:

    1. A tiling with the highest density of any tiling of order n is called superdense.

    2. A tiling with the highest density of any tiling of order n of its type is called dense.

    3. A tiling with density less than the highest density of any tiling of order n of its type is called sparse.




    The tilings for n = 2 and n = 3 in Figs. C4.0.1b and C4.0.2b, respectively, are superdense, since their density is equal to 1.
    The tiling for n = 4 in Fig. C4.0.3b is superdense. Its density is equal to 2 (√ 2 − 1) ≅ .828427, which is maximal. (It is easily proved that there exists no tiling by regular octagons with higher density.)
    The tiling for n = 6 in Fig. C4.0.4b is sparse, since its density of 1/√ 3 ≅ .577350 is less than the density (8√ 3 − 12)/3 ≅ .618802 of the kagome tiling, shown in Fig. C4.0.5, which is conjectured to be superdense.

    Although all of the tilings for n<9 conjectured to be superdense have reflection symmetries, the tiling for n = 9 (cf. Figs. C5.9.1a and b) that is conjectured to be superdense does not — it has only p3 symmetry. Its density is almost 30% larger than that of the tiling with p3m1 symmetry in Fig. C4.10.2.



    Every RW tiling treated here is characterized as belonging to either the row class or the dispersed class.

    Within the row class, there are two sub-classes: straight and zig-zag.
    Within each of these two sub-classes, there are four types.

    Within the dispersed class, there are three sub-classes: square lattice, hexagonal lattice, and other lattices.

    This classification scheme is by no means exhaustive. The kagome tiling in Fig. C4.0.5 belongs to neither dispersed nor row class. Neither do tilings in which all the rosettes are arranged in closed rings, with every rosette incident at each of two neighbors. In another example, some rosettes are incident at no other rosettes, and still other rosettes are arranged in rows.

    I have not investigated examples of every one of the types listed below. Except for a few small values of n, I have not proved maximal density. The labels 'dense' and 'superdense' should be regarded as tentative except where stated otherwise.



    THE CLASSES, SUB-CLASSES, AND TYPES OF RHOMBIC WALLPAPER TILINGS

    Row tiling
    The rosettes are arranged in parallel rows.
    Every rosette is contiguous to exactly two other rosettes in the same row.
    No rosette shares a vertex with a rosette in another row.
      Straight row
      The rosette centers lie at equal intervals on a straight line.
        Vertex-sharing
        Contiguous rosettes share a vertex, but not an edge.
        Edge-sharing
        Contiguous rosettes share an edge.
        Aligned
        The line between the centers of two contiguous rosettes is perpendicular to the line between the center of either of them and the center of the nearest rosette in an adjacent row.
        Staggered
        Non-aligned
      Zig-zag
      The rosette centers lie at the vertices of a symmetrical sawtooth.
        Vertex-sharing
        Contiguous rosettes share a vertex, but not an edge.
        Edge-sharing
        Contiguous rosettes share an edge.
        Aligned
        The line between the centers of two contiguous rosettes is perpendicular to the line between the center of either of them and the center of the nearest rosette in an adjacent row.
        Staggered
        Non-aligned
    Dispersed tiling
    No two rosettes are contiguous
      Square lattice
      Hexagonal lattice
      Other lattices


    A RW tiling may also be categorized according to the residue class of n, e.g., even, odd, congruent to 3 (mod 6), etc. For every tiling, whether it is of the dispersed or row class, the wallpaper group is identified. But at every point of intersection of three or more lines in a star grid, the orientation in the tiling plane and — for four or more intersecting lines — the symmetry of the arrangement of rhombs inside the convex polygon ('oval') dual to the point of intersection is indeterminate. An arbitrary choice of orientation (and also of symmetry, when four or more lines are involved) must be made in each such case. The wallpaper group of the tiling depends on precisely which choices are made.

    For some families of dense tilings of a particular type that are parametrized by the order n of the tiling, the dependence of density on n — for small n — can be described by an algebraic expression conjectured to hold for all n. In some cases, it is not difficult to guess an asymptotic form for this expression that is confirmed by numerical calculations, but I have not proved any of these results. Some of these asymptotic expressions appear following Fig. C4.10.2.



