## Back to GEOMETRY GARRET

INFINITE TILINGSAlan H. Schoen

Comments are welcome!

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

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

Dan Shechtman

Awarded 2011 chemistry Nobel prize for his discovery of quasicrystals

Paul Steinhardt

photo by Tony Rinaldo

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-dimensional tilings.two

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

quasiperiodictilings in his legendary monthly column,

, inMathematical RecreationsScientific Americanmagazine. There he explained that one is free to choose either

the so-calledkiteanddartpolygons or a pair ofthinandthick 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 C1.2matchedif the tiling

is edge-to-edge and the markings on the tiles are continuous across every edge. Penrose proved that(a) there is no

periodicmatched tiling of the plane by the prototiles of either Fig. C1.1 or C1.2, and

(b) there is an uncountable infinity ofaperiodicmatched tilings by these prototiles. Such tilings are called quasiperiodic.

Fig. C1.1

Fig. C1.2The

periodicityof 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 remarkableinflation/deflationmechanism (so named by John Conway).

Chapter 7 of Martin Gardner's

The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and, W. W. Norton & Co. (2001), contains a description of inflation/deflation and how it leads to a

Problems

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:

- David Austin's Penrose Tiles Talk Across Miles

- David Austin and Matthew Stamps' Circle Packings from Penrose Tilings

- Donald L. D. Caspar and Eric Fontano's Five-fold symmetry in crystalline quasicrystal lattices

- Makowski's 1998 Biophysical Journal article about Donald Caspar An unreasonable man in a quasi-equivalent world

- Martin Gardner's Penrose tiles, pp. 73-89, excerpted from Chapter 7 of his
The Colossal Book of Mathematics

- E. Harriss and D. Frettlöh's Tiling Encyclopedia (1), Tiling Encyclopedia (2), Tiling Encyclopedia (Socolar)

- Eric Hwang's Penrose Tilings and Quasicrystals

- Ron Lifshitz's Quasicrystals

- Michael S. Longuet-Higgins' Nested icosahedral shells, or How to grow a quasi-crystal, Math. Intelligencer
25, 25-43 (2002)

- Amy Qualls-McClure's Penrose Quilts

- Maurizio Paolini and Alessandro Musesti's Animated Penrose Tilings (videos)

- John Savard's Pentagonal tesselations

- Marjorie Senechal's Quasicrystals' Gifts to Mathematics (video)

- Dany Shechtman's Quasicrystals, a New Form of Matter (video)

- Paul J. Steinhardt's A New Paradigm for the Structure of Quasicrytals

- Walther Steurer's Fascinating Quasicrystals (video)

- Russell Towle's Zonotiles and Rhombic Tilings (videos)

- Eric Weisstein's Penrose tiles

- Michael Widom's Quasicrystals

- Norman J. Wildberger's WildTrig25: Pentagons and 5-fold symmetry (video)

- Stephen Wolfram's Penrose tiles (video)

(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.

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,

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

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

singleprototile ('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 acoveringrather than atiling. 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 Saitohet 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 ofpentagrids, 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 pentagridEvery pentagrid is identified by a set of five

shiftsγ_{i}(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 ifΓ, the sum of the five shifts, is equal to an integer. Among Penrose tilings for which the five shifts areequal, the one for whichΓ= 1 is called SUN and the one for whichΓ= 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 pairaandbof these six pentagrids for which γ_{a}+ γ_{b}= 1

are related byinversionin 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 γ_{i}≤ 1/2, they are the only two Penrose tilings withd5symmetry.The remarkable CARTWHEEL tiling, for which all five shifts are zero, has only

d1symmetry. 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 ratioφ(≅ 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 outwardreflections in the enclosingnecklaces

(pentagonal rings of Conway mirror worms)

of successively larger central regions,

followed by

(b)lateralreflections 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 triangulargapin stage three

(and in all subsequent stages, according to the Recursion Conjecture)

is filled by an image of the nearbytriangular 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 triangulargapin stage three

(and in all subsequent stages, according to the Recursion Conjecture)

is filled by an image of the nearbytriangular 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

f_{5}^{2}= (1 + 2 cosπ/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+0 φ

2+3φ

13+21φ

89+144φ

...Let S

_{1}= {1, 2, 13, 89,...} and S_{2}= {0, 3, 21, 144,...}.

Since the Fibonacci sequence is

{.., 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...}, where

F_{1}= 1,

F_{2}= 0,

F_{3}= 1,

F_{4}= 1,

F_{5}= 2,

F_{6}= 3,

etc., then

S_{1}= {F_{1}, F_{5}, F_{9}, F_{13}, ...} and

S_{2}= {F_{2}, F_{6}, F_{10}, F_{14}, ....}.

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 betweenrhombsandpoints of intersection.

The amplitude of each vector in the centralshift starfor this pentagrid is equal to 1/4.CARTWHEEL (

Γ= 0)Here is the pentagrid for CARTWHEEL.

Next — two differently shaded images of each of thed5-symmetric Penrose tilings, SUN and STAR. In the second image of each pair, concentric pentagonal rings ofConway mirror-wormsare distinguished by color. The role of these mirrors in the conjectured recursive structure is explained below.

SUN (de Bruijn shift = 2/10). (TWOimages:for the second image.)Scroll down

STAR (de Bruijn shift = 4/10).(TWOimages:for the second image.)Scroll down

Here are the pentagrids for SUN and STAR.

pentagrid for SUNpentagrid for STAR Below are images of

Generalized Penrose tilings are de Bruijn tilings with equal grid shifts for which

- the three
specialPenrose tilings —SUN, STAR, and CARTWHEEL—and their dual de Bruijn pentagrids- three
generalizedPenrose tilings and their dual de Bruijn pentagrids.Γis ahalf-integer. Each of the first two hasd5symmetry, and the third hasd10symmetry. They arenotrecursively structured, and they arenotgoverned by Penrose-like matching rules. A smaller portion of the third one, which hasd10symmetry, 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) Here are the pentagrids for these three tilings. Note the emergence of

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

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

d10symmetry (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. RPntilings (recursive rhombic tilings ofdnsymmetry)

- C3.0 Introduction
- C3.1 The recursive structure of RP7 tilings
- C3.2 The structural components of RP7 tilings
- C3.3 Pointed tilings
vs.blunt tilings- C3.4 Beadstring signatures and the linear expansion matrix
- C3.5 Rhomb population totals in consecutive generations
- C3.6 Allowed initial signatures
- C3.7 Beadstring signature parity

- Catalog of initial signatures
- Compound beads
- C3.8 Uniqueness of beadstring signatures
- C3.9 Orientation of beadstring rhombs
- C3.10 Tiling wedges
- C3.11 Rings of 14 stars in RP7 tilings of
d7symmetry- C3.12 Gallery of RP7 tilings
- C3.13 Gallery of RP
ntilings (n≠ 7)- C3.14 Gallery of RP
n^{∗}tilings- C3.15 Some history

- Detour into rhombic rosettes, ROMBIX, and other things
- Back to RP7 tilings
- C3.18 Stereoscopic view of (1,0,0)B stepped pyramid.