    EXAMPLES

    Let's now examine examples of tilings of several types, in increasing order of n. The tiles are defined to have unit edge length. In several cases, I have included the lattice star or star grid (or both). In addition to density ρn, I record the wallpaper group and — for row tilings — λn, the distance between the center-lines of adjacent rows. Some types have no representatives, because I am considering only edge-to-edge tilings, i.e., tilings in which the corners and sides of the tiles coincide with vertices and edges of the tiling (cf. Tilings and Patterns, by Grunbaum and Shephard, p. 18).


    Fig. C4.0.9
    Examples of straight row tiling lattice stars for small n.
    Top row: lattice stars for the square lattice.
    Bottom row: lattice stars for the hexagonal lattice.



    C4.1 Straight row tilings, aligned: even n


    Fig. C4.1.1

    n = 2





    Fig. C4.1.2
    n = 4





    Fig. C4.1.3
    n = 6





    Fig. C4.1.4a
    n = 8
    Straight row aligned edge-sharing
    ρ8 ≅ .394591
    λ8 ≅ 10.1371
    pmm
    (This is the same tiling as the one at the right in Fig. C4.1.6.)


    Fig. C4.1.4b
    n = 8
    Lattice star for the tiling of Fig. C4.1.4a


    Fig. C4.1.4c
    n = 8
    Star grid for the tiling of Fig. C4.1.4a
    (This star grid is drawn on the same scale as the lattice star in Fig. C4.1.4b.)





    Fig. C4.1.5a
    n = 8
    Straight row aligned vertex-sharing
    ρ8 ≅ .253965
    λ8 ≅ 15.4476
    pmm
    (This is the same tiling as the one in Fig. C4.1.5d
    and is a slightly rearranged version of the tiling at the left in Fig. C4.1.6.)




    Figs. C4.1.5b and C4.1.5d demonstrate that although the angular distribution of the 2n rays in a lattice star is not uniform, those rays represent 2n radial lines that are uniformly distributed: each side of every rhomb in the tiling is perpendicular to one of the two radial lines associated with the pair of intersecting rays whose intersection is dual to the rhomb.





    Fig. C4.1.5b
    n = 8
    Lattice star for the tiling of Fig. C4.1.5a


    Fig. C4.1.5c
    n = 8
    Star grid for the tiling of Fig. C4.1.5a
    (This star grid is drawn on the same scale as the lattice star in Fig. C4.1.5b.)


    Fig. C4.1.5d
    n = 8

    Straight row aligned vertex-sharing
    Ladders 0 (red), 1 (green), 2 (blue), 3 (violet)
    (cf. rays 0, 1, 2, 3 in the lattice star of Fig. C4.1.5b)
    Each thick black radial segment in the central rosette is perpendicular to the rungs of its associated ladder.





    Fig. C4.1.6
    n = 8





    Fig. C4.1.7
    n = 10




    C4.2 Straight row tilings, aligned: odd n


    Fig. C4.2.1
    n = 3
    ρ3 ≅ .75000
    λ3 ≅ √ 3
    pmm





    Fig. C4.2.2
    n = 5
    ρ5 ≅ .5414                                                                       ρ5 ≅ .5150
    λ5 ≅ 4.9798                                                                       λ5 ≅ 4.8541
    pmm





    Fig. C4.2.3
    n = 7
    ρ7 ≅ .3466                                                                       ρ7 ≅ .3745
    λ7 ≅ 9.8447                                                                       λ7 ≅ 9.3488
    pmm




    C4.3 Straight row tilings, staggered: even n


    Fig. C4.3.1
    n = 2





    Fig. C4.3.2
    n = 4





    Fig. C4.3.3
    n = 6





    Fig. C4.3.4
    n = 8




    C4.4 Straight row tilings, staggered: odd n


    Fig. C4.4.1
    n = 3
    ρ3 ≅ 1
    λ3 ≅ 1.5
    pmm





    Fig. C4.4.2
    n = 5
    ρ5 = (1 - 8√ 5)[25 - 10√(6 - 2√ 5)] ≅ .669153
    λ5 ≅ √ 5 + 1.5 ≅ 3.7361
    pmm





    Fig. C4.4.3
    n = 7
    ρ7 ≅ .460658
    λ7 ≅ 7.5978
    pmm




    C4.5 Zig-zag row tilings, aligned: odd n


    Fig. C4.5.1a
    n = 5
    Lattice star for sites of type A in zigzag-row tiling of Fig. C4.5.1d.
    A and B are the two inequivalent types of sites in each lattice fundamental domain.