The seven liberal arts,

in the Hortus deliciarum of Herrad von Landsberg

(from Wikipedia)

C3.0 IntroductionI call any pseudo-Penrose tiling for

n≥ 3 anRPtiling if itn

(a) contains every shape of rhomb in SRI_{n}(cf.§ B1),viz.,n⁄ 2 shapes for evennand (n− 1) ⁄ 2 shapes for oddn;

(b) is recursive in the restricted sense described below in § C3.1; and

(c) has dihedral symmetrydn.

AnRPtiling is a tiling for evenn^{∗}nthat satisfies both (b) and (c), but not (a), because it omits one or more of the shapes of rhomb in SRI_{n}.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.theblunttiling (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 RPntilings forn= 3, 4, 5, 6, 8, and 9 are in § C3.13, and § C3.14 shows examples of RPn^{∗}tilings forn= 4, 6, 8, 10, 12, 14, and 16.A particular subset of the RP

n^{∗}tilings in § C3.14 allows tilings constructed by a common algorithm. It has the following properties:

The prototiles of the tiling, which exists for everyn> 5, are thetwo smallest rhombsin SRI_{n}—ρ_{1}andρ_{2}. Aside from a central star-like core comprised of 2nspecimens ofρ_{1}, the tiling consists of 2ncentral 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 ofρ_{1}and one specimen ofρ_{2}.It is somewhat misleading to apply the term 'recursive' to RP

ntilings forn> 7 and RPn^{∗}tilings forn> 6, because — as Fig. C3.0.1 demonstrates up ton= 12 and Fig. C3.0.2 demonstrates up ton= 42 — with increasingn, the fraction of the tiling inherited from the previous stage that is unaltered in the current stage decreases significantly because of increasingoverlap(cf.§ C3.1).

Fig. C3.0.1

The geometrical basis ofrecursionin RPnand RPn^{∗}tilings (n≥ 3)

(For details,cf.§ C3.1)For example: in the RP

n^{∗}tiling forn= 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 RP

nand RPn^{∗}tilings recursive is that thenecklacesthat 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 RP

ntilings for oddn≠ 7.RP7 tilings were an afterthought of a 1978 conjecture (

cf.§ C1) suggesting that thed5-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 2jπ⁄ 7.

For alln≥ 3, the ⌊(n− 1) ⁄ 2⌋ prototile rhombs are defined in similar fashion: rhombj(j= 1, 2, ⌊(n− 1) ⁄ 2⌋) has upper and lower face angles 2jπ⁄n. For evenn, 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 RPntilings for evenn(cf.§ C3.13).

Fig. C3.0.3

The three prototile rhombs forn= 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 tilingsRP7 tilings are created by iterating the reflection of a finite

d7-symmetric tiled region, beginning with acoretiling, in the edges of its heptagonal boundarymirror. Each stage of iteration produces a heptagonal annulus that encloses — and thereby enlarges — the tiling of the previous stage. The mirrorsfor 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.M1,_{}M2,_{}M3_{}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 orderlyad hocarrangement of tiles called awedge, 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 theremainder.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.

,M_{1}, andM_{2}are the boundary mirrors of these three stages.M_{3}

Fig. C3.1.2 illustrates the difference between odd and evennin the expansion of thek^{th}central polygonal region.

Fig. C3.1.2

oddnevenn

Ris the circumradius and_{k}ris the inradius of the_{k}k^{th}central polygonal region.The

linear expansion ratiof_{n}=R_{k}_{+1}/R=_{k}r_{k}_{+1}/r_{k}

for the concentric polygonal regions of consecutive stages

is equal to

1 + 2 cosπ/nforoddn≥ 3

and

3 forevenn.For

n= 7,f_{7}≅ 2.8019Although 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

g_{7}=f_{7}^{2}≅ 7.85. (Note thatg_{7}^{2}≅ 62,g_{7}^{3}≅ 484,g_{7}^{4}≅ 3799, andg_{7}^{5}≅ 29,825.)

C3.2 The structural components of RP7 tilingsAt the center of every RP7 tiling (

e.g.,pointed tiling (1,1,0)P) lies ad7-symmetriccoreenclosed by a heptagonalnecklacecomposed of sevenbeadstrings(Conway worms), each of which is a string of one or morebeads(cf.Figs. C3.3.1 and C3.3.2). A bead is ad2-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 calledminimal. (Central beads are discussed in § C3.7.) The regular heptagon whose edges bisect the beadstrings is called amirror(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 curveof every beadstring (the 'beadstring polygon') necessarily hasd2symmetry. 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

requiredto be symmetrical, so long as it is related by reflection to the tiling of its conjugates, it is alwayspossiblefor the tiling of a bead to be symmetrical by reflection in a line perpendicular to the mirror edge associated with the bead (d1symmetry). Only in central beads is the tilingforced— by thed7symmetry of the tiling — to haved1symmetry. Central beads are found only in beadstrings whosesignatures(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 eithersimple,i.e.,tiled by a single rhomb, orcompound,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

associated with it, the tiling on opposite sides of the beadstring would not be related by reflection ine. But it is not necessary for the boundary of every individual bead to haveed2symmetry, so long as every beadstring is symmetrical by reflection in.e

C3.3 Pointed tilingsvs.blunt tilingsI consider here principally two types of RP

nand RPn^{∗}tilings —andpointed.blunt

In

pointedRP7 tilings (cf.Fig. C3.3.1), beadstrings are terminated at each end by a beadvertex(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

bluntRP7 tilings (cf.Fig. C3.3.2), acorner rhombthat 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 tanπ/n(cf.Fig. C3.3.3).

Fig. C3.3.1The core (green), mirror (red heptagon), and necklace (closed chain of seven 6-rhomb convex beadstrings)

for thetiling (1,1,0)Ppointed

Fig. C3.3.2

The core (orange and blue), mirror (red heptagon), and necklace (closed chain of seven 3-rhomb convex beadstrings)

for thetiling (0,0,1)Bblunt

Acorner rhomb(blue) is inserted at each end of every blunt beadstring.

Fig. C3.3.3

Thecorner rhombρ_{0}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 typeV*) 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 calledoverlapping blunt.

C3.4 Beadstring signatures and the linear expansion matrixA 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

k^{th}stage of recursion is specified by thek^{th}signature

σ_{k}= (σ_{k}(1),σ_{k}(2),σ_{k}(3)),

whereσ_{k}(1),σ_{k}(2), andσ_{k}(3) are non-negative integers.σ_{1}is called theinitial signature. Ifσ_{1}= (0,0,0), the tiling is called the null tiling.For both pointed and blunt beadstrings, the three components of the signature

σ_{k}are defined as follows:

In either the left or right halfof a [horizontally oriented] beadstring of thek^{th}necklace,

σ_{k}(j) is the number of boundary edges withprojected lengthsinjπ/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 forn= 7

In a beadstring composed of single-rhomb beads,

rhombρis oriented so that it contributes 2sin_{j}jπ/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 3

π/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

= (sinSπ/7, sin 2π/7, sin 3π/7).

Inpointedtilings, (λ_{k})_{pointed}— half the length of an edge of thek^{th}mirror — is equal to the scalar product ofσ_{k}and:S

( λ_{k})_{pointed}=σ_{k}⋅,SIn

blunttilings, (λ_{k})_{blunt}— half the length of an edge of thek^{th}mirror — is equal to the scalar product ofσ_{k}andplus the term (1/2) tanS/7 (π≅0.2403) contributed by one corner rhomb (cf.Fig. C3.5.2).( λ_{k})_{blunt}=σ_{k}⋅+ (1/2) tanSπ/7.

Fig. C3.4.2

The contribution (1/2) tanπ/7 of a corner rhomb to

the half-edge length (λ_{k})_{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 (

λ_{k})_{blunt}, so that both pointed and blunt tilings can be treated in a unified way:

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

⋅S

=δ⋅,S= (1,1,− 1).δHence

( whereλ_{k})_{blunt}= (σ_{k}+)δ⋅S

=τ_{k}⋅,S

τ_{k}=σ_{k}+.δτ_{k}is called thekth^{}expanded signature.

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

whereFor

n= 3, 5, 7, 9, 11, …, this expansion takes the forms(1/2) tan

π/3 = sinπ/3

(1/2) tanπ/5 =−sinπ/5+sin 2π/5

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

(1/2) tanπ/9 =−sinπ/9+sin 2π/9+sin 3π/9−sin 4π/9

(1/2) tanπ/11 = sinπ/11+sin 2π/11 − sin 3π/11−sin 4π/11+sin 5π/11

(1/2) tanπ/13 = − sinπ/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 signaturesσ_{k}andσ_{k+1}of consecutive beadstrings is expressed by the symmetric tri-diagonallinear expansion matrixE_{3}shown in Fig. C3.4.3. For oddn, letm= (n− 1)/2.E_{3}is then= 7 version of themxmlinear expansion matrixE_{m}that is applicable to all RPntilings for oddn. All of the non-zero elementse_{ij}ofE_{m}are equal to 1, except fore= 2._{mm}

Fig. C3.4.3

The linear expansion matrixE_{3}forn= 7

For pointed tilings,σ_{k+1}^{T}=E_{3}σ_{k}^{T},

and for blunt tilings,

σ_{k+1}^{T}=E_{3}(σ_{k}+)δ^{T}−δ^{T}

= E_{3}τ_{k}^{T }−δ^{T}.Here is proof that for linear recursion via the matrix

E_{m}— withm= (n− 1)/2 — between signatures of consecutivepointedbeadstrings for oddn, the ratio of the edgelengths of the associated mirrors is equal to the linear expansion ratiof_{n}(cf.§ C3.1).And here is proof that for linear recursion between

expandedsignatures of consecutiven= 7bluntbeadstrings via the matrixE_{3}, the ratio of the edgelengths of the associated mirrors is also equal to the linear expansion ratiof_{n}(cf.§ C3.1).

C3.5 Rhomb population totals in consecutive generationsLet

ρ_{k}= (ρ_{k}(1),ρ_{k}(2),ρ_{k}(3)),

whereρ_{k}(j) = the number of specimens of rhombj(j= 1,2,3) in the interior of the mirrorof generationMk_{}k(cf.Fig. C3.1.1),and let

A_{k}= the area in the interior of.Mk_{}Here's my proof that if

where ρ_{k+1}^{T}=E_{3}^{2}ρ_{k}^{T},E_{3}^{2}=E_{3}E_{3}

which is equal toor

,

Fig C3.5.1

then

A_{k+1}/A_{k}=g_{n}(cf.§ C3.1).

C3.6 Allowed initial signaturesIt 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

σ_{1}= (σ_{1}(1),σ_{1}(2),σ_{1}(3)) is calledallowedif and only ifthe beadstring with signature

σ_{2}= (σ_{2}(1),σ_{2}(2),σ_{3}(3)) contains two replicas of the beadstring with signatureσ_{1}— one at either end.Hence

σ_{2}(1) ≥ 2σ_{1}(1),σ_{2}(2) ≥ 2σ_{1}(2), andσ_{2}(3) ≥ 2σ_{1}(3).

An initial signature that is not allowed is calledforbidden.I have proved that

for

pointedtilings, the initial signatureσ_{1}= (σ_{1}(1),σ_{1}(2),σ_{1}(3)) is allowed if and only if

σ_{1}(1) +σ_{1}(3) ≥σ_{1}(2) ≥σ_{1}(1),and for

blunttilings, the initial signatureσ_{1}= (σ_{1}(1),σ_{1}(2),σ_{1}(3)) is allowed if and only if

σ_{1}(1) +σ_{1}(3) ≥σ_{1}(2) ≥σ_{1}(1) − 1.For both pointed and blunt tilings, it is readily proved that if a given signature

σ_{k}is allowed, its successorσ_{k+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 ≤

σ_{1}(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 ≤

σ_{1}(i) ≤n(i=1,2,3) is (n+1)^{3}, it follows that only one-third ofall possibleinitial pointed signatures are allowed. A similar result holds for initial blunt signatures.

C3.7 Beadstring signature parityThe

parityΠof a beadstring signatureσ_{k}= (σ_{k}(1),σ_{k}(2),σ_{k}(3)) is defined as

Π(σ_{k})= (σ_{k}(1),σ_{k}(2),σ_{k}(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 issimpleorcompound.

Π(σ_{k}) has eight possible values. Parity (0,0,0) is calledeven; the seven other parities are calledodd.(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)

pointedand (b)bluntbeadstrings 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

simplebead 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

σ_{1}— pointed or blunt, allowed or forbidden — for which 0 ≤σ_{1}[i] ≤ 2 (i= 1,2,3). Each signatureσ_{1}is followed by the signature of its daughterσ_{2}. Ifσ_{1}is forbidden,σ_{2}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

bluntbead, whether simple or compound, can be constructed simply by attaching a string of parallel rhombs to the boundary of the bottom half of the correspondingpointedbead.

Fig. C3.7.1

The smallest possible pointed compound beads for beadstrings of odd parity

For

pointedsignatures(a) if the initial signature has

evenparity (0,0,0),

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

oddparity,

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

bluntsignatures,(a) if the initial signature has

oddparity (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 signaturesI conjectured as a result of numerical experiments that iff

nis prime, the sum

c_{1}sinπ/n+c_{2}sin 2π/n+ … +c(_{}n-1)/2 sin [(n-1)/2]π/n,

wherec_{1},c_{1}, …c(_{}n-1)/2 are rational numbers, is equal to zero iffIf true, this conjecture would imply that for prime c_{1}=c_{2}= … =c(_{}n-1)/2 = 0.

n,

no two andpointedbeadstrings with different signatures have the same length

no two bluntbeadstrings 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

nis prime, for a given initial signature, the composition of beadstrings that are recursively related via the expansion matrixE_{3}(cf.§ C3.4) is uniquely determined. But because of the peculiar coincidence that not only theedge lengthbut also theareaof 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 forn= 7In a beadstring composed of single-rhomb beads,

rhombρis oriented so that it contributes 2sin_{j}jπ/7 to the total length of a mirror edge.