    Fig. C4.5.1b
    n = 5
    Lattice star for sites of type B in zigzag-row tiling of Fig. C4.5.1d.
    A and B are the two inequivalent types of sites in each lattice fundamental domain.



    Fig. C4.5.1c
    n = 5
    Star grid for the zigzag-row tiling of Fig. C4.5.1d
    (This star grid is drawn on the same scale as the lattice stars in Figs. C4.5.1a and b.)



    Fig. C4.5.1d
    n = 5
    Zigzag-row tiling
    ρ5 = 5 − 2√ 5 ≅ .527864
    λ5 ≅ 4.9798




    (Skeletons of zig-zag rows of non-overlapping rosettes)




    In Figs. C4.5.2 and C4.5.3 are two zig-zag row tilings for n = 7. Each of them is a poly-lattice tiling, with m = 2.


    Fig. C4.5.2a
    n = 7
    Lattice stars A and B for zigzag-row tiling no. 1 of Fig. C4.5.2c


    Fig. C4.5.2b
    n = 7
    Star grid for zigzag-row tiling no. 1 of Fig. C4.5.2c


    Fig. C4.5.2c
    n = 7
    Zigzag-row tiling no. 1
    ρ7 ≅ .4655
    λ7 ≅ 7.8948





    Fig. C4.5.3a
    n = 7
    Lattice stars A and B for zigzag-row tiling no. 2 of Fig. C4.5.3c



    Fig. C4.5.3b
    n = 7
    Star grid for zigzag-row tiling no. 2 of Fig. C4.5.3c



    Fig. C4.5.3c
    n = 7
    Zigzag-row tiling no. 2
    ρ7 ≅ .4547
    λ7 ≅ 8.5429




    Now let us look at some examples of lattice tilings — mostly, but not all — dense tilings. We begin with

    C4.6 Square lattice tilings: even n


    Fig. C4.6.1
    n = 4
    ρ4 = 2 (√ 2 − 1) ≅ .828427





    Fig. C4.6.2
    n = 4
    ρ4 = 2 (√ 2 − 1) ≅ .828427


    Fig. C4.6.3
    n = 4
    ρ4 = 2 (√ 2 − 1) ≅ .828427





    Fig. C4.6.4a
    n = 8
    If you fill these 81 holes (and half-holes) ...



    Fig. C4.6.4b
    n = 8
    with these 81 polka dots ...



    Fig. C4.6.4c
    n = 8
    you get this superdense tiling.
    ρ8 ≅ .425442





    Fig. C4.6.5
    n = 8
    ρ8 ≅ .0143164
    Compare this quite sparse tiling to the superdense tiling in Fig. C4.6.4c.
    (To see this one in brighter colors, look here.)





    Fig. C4.6.6a
    n = 10
    ρ10 ≅ .226550
    Introducing a non-convex prototile into this tiling, which has no reflection symmetries,
    changes it into the one shown in Fig. C4.6.6b, which does have reflection symmetries.


    Fig. C4.6.6b
    n = 10
    This tiling has reflection symmetries
    and a translational fundamental domain only half as large as in Fig. C4.6.6a.





    Fig. C4.6.7
    n = 12
    ρ12 ≅ .261899
    For a pdf version of this tiling, look here.

    In addition to the very prominent d3 rosettes of order 12, smaller rosettes of orders 3, 4, and 6 appear in this pattern. You can count a total of sixteen rosettes in each translational fundamental region of the lattice. (The symmetry of this tiling is reduced by errors in the orientation of the rosettes.)



    C4.7 Square lattice tilings: odd n



    C4.8 Rectangular lattice tilings: even n


    Fig. C4.8.1a
    n = 8
    Lattice star for rectangular lattice tiling in Fig. C4.8.1c
    Star magnitude = 52



    Fig. C4.8.1b
    n = 8
    Star grid for rectangular lattice tiling in Fig. C4.8.1c



    Fig. C4.8.1c
    n = 8
    Star magnitude = 52





    Fig. C4.8.2a
    n = 8
    Lattice star for rectangular lattice tiling in Fig. C4.8.2c
    Star magnitude = 44



    Fig. C4.8.2b
    n = 8
    Star grid for rectangular lattice tiling in Fig. C4.8.2c



    Fig. C4.8.2c
    n = 8
    Star magnitude = 44





    Fig. C4.8.3
    n = 8
    Splitting (left) and augmenting (right) the tiling in Fig. C4.8.2c





    Fig. C4.8.4a
    n = 8
    Lattice star for rectangular lattice tiling in Fig. C4.8.4c
    Star magnitude = 44



    Fig. C4.8.4b
    n = 8
    Star grid for rectangular lattice tiling in Fig. C4.8.4c



    Fig. C4.8.4c
    n = 8
    Star magnitude = 44





    Fig. C4.8.5a
    n = 8
    Star magnitude = 36



    Fig. C4.8.5b
    n = 8



    C4.9 Hexagonal lattice tilings: even n


    Fig. C4.9.1
    n = 6
    ρ6 =1/√ 3 ≅ .577350
    superdense




    C4.10 Hexagonal lattice tilings: odd n


    Fig. C4.10.1a
    n = 9
    ρ9 ≅ .341644
    superdense
    p3


    Fig. C4.10.1b
    (Another coloring of the superdense tiling of Fig. C4.10.1a)



    Fig. C4.10.2
    n = 9
    ρ9 ≅ .263346
    sparse
    p3m1



    Below are:
           C4.11 Additional examples of RWn
           C4.12 How to construct a RWn
           C4.13 Origins of RWn



    C4.11 Additional examples of RWn

    Below are examples of both dense and sparse RWn tilings for n in the interval [2,22].

    In many of the the RW tilings shown here, each shape of rhomb has a characteristic color (or shading). Some color schemes produce striking subliminal patterns—approximations of circles, triangles, hexagons, etc. The scale of such patterns is sometimes so large that an assembly of several unit cells of the tiling may be required to reveal them.

    It is characteristic of RWn tilings that along lines of reflection of the tiling lie infinite linear strings of rhombs that are analogs of minimal pointed beadstrings (cf. § C3.2): each bead is composed of a single rhomb. Some tilings in addition contain finite strings composed of blunt beads, aligned in directions that do not coincide with lines of reflection of the tiling.

    RW21(4) is an instructive example. The axes of three pointed strings of infinite length, intersecting at the center of the image, lie on lines of reflection of the tiling at 0, 60, and 120 degrees from the vertical. In addition every pair of nearest neighbor rosettes is joined by a finite string of blunt beads. These blunt strings can only partially mimic the mirroring effects of blunt beadstrings in RPn tilings, described in § 3.2 and § 3.3. The tiling would be perfectly symmetrical by reflection in the longitudinal axis of each such blunt string if it were not for the fact that neither

          (a) the arrangement of rhombs in the interior of the beads of each string
    nor
          (b) the tiling of the rosettes

    is symmetrical by reflection in those axes. In RW21(4), the rosettes would require d6 symmetry, but symmetry of even order is impossible for rosettes (cf. § B1).

    RW9, illustrates the same effects.



    n = 2: Regular tiling by squares (square lattice)



    n = 3: Regular tiling by hexagons, each tiled by three congruent rhombs (hexagonal lattice)



    RW6(1)




    n = 7: Not every rosette is centered at a lattice point (rectangular lattice).

    Fig. C4.11.1
    RW7(1)
    ρ7 ≅ 0.5289856

    I will soon post a picture of the 25" x 38" RHOMBBURST poster, which includes an article at the bottom summarizing the state of knowledge in 1995 about Penrose tilings. Unlike RW8(2) (above), the tiling of the RHOMBBURST poster is not a wallpaper pattern—it is a centro-symmetric tiling with a single center of symmetry. It has 16 lines of reflection and therefore has d16 symmetry. How many lines of reflection do you find in a unit cell (fundamental domain under translation) of the periodic RW8(2) (above)?


    RW9(1) is a more colorful version of RW9. Note that the image is rotated, relative to RW9, by one-sixth of a turn around its center. In this orientation, it is symmetrical by reflection in a vertical line through the center.




    RW10(1) is a row tiling.

    Fig. C4.11.2
    Two colorings of RW10(1)

    For other images of RW10(1), see RW10(2) , RW10(3) or RW10(4) .

    Note that exactly halfway between adjacent rows of large rosettes in RW10(1), there is a string of strawberry rosettes of order 5. In row tilings of odd order, there is a medial string of convex polygons, in a regular alternating sequence, with n-1 and n+1 sides, respectively. I will soon add an example or two.