Let us call a vertex of the prototile rhombρ(_{j}j= 1, 2, 3)oddorevenaccording to whether the face angle ofρat that vertex is an odd or even multiple of_{j}π ⁄7. Ifπ ⁄7 is the unit of measurement for angles, the value of the angle is called theangle index. Two of the angle indices of eachρare even and two are odd. As shown in Fig. C3.9.1, if a beadstring rhomb_{j}ρis oriented so that its_{bead}oddvertices 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 2sinjπ ⁄7. Let us call this orientation thestandardorientation ofρ. No other orientation is possible for such a rhomb in a RP7 tiling._{bead}If a rhomb in a RP7 tiling is bisected by a radial line of reflection, it must be oriented so that its

evenvertices lie on that line. Its contribution to the length of the line is 2cosjπ ⁄7, and it too is described as being instandardorientation.In a blunt RP7 tiling, non-standard orientation for a beadstring rhomb

ρcan be ruled out by a parity argument: if the even vertices of_{bead}ρlie either on a mirror edge or on a line parallel to a mirror edge, the odd vertices lie on a perpendicular line (_{bead}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),ρcannot be incorporated into a bead._{bead}

Fig. C3.9.1

Hypothetical non-standard orientation for a beadstring rhombρin a_{bead}blunttiling

iis the odd angle index of the face angle of the rhomb at its bottom vertex_{odd}(b)

PointedRP7 tilingsIf the rhomb

ρin 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._{bead}

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.)

Fig. C3.9.2

Hypothetical non-standard orientation for a pair of adjacent beads in apointedtiling

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

d7Note the trapezoidal tiles at the center (

cf.the 'pseudo-compound rosette' forn= 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 thenon-standardorientation 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

d5Note the trapezoidal tiles at the center (

cf.the 'pseudo-compound rosette' forn= 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 thenon-standard orientationof the rhombs that are bisected by a radial line of reflection symmetry.

pdf version

C3.10 Tiling wedgesThe 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

essentialto 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 ofd7symmetryThe concentric regular heptagons

,M_{1}, andM_{2}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.)M_{3}

Fig. C3.11.1

'Proof [almost] without words'The images of uniformly spaced starsseven

produced by reflection in the mirror edges at each stage

define a ring ofstarsfourteen

with centers at the vertices of a regular 14-gon.

C3.12 Gallery of RP7 tilingsWhen 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

overlapslook like before they are replaced bywedges. (Pointed tiling (1,1,0)P-8 is a good example. After the image is displayed, toggle between zoom up and zoom down.)

C3.13 Gallery of RP(ntilings (n≠ 7)cf.§ C3.12 for RP7 tilings)Recall (from § 3.4) that the components

σ_{k}(j) (j= 1, 2, …, ⌊n⁄ 2 ⌋) of the signatureσ_{k}of thek^{th}stage of the tiling are defined as follows:

In either the left or righthalfof a [horizontally oriented] beadstring of thek^{th}necklace,

σ_{k}(j) is the number of boundary edges withprojected lengthsinjπ/non 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

f_{n}is equal to three for evenntilings and to 1 + 2 cosπ⁄nfor oddntilings (cf.Fig C3.1.2).A necessary condition for the existence of

bluntRPntilings is that tanπ/ncan be expressed as a linear combination, with integer coefficients, of sines of integer multiples ofπ/n(cf.§ C3.4). This requirement is met by oddnbut not by evenn. Hence there are no blunt RPntilings for evenn.For the 'crystallographic integers'

n= 2, 3, 4, and 6, RPntilings 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 edgesare 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,

gapsare green andoverlapsare red (cf.§ C3.1) .

Fig. C3.13.1

RP3

n= 3

f_{3}= 2

σ_{1}= (2)P

σ_{2}= (4)P

σ_{3}= (8)P

σ_{4}= (16)P

four stages

p3m1

pdf version

Fig. C3.13.2

RP4

n= 4

f_{4}= 3

σ_{1}= (1,1)P

σ_{1}= (3,3)P

σ_{1}= (9,9)P

three stages

p4m

pdf version

Fig. C3.13.3

RP5

n= 5

f_{5}= 1 +φ(≅ 2.618)

σ_{1}= (1,2)P

σ_{2}= (3,5)P

σ_{3}= (8,13)P

three stages

d5

pdf version

Fig. C3.13.4

RP5

n= 5

f_{5}= 1 +φ(≅ 2.618)

σ_{1}= (1,2)B

σ_{2}= (3,5)B

σ_{3}= (8,13)B

three stages

d5

pdf version

Fig. C3.13.5

RP6

n= 6

f_{6}= 3

σ_{1}= (1,3,1)P

σ_{2}= (3,9,3)P

σ_{3}= (9,27,9)P

three stages

p6m

pdf version

Fig. C3.13.6

RP6

n= 6

f_{6}= 3

σ_{1}= (2,3,2)P

σ_{2}= (6,9,6)P

σ_{3}= (18,27,18)P

three stages

p6m

pdf version

Fig. C3.13.7

RP8

n= 8

f_{8}= 3

σ_{1}= (0,2,1,1))P

σ_{2}= (0,6,3,3))P

σ_{3}= (0,18,9,9))P

two stages

p8m

pdf version

Fig. C3.13.8

RP9

n= 9

f_{9}≅ 2.879

Applying the expansion matrixE_{4}, withδ_{9}= (-1,1,1,-1) (cf.§ C3.4), to the intial signature

σ_{1}= (0,0,1,2)B

to obtain subsequent signatures yields

σ_{2}= (1,1,3,5)B

and

σ_{3}= (3,5,9,13)B,

but the second signature here is

σ_{2}= (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 forσ_{2}define beadstrings of the same length.

(There is a similar degeneracy foreveryoddnthat is composite.)two stages

d9

pdf version

C3.14 Gallery of RPn^{∗}tilingsA necessary condition for the existence of

bluntRPn^{∗}tilings is that tanπ/ncan be expressed as a linear combination, with integer coefficients, of sines of integer multiples ofπ/n(cf.§ C3.4). This requirement is met by oddnbut not by evenn. Hence there are no blunt RPn^{∗}tilings for evenn. But ifnis equal to twice an odd integer and the prototiles of the tiling are restricted to the rhombs of SRI_{n ⁄ 2}, then a blunt tiling is possible.If

nis equal to twice an odd integer, the rhombs that are bisected by radial lines of reflection tocornersof the tiling are oriented oppositely to those that are bisected by radial lines of reflection toedge midpointsof 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 —ρ_{2}— of the two —ρ_{1},ρ_{2}— in SRI_{4}

f_{2(2)}= 3

σ_{1}= (0,1)P

σ_{2}= (0,3)P

σ_{3}= (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 —ρ_{2}— of the three —ρ_{1},ρ_{2},ρ_{3}— in SRI_{6}

f_{2(3)}= 3

σ_{1}= (0,1,0)P

σ_{2}= (0,3,0)P

σ_{3}= (0,9,0)P

three stages

p6m

pdf version

Fig. C3.14.3

RP6^{∗}

n= 6

Has one prototile —ρ_{2}— of the three —ρ_{1},ρ_{2},ρ_{3}— in SRI_{6}

f_{6}= 3

σ_{1}= (0,1,0)P

σ_{2}= (0,3,0)P

σ_{3}= (0,9,0)P

three stages

p6m

pdf version

Fig. C3.14.4

RP8^{∗}

n= 8

Has two prototiles —ρ_{2},ρ_{4}— of the four —ρ_{1},ρ_{2},ρ_{3},ρ_{4}— in SRI_{8}

f_{2(4)}= 3

σ_{1}= (0,1,0,0)P

σ_{2}= (0,3,0,0)P

σ_{3}= (0,9,0,0)P

three stages

d8

(1,0)P-1

Fig. C3.14.5

RP8^{∗}

n= 8

Has two prototiles —ρ_{2},ρ_{4}— of the four —ρ_{1},ρ_{2},ρ_{3},ρ_{4}— in SRI_{8}

f_{2(4)}= 3

σ_{1}= (0,1,0,0)P

σ_{2}= (0,3,0,0)P

σ_{3}= (0,9,0,0)P

three stages

d8

(1,0)P-2

Fig. C3.14.6

RP8^{∗}

n= 8

Has two prototiles —ρ_{2},ρ_{4}— of the four —ρ_{1},ρ_{2},ρ_{3},ρ_{4}— in SRI_{8}

f_{2(4)}= 3

σ_{1}= (0,1,0,0)P

σ_{2}= (0,3,0,0)P

σ_{3}= (0,9,0,0)P

three stages

d8

(1,0)P-3

Fig. C3.14.7

RP8^{∗}

n= 8

Has two prototiles —ρ_{2},ρ_{4}— of the four —ρ_{1},ρ_{2},ρ_{3},ρ_{4}— in SRI_{8}

f_{2(4)}= 3

σ_{1}= (0,1,0,0)P

σ_{2}= (0,3,0,0)P

σ_{3}= (0,9,0,0)P

three stages

d8

pdf version

Fig. C3.14.8

RP8^{∗}

n= 8

Has two prototiles —ρ_{2},ρ_{4}— of the four —ρ_{1},ρ_{2},ρ_{3},ρ_{4}— in SRI_{8}

f_{2(4)}= 3

σ_{1}= (0,0,1,0)P

σ_{2}= (0,0,3,0)P

σ_{3}= (0,0,9,0)P

three stages

d8

pdf versionThis 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 —ρ_{2},ρ_{4}— of the five —ρ_{1},ρ_{2},ρ_{3},ρ_{4},ρ_{5}— in SRI_{10}

f_{2(4)}= 3

σ_{1}= (0,1,0,0,0)P

σ_{2}= (0,3,0,0,0)P

σ_{3}= (0,9,0,0,0)P

three stages

d10

pdf version

Fig. C3.14.10

RP10^{∗}

n= 10

Has two prototiles —ρ_{2},ρ_{4}— of the five —ρ_{1},ρ_{2},ρ_{3},ρ_{4},ρ_{5}— in SRI_{10}

f_{2(5)}= 3

σ_{1}= (0,0,1,0,0)P

σ_{2}= (2,0,1,0,1)P

σ_{3}= (4,0,5,0,2)P

three stages

d10

pdf version

Fig. C3.14.11

RP12^{∗}

n= 12

Has two prototiles —ρ_{2},ρ_{4}— of the six —ρ_{1},ρ_{2},ρ_{3},ρ_{4},ρ_{5},ρ_{6}— in SRI_{12}

f_{2(6)}= 3

σ_{1}= (0,1,0,0,0,0)P

σ_{2}= (0,3,0,0,0,0)P

σ_{3}= (0,9,0,0,0,0)P

three stages

d12

pdf version

Fig. C3.14.12

RP12^{∗}

n= 12

Has three prototiles —ρ_{2},ρ_{4},ρ_{6}— of the six —ρ_{1},ρ_{2},ρ_{3},ρ_{4},ρ_{5},ρ_{6}— in SRI_{12}

f_{2(6)}= 3

σ_{1}= (0,0,1,0,0,0)P

σ_{2}= (0,0,3,0,0,0)P

σ_{3}= (0,0,9,0,0,0)P

three stages

d12

pdf version

Fig. C3.14.13

RP14^{∗}

n= 14

Has two prototiles —ρ_{2},ρ_{4}— of the seven —ρ_{1},ρ_{2},ρ_{3},ρ_{4},ρ_{5},ρ_{6},ρ_{7}— in SRI_{14}

f_{2(7)}= 3

σ_{1}= (0,1,0,0,0,0,0)P

σ_{2}= (0,3,0,0,0,0,0)P

σ_{3}= (0,9,0,0,0,0,0)P

three stages

d14

pdf version

Fig. C3.14.14

RP14 tiling

n= 14

Has three prototiles —ρ_{2},ρ_{4},ρ_{6}— of the seven —ρ_{1},ρ_{2},ρ_{3},ρ_{4},ρ_{5},ρ_{6},ρ_{7}— in SRI_{14}

f_{2(7)}= 3

σ_{1}= (0,0,0,0,0,1,0)P

σ_{2}= (0,0,0,0,0,3,0)P

σ_{3}= (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 —ρ_{2},ρ_{4}— of the eight —ρ_{1},ρ_{2},ρ_{3},ρ_{4},ρ_{5},ρ_{6},ρ_{7},ρ_{8}— in SRI_{16}

f_{2(8)}= 3

σ_{1}= (0,1,0,0,0,0,0,0)P

σ_{2}= (0,3,0,0,0,0,0,0)P

σ_{3}= (0,9,0,0,0,0,0,0)P

three stages

d16

pdf version

C3.15 Some historyIn his legendary 1977 essay, Martin Gardner introduced Penrose tilings in his

Scientific Americancolumn,Mathematical Recreations. SUN and STAR show the central regions of the twod5-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 γ_{j}of the five grids of the pentagrid is identical to a Penrose tiling created by arranging rhombs constrained by Penrose'smatching 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 (

d5symmetry), STAR (d5symmetry), 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

provedanything!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 forn= 7. I was intrigued by tilings by the three rhombs of SRI_{7}, but I recognized that tilings by those rhombs — or anyothernon-Penrose rhombs — lack the special charm of authentic Penrose tilings based on the magic number five.