    In RW10(1), the large rosettes in adjacent rows are 'staggered' (offset) by a rosette circumradius. There also exist 'non-staggered' row tilings, for both odd and even n. In row tilings, the convex 'wave fronts' of tiles that border large rosettes in one row are transformed into concave 'wave fronts' by the time they reach the large rosettes in an adjacent row. I call the pattern elements that mediate this change ginkgo leaves.

    RW16_row and RW22_row demonstrate the hierarchical arrangement of ginkgo leaves in row tilings. Ginkgo leaves bear a superficial resemblance to the arbelos (shoemaker's knife) of Archimedes, but while the circular arcs that define the boundary of the arbelos do not all have the same curvature, all three boundary 'curves' of a ginkgo leaf have the same curvature.



    n=15: RW15 has a hexagonal lattice.

    Fig. C4.11.3
    RW15
    For a pdf version, look here.
    RW15(1) is another coloring of RW15.



    n=16: RW16(1) illustrates how broken symmetries unavoidably appear when you embed a rosette—which necessarily has symmetry of odd order—in a RW tiling with no symmetries of odd order. (In its present form, this tiling has no symmetries, but a half-turn rotation of every rosette in alternate horizontal rows would introduce horizontal lines of reflection.)



    In RW16_row, a row of n=8 small rosettes lies halfway between each pair of adjacent rows of n=16 large rosettes:


    Fig. C4.11.4
    RW16_row

    The horizontal center-line of each row of n=16 rosettes is a line of reflection, but the horizontal center-line of each row of n=8 rosettes is not a line of reflection. The enlarged image below shows that the 16-gon boundaries of adjacent n=8 rosettes are oppositely rotated around their centers.



    Fig. C4.11.5
    Adjacent n = 8 rosetttes in RW16_row are oppositely rotated.




    n=21: RW21(1) is an ambitious example of a RWn tiling. To see the smallest rhomb clearly, you may have to enlarge the image.

    It's apparent that d3 rosettes fit more harmoniously on a hexagonal lattice than on a square lattice. I'll explain below how the overall symmetry is affected by the parity — (even n vs. odd n) — of the SRIn.

    To see how color choices for the rhombs affect RW21, see RW21(2),   RW21(3),     RW21(4),     RW21(5),  and   RW21(6).  
    RW21(6) is a monstrously large piece of RW21. It contains seven times as many rhombs as the other versions. I didn't attempt to find color choices that would emphasize 'subliminal' image effects, but you can see suggestions of such effects in the roughly circular 'watermarks' embedded in the pattern.

    Here is an ordered sequence of ten images of RW21. In each image only one shape of rhomb is highlighted. The sequence begins with the smallest rhomb and ends with the largest rhomb. By closely examining image sets like these, one could probably discover how to enhance the strength of particular subliminal images. (I have no plans to do that!)



    n=22: RW22 is a row tiling of order 22 that shows the 'mortar' between rosettes but not the rosettes themselves.



    C4.12 How to construct a RWn

    Let's first recall that a de Bruijn multigrid is a set of n overlapping uniformly rotated grids of parallel lines. Here's an example for n = 5:


    Fig. C4.12.1
    A de Bruijn pentagrid

    If no point of a multigrid belongs to more than two of its n grids, de Bruijn calls the multigrid regular; otherwise he calls it singular.

    A star grid of order n is the periodic counterpart of a de Bruijn multigrid of order n.

    DEFINITIONS:

    (1) A lattice star of order n is a set of 2n rays (line segments) associated with the lattice L. One end of every ray is incident at the common lattice point P0; the other end is incident at a distinct one of the lattice points Pk (k=0, 1, 2, ..., n-1).
    The 2n rays of the lattice star are labelled CCW by their n integer indices 0, 1, 2, ..., n, n+1, ..., n+2, ..., 2n-1 (mod n).
    A lattice star has the same point symmetry as its root lattice point P0.

    (2) A star grid of order n is the union of congruent parallel lattice stars of order n, one specimen of which is rooted at every point of the lattice L.

    In a star grid of order n, n lines intersect at every lattice point. Mimicking de Bruijn, we call any point of a star grid at which only two lines intersect a regular point, and any point at which more than two lines intersect a singular point.