Detour into rhombic rosettes, ROMBIX, and other thingsOne 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 ofn(n− 1)/2 rhombs, as described by Donald Coxeter. I decided to experiment by constructing atwin, in every possible edge-to-edge configuration, from every possiblepair of rhombsin Coxeter's set ofn(n− 1)/2 rhombs. When I tested this idea for everyn≤ 10, I was startled to discover that so long as one adds exactlyonespecimen of eachsingle 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 integern≥ 3.I found that I was able to tile the interior of the regular 2

n-gon with some arrangement of the pieces of every set of ordern≤ 12. Of course it took a lot longer for the larger values ofn. 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 ofKADONmade a laser-cut acrylic version of ROMBIX that is still sold today.

Back to RP7 tilingsImmediately 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 binennialGatherings for Gardner. Tom told me that he had seen ceramic versions ofd7rhombic 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-upWhen 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

d7symmetry 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 primenthe 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

physicaltilings (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-similarcentro-symmetric tilings of the Euclidean plane by rhombs. The three prototile rhombs of RP7 tilings areunmarked, andthere are no matching rules. The two prototile rhombs of Penrose tilings, by contrast, are marked in such a way as to preventperiodictilings and allow onlyaperiodictilings, 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, thed5-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
trivialexamples 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 RW
_{n}- C4.12 How to construct a RW
_{n}- C4.13 Origins of RW
_{n}

C4.0 IntroductionUsing the

n(n− 1)/2 rhombs of SRI_{n}as prototiles, it's easy to construct examples of periodic tilings in which there areno embedded rosettes(aside from the rosette composed of a single square whennis even). Here is an example of such a tiling forn= 7:

I will ignore all such tilings, and I'll call periodic tilings by the rhombs of SRI

Fig. C4.0.1

Periodic tiling forn= 7 in which

there are no embedded rosettes

_{n}in which rosettes of ordernare embedded

rhombic wallpaper(orRW)of order.n

For

n= 2, 3, 4, and 6, there exist RW tilings that I call thetrivialexamples. They are based ontwo of the three

regulartilings:

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.

Now consider the following admittedly frivolous question:

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 fourtrivial examplesof RW tilings based on them (right)

How densely can non-overlapping rosettes of ordernbe embedded in a RW tiling of ordern?This question is related to a slightly more restricted one

:Imagine an infinite orchard of identical trees arranged on a square lattice. Let

ρ_{tree}= the tree density (number of trees per unit area). A straight cable is stretched from each tree to 2nother 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 2nconnected 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 whichmcables cross is equal tom(m− 1)/2. Letρ_{node}= 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 ratioρ_{node}/ρ_{tree}thetree-normalized node density.

For givenn, 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

nhavemaximal 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 forn= 6 in Fig. C4.0.4b is almost seven percent less than that of thekagometiling in Fig. C4.0.5, which is conjectured to have maximal density.

Fig. C4.0.5

density = (8√ 3 − 12)/3 ≅ .618802For small

n, there are a few empirical rules — described below — that predict which RW tiling has maximal density. For everyn>2, there is an uncountable infinity of RW tilings in which the density of the rosettes of ordernis 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. Forn≠ 2, the density can be made arbitrarily close to any value less than or equal to the maximal density, by using thesplitting and augmentingoperation illustrated in § 4.8.* * * * * (For a concise description ofde 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

aperiodictiling forn≥ 3 that is the dual of a de Bruijn multigrid, one cannot help noticing the not necessarily symmetrical rosettes of ordernthat are embedded here and there. (Here is an example forn= 7.) The occurrence of rosettes is statistically inevitable in these aperiodic tilings. A rosette is the dual of thek(k− 1)/2 points of intersection of any set ofklines in which each line intersects every other at a unique point. In every de Bruijnpentagrid, arrangements of five lines that satisy this condition occur infinitely often, since Conway's town theorem guarantees that the portionPof the tiling dual to every such arrangement is no farther from its nearest replica than slightly more than twice the diameter ofP. 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 5These six line arrangements define the following six rosettes:

Fig. C4.0.7

The six rosettes forn= 5For

n> 5, without analogs of Conway'sn= 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, forn> 5, the number of ways a rosette of orderncan be tiled by the rhombs of SRI_{n}is unknown. But I have found that forn= 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 ordern. 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

tilings, in which a rosette of ordersingle-latticenis 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 fortilings, in which a rosette of orderpoly-latticenis centered at each ofpoints (mm>1) in the lattice fundamental domain. In this case, the rosettes are centered at the lattice points ofmcongruentsub-latticesof the tiling. (The kagome tiling in Fig. C4.0.5 is an example of a poly-lattice tiling withm= 2.)Scheme for

single-latticetilings(i) Choose a lattice

L.

(ii) Construct alattice starcomposed of 2ndistinctrays(line segments), each of which extends from a common root-lattice point P_{0}to a terminal lattice point P_{k}(k= 1, 2, ..., 2n). The 2nterminal lattice points are all distinct.

(iii) Generate astar gridby translating the lattice star to each lattice point ofL. The rhombs of the tiling aredualto the vertices of the star grid.Scheme for

poly-latticetilings (mcongruentsub-lattices)(i) Choose a lattice

L.

(ii) Constructmlattice stars—one for each ofmsub-latticesL_{k}(k= 1, 2, ...,m) congruent toL, each composed of 2ndistinctrays. In each lattice star, every ray extends from a common root-lattice point P_{0}to a terminal lattice point P_{k}(k= 1, 2, ..., 2n), which may belong either to the same sub-lattice or to a different sub-lattice. The 2nterminal lattice points are all distinct.

(iii) Generate astar gridby translating the lattice star for each sub-latticeL_{k}to every other point ofL_{k}. The rhombs of the tiling aredualto the vertices of the star grid.

For the unit square lattice (integer lattice), we define the

magnitudeof a lattice star — using the conventions of taxicab geometry — as the sum of the 'manhattan lengths' of the 2nrays 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 = 44For RW tilings for small

non 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 samenhave 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 aweightedcount of nodes, each weight being the area of the rhomb dual to each node, then the tree-normalized node density would beexactlyinversely correlated.)

It is convenient to identify RW tilings of ordernbyclassandtype. 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

nis calledsuperdense.2. A tiling with the highest density of any tiling of order

nof its type is calleddense.3. A tiling with density less than the highest density of any tiling of order

nof its type is calledsparse.

The tilings forn= 2 andn= 3 in Figs. C4.0.1b and C4.0.2b, respectively, aresuperdense, since their density is equal to 1.

The tiling forn= 4 in Fig. C4.0.3b issuperdense. 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 forn= 6 in Fig. C4.0.4b issparse, 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 besuperdense.Although all of the tilings for

n<9 conjectured to be superdense have reflection symmetries, the tiling forn= 9 (cf.Figs. C5.9.1a and b) that is conjectured to be superdense does not — it has onlyp3symmetry. Its density is almost 30% larger than that of the tiling withp3m1symmetry in Fig. C4.10.2.

Every RW tiling treated here is characterized as belonging to either the

row classor thedispersed class.Within the

row class, there are twosub-classes:straightandzig-zag.