    The dual of every regular point in a star grid is a rhomb whose face angles are defined by the difference between the indices of the two rays that intersect there. The set of rhombs dual to a singular point at which m lines intersect (m=3, 4,...,n) is a convex assembly of m(m-1)/2 rhombs that tile a 2m-gon. The precise arrangement of these rhombs is indeterminate. It is appropriate, whenever possible, to arrange them in a tiling of the 2k-gon that has the point symmetry of the singular point. If the point symmetry of the singular point is dihedral of odd order, it is always possible to find an arrangement of these rhombs with the same symmetry. But if the point symmetry of the singular point is dihedral of even order, there exists no arrangement of the rhombs with the same symmetry. For these reasons, RWn tilings of odd order are somewhat more 'harmonious' in appearance than those of even order — their space group is likely to contain more symmetries.

    For large separation of the rosettes (sparse dot packing) in a RWn tiling, the unit cell of the lattice is large, and the tiling may at first seem indistinguishable from a pseudo-Penrose tiling derived from a de Bruijn multigrid (cf. de Bruijn 2/14(1), for example).



    A lattice star for n=9 is shown below. Because this lattice star produces a RW9 tiling with higher density than any alternative set of lattice points of the same symmetry, it is called dense. For illustration purposes, circles have been centered at each lattice point to indicate the positions (although not the sizes) of the rosettes. In the associated star grid, nine lines intersect at the center of every circle. These lines are truncated here by the circle boundaries.


    Fig. C4.12.2


    Fig. C4.12.3
    This extremal star grid for n = 9 is a periodic array of replicas of the lattice star in Fig. C4.12.2.

    The dual of this star grid is the tiling in Fig. C4.10.2, which is shown in color here.
    (Note that a chain of three-bead Conway worm segments connects each pair of nearest-neighbor rosettes.)

    It is instructive to compare an edge between a particular pair of lattice points in the star grid with the corresponding chain of rhombs in the tiling. In the star grid above, there are only two kinds of edges—those between nearest-neighbor dots and those between fourth-nearest-neighbor dots. Choose a particular edge and then follow the chain of rhombs ('ladder' with parallel rungs) between the two rosettes in RW9(1) that correspond to the pair of lattice points joined by the edge.




    Fig. C4.12.4
    An extremal star grid for n = 15, which is the dual of the tiling RW15(1).


    Fig. C4.12.5
    A zoom shot of the region just below the center of the image in Fig. C4.12.4






    Fig. C4.12.6
    A single lattice unit cell of the extremal star grid for n = 16, which is the dual of RW16(1)



    .
    Fig. C4.12.7
    The lattice star for the extremal tiling RW21(1).

    .
    Fig. C4.12.8
    Star grid for the extremal tiling RW21(1)
    The star grid is a superposition of the lattice stars in Fig. C4.12.7.
    (Lattice points here are approximately six times farther apart than in Fig. C4.12.7.)

    C4.13 Origins of RWn

    In 1991, in the first edition of the manual for the tiling puzzle ROMBIX , I posed a special puzzle challenge called 'POLKA DOTS'. It is reformulated here as a puzzle for RWn.

    Suppose you are required to tile a vast flat area—like the state of Rhode Island, for example—with Rhombic Wallpaper. Suppose further that there are rosettes ('dots') embedded in this tiling. All of the spaces (the 'mortar') between the dots must be tiled by the rhombs of RWn.

    HOW DENSELY CAN YOU PACK THE 'DOTS'?

    I'll call this rather messy problem connecting the dots. It's an example of a so-called extremal problem. In 1991 I didn't have a general solution and I still don't have one. For n = 2, 3, and 4, the problem is trivial, because solutions follow immediately from properties of regular and semi-regular tilings of the plane by squares, hexagons, and octagons.

    When I revisited this problem in 2002, I replaced the rombiks of order eight by the rhombs of SRIn. Perhaps I was guided by what Halmos called "Polya's dictum":

    "If you can't solve a problem, then there is an easier problem you can't solve—find it!"

    I don't know whether the solution for every n is a periodic tiling, but if it is, then it can be found from an algorithm I adapted from the Gessel-de Bruijn method of associating rhombs with their duals—the points of intersection in configurations of lines.

    The algorithm makes it possible to investigate candidate solutions for any particular n by using a periodic variant of the multigrids invented by de Bruijn for his analysis of Penrose tilings. I call these modified grids star grids. I examined not only dense dot packings, but also sparse ones in which the rosettes are quite widely separated. The density of the rosettes in a tiling ('density') is determined by the design of its lattice star—the basic structural unit of the star grid dual of the tiling. Star grids and lattice stars are defined below.

    For a partial summary of results obtained so far, see dot spacing data. These results have been confirmed for small values of n, but they have not yet been proved.



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