Within each of these twosub-classes, there are fourtypes.Within the

dispersed class, there are threesub-classes:square lattice,hexagonal lattice, andother lattices.

This classification scheme is by no means exhaustive. The kagome tiling in Fig. C4.0.5 belongs to neither

dispersednorrowclass. Neither do tilings in which all the rosettes are arranged in closed rings, with every rosette incident at each oftwoneighbors. 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 latticeHexagonal latticeOther 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 thedispersedorrowclass, the wallpaper group is identified. But at every point of intersection of three or more lines in a star grid, theorientationin the tiling plane and — for four or more intersecting lines — thesymmetryof 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

familiesof dense tilings of a particular type that are parametrized by the ordernof the tiling, the dependence of density onn— for smalln— can be described by an algebraic expression conjectured to hold for alln. 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.

EXAMPLESLet's now examine examples of tilings of several types, in increasing order of

n. The tiles are defined to haveunit edge length. In several cases, I have included the lattice star or star grid (or both). In addition to density ρ_{n}, I record the wallpaper group and — for row tilings —λ_{n}, the distance between the center-lines of adjacent rows. Some types have no representatives, because I am considering onlyedge-to-edgetilings,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 smalln.

Top row: lattice stars for the square lattice.

Bottom row: lattice stars for the hexagonal lattice.

C4.1 Straight row tilings, aligned:evenn

Fig. C4.1.1n= 2

Fig. C4.1.2n= 4

Fig. C4.1.3n= 6

Fig. C4.1.4an= 8

Straight row aligned edge-sharingρ_{8}≅ .394591λ_{8}≅ 10.1371pmm

(This is the same tiling as the one at the right in Fig. C4.1.6.)

Fig. C4.1.4bn= 8

Lattice star for the tiling of Fig. C4.1.4a

Fig. C4.1.4cn= 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. C4.1.5an= 8

Straight row aligned vertex-sharingρ_{8}≅ .253965λ_{8}≅ 15.4476pmm

(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 2nrays in a lattice star is not uniform, those rays represent 2nradial lines thatareuniformly 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. C4.1.5bn= 8

Lattice star for the tiling of Fig. C4.1.5a

Fig. C4.1.5cn= 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. C4.1.5dn= 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 therungsof its associated ladder.

Fig. C4.1.6n= 8

Fig. C4.1.7n= 10

C4.2 Straight row tilings, aligned:oddn

Fig. C4.2.1n= 3ρ_{3}≅ .75000λ_{3}≅ √ 3pmm

Fig. C4.2.2n= 5ρ_{5}≅ .5414ρ_{5}≅ .5150

λ_{5}≅ 4.9798λ_{5}≅ 4.8541pmm

Fig. C4.2.3n= 7ρ_{7}≅ .3466ρ_{7}≅ .3745

λ_{7}≅ 9.8447λ_{7}≅ 9.3488pmm

C4.3 Straight row tilings, staggered:evenn

Fig. C4.3.1n= 2

Fig. C4.3.2n= 4

Fig. C4.3.3n= 6

Fig. C4.3.4n= 8

C4.4 Straight row tilings, staggered:oddn

Fig. C4.4.1n= 3ρ_{3}≅ 1λ_{3}≅ 1.5pmm

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

Fig. C4.4.3n= 7ρ_{7}≅ .460658λ_{7}≅ 7.5978pmm

C4.5 Zig-zag row tilings, aligned:oddn

Fig. C4.5.1an= 5

Lattice star for sites of typeAin zigzag-row tiling of Fig. C4.5.1d.AandBare the two inequivalent types of sites in each lattice fundamental domain.

Fig. C4.5.1bn= 5

Lattice star for sites of typeBin zigzag-row tiling of Fig. C4.5.1d.AandBare the two inequivalent types of sites in each lattice fundamental domain.

Fig. C4.5.1cn= 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. C4.5.1dn= 5

Zigzag-row tilingρ_{5}= 5 − 2√ 5 ≅ .527864λ_{5}≅ 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 forn= 7. Each of them is a poly-lattice tiling, withm= 2.

Fig. C4.5.2an= 7

Lattice starsAandBfor zigzag-row tiling no. 1 of Fig. C4.5.2c

Fig. C4.5.2bn= 7

Star grid for zigzag-row tiling no. 1 of Fig. C4.5.2c

Fig. C4.5.2cn= 7

Zigzag-row tiling no. 1ρ_{7}≅ .4655λ_{7}≅ 7.8948

Fig. C4.5.3an= 7

Lattice starsAandBfor zigzag-row tiling no. 2 of Fig. C4.5.3c

Fig. C4.5.3bn= 7

Star grid for zigzag-row tiling no. 2 of Fig. C4.5.3c

Fig. C4.5.3cn= 7

Zigzag-row tiling no. 2ρ_{7}≅ .4547λ_{7}≅ 8.5429

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

C4.6 Square lattice tilings:evenn

Fig. C4.6.1n= 4ρ_{4}= 2 (√ 2 − 1) ≅ .828427

Fig. C4.6.2n= 4ρ_{4}= 2 (√ 2 − 1) ≅ .828427

Fig. C4.6.3n= 4ρ_{4}= 2 (√ 2 − 1) ≅ .828427

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

Fig. C4.6.4bn= 8with these 81 polka dots ...

Fig. C4.6.4cn= 8you get this superdense tiling.ρ_{8}≅ .425442

Fig. C4.6.5n= 8ρ_{8}≅ .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. C4.6.6an= 10ρ_{10}≅ .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, whichdoeshave reflection symmetries.

Fig. C4.6.6bn= 10

This tiling has reflection symmetries

and a translational fundamental domain only half as large as in Fig. C4.6.6a.

Fig. C4.6.7n= 12ρ_{12}≅ .261899

For a pdf version of this tiling, look here.In addition to the very prominent

d3rosettes 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:oddn

C4.8 Rectangular lattice tilings:evenn

Fig. C4.8.1an= 8

Lattice star for rectangular lattice tiling in Fig. C4.8.1c

Star magnitude = 52

Fig. C4.8.1bn= 8

Star grid for rectangular lattice tiling in Fig. C4.8.1c

Fig. C4.8.1cn= 8

Star magnitude = 52

Fig. C4.8.2an= 8

Lattice star for rectangular lattice tiling in Fig. C4.8.2c

Star magnitude = 44

Fig. C4.8.2bn= 8

Star grid for rectangular lattice tiling in Fig. C4.8.2c

Fig. C4.8.2cn= 8

Star magnitude = 44

Fig. C4.8.3n= 8

Splitting (left) and augmenting (right) the tiling in Fig. C4.8.2c

Fig. C4.8.4an= 8

Lattice star for rectangular lattice tiling in Fig. C4.8.4c

Star magnitude = 44

Fig. C4.8.4bn= 8

Star grid for rectangular lattice tiling in Fig. C4.8.4c

Fig. C4.8.4cn= 8

Star magnitude = 44

Fig. C4.8.5an= 8

Star magnitude = 36

Fig. C4.8.5bn= 8

C4.9 Hexagonal lattice tilings:evenn

Fig. C4.9.1n= 6ρ_{6}=1/√ 3 ≅ .577350

superdense

C4.10 Hexagonal lattice tilings:oddn

Fig. C4.10.1an= 9ρ_{9}≅ .341644superdensep3

Fig. C4.10.1b

(Another coloring of thesuperdensetiling of Fig. C4.10.1a)

Fig. C4.10.2n= 9ρ_{9}≅ .263346

sparsep3m1

Below are:

C4.11 Additional examples of RW_{n}

C4.12 How to construct a RW_{n}

C4.13 Origins of RW_{n}

C4.11 Additional examples of RW_{n}Below are examples of both dense and sparse RW

_{n}tilings fornin 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

subliminalpatterns—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 RW

_{n}tilings that along lines of reflection of the tiling lie infinite linear strings of rhombs that are analogs ofminimal 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.RW

_{21}(4) is an instructive example. The axes of threepointedstrings 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 ofbluntbeads. These blunt strings can only partially mimic the mirroring effects of blunt beadstrings in RPntilings, 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

interiorof the beads of each string

nor

(b) the tiling of the rosettesis symmetrical by reflection in those axes. In RW

_{21}(4), the rosettes would required6symmetry, but symmetry of even order is impossible for rosettes (cf.§ B1).RW

_{9}, illustrates the same effects.

:n= 2Regulartiling by squares (square lattice)

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

RW _{6}(1)

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

Fig. C4.11.1

RW_{7}(1)

ρ_{7}≅ 0.5289856

I will soon post a picture of the 25" x 38"

RHOMBBURSTposter, which includes an article at the bottom summarizing the state of knowledge in 1995 about Penrose tilings. Unlike RW_{8}(2) (above), the tiling of theRHOMBBURSTposter isnota wallpaper pattern—it is acentro-symmetrictiling 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 RW_{8}(2) (above)?

RW_{9}(1) is a more colorful version of RW_{9}. Note that the image is rotated, relative to RW_{9}, by one-sixth of a turn around its center. In this orientation, it is symmetrical by reflection in averticalline through the center.

RW

_{10}(1) is a row tiling.

Fig. C4.11.2

Two colorings of RW_{10}(1)For other images of RW

_{10}(1), see RW_{10}(2) , RW_{10}(3) or RW_{10}(4) .

Note that exactly halfway between adjacent rows of large rosettes in RW

_{10}(1), there is a string of strawberry rosettes of order 5. In row tilings ofoddorder, there is a medial string of convex polygons, in a regular alternating sequence, withn-1 andn+1 sides, respectively. I will soon add an example or two.In RW

_{10}(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 evenn. In row tilings, the'wave fronts' of tiles that border large rosettes in one row are transformed intoconvex'wave fronts' by the time they reach the large rosettes in an adjacent row. I call the pattern elements that mediate this changeconcaveginkgo leaves.RW

_{16}_row and RW_{22}_row demonstrate the hierarchical arrangement of ginkgo leaves in row tilings. Ginkgo leaves bear a superficial resemblance to thearbelos(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.

: RWn=15_{15}has alattice.hexagonal

Fig. C4.11.3

RW_{15}

For a pdf version, look here.

RW_{15}(1) is another coloring of RW_{15}.

: RWn=16_{16}(1) illustrates howunavoidably appear when you embed a rosette—which necessarily has symmetry ofbroken symmetriesoddorder—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 RW

_{16}_row, a row ofn=8 small rosettes lies halfway between each pair of adjacent rows ofn=16 large rosettes:

The horizontal center-line of each row of

Fig. C4.11.4

RW_{16}_row

n=16 rosettes is a line of reflection, but the horizontal center-line of each row ofn=8 rosettes isnota line of reflection. The enlarged image below shows that the 16-gon boundaries of adjacentn=8 rosettes are oppositely rotated around their centers.

Fig. C4.11.5

Adjacentn= 8 rosetttes in RW_{16}_row are oppositely rotated.

: RWn=21_{21}(1) is an ambitious example of a RW_{n}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

hexagonallattice than on asquarelattice. I'll explain below how the overall symmetry is affected by theparity— (evenn vs. odd n) — of the SRI_{n}.To see how color choices for the rhombs affect RW

_{21}, see RW_{21}(2), RW_{21}(3), RW_{21}(4), RW_{21}(5), and RW_{21}(6).

RW_{21}(6) is a monstrously large piece of RW_{21}. 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 RW

_{21}. 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!)

: RWn=22_{22}is a row tiling of order 22 that shows the 'mortar' between rosettes but not the rosettes themselves.

C4.12 How to construct a RW_{n}Let's first recall that a de Bruijn multigrid is a set of

noverlapping uniformly rotated grids ofparallellines. Here's an example forn= 5:

Fig. C4.12.1

A de Bruijn pentagridIf no point of a multigrid belongs to more than two of its

ngrids, de Bruijn calls the multigridregular; otherwise he calls itsingular.A

is thestar grid of order nperiodiccounterpart of a de Bruijn multigrid of ordern.

DEFINITIONS:(1) A

is a set of 2lattice star of order nnrays (line segments) associated with the lattice. One end of every ray is incident at the common lattice pointLP_{0}; the other end is incident at a distinct one of the lattice pointsP(_{k}k=0, 1, 2, ...,n-1).

The 2nrays of the lattice star are labelled CCW by theirninteger0, 1, 2, ...,indicesn,n+1, ...,n+2, ..., 2n-1 (modn).

A lattice star has the same point symmetry as its root lattice pointP_{0}.(2) A

is the union of congruent parallel lattice stars of orderstar grid of order nn, one specimen of which is rooted at every point of the lattice.L

In a star grid of order

n,nlines intersect at every lattice point. Mimicking de Bruijn, we call any point of a star grid at which only two lines intersect aregular point, and any point at which more than two lines intersect asingular point.The dual of every

regularpoint 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 asingularpoint at whichmlines intersect (m=3, 4,...,n) is a convex assembly ofm(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 ofoddorder, 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 ofevenorder, there exists no arrangement of the rhombs with the same symmetry. For these reasons, RW_{n}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 (

sparsedot packing) in a RW_{n}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 RW_{9}tiling with higher density than any alternative set of lattice points of the same symmetry, it is calleddense. 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 forn= 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 RW

_{9}(1) that correspond to the pair of lattice points joined by the edge.

Fig. C4.12.4

An extremal star grid forn= 15, which is the dual of the tiling RW_{15}(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 forn= 16, which is the dual of RW_{16}(1)

.

Fig. C4.12.7

The lattice star for the extremal tiling RW_{21}(1).

.

Fig. C4.12.8

Star grid for the extremal tiling RW_{21}(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 RW_{n}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 RW

_{n}.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 RW

_{n}.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. Forn= 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 eightby therhombs of SRI. Perhaps I was guided by what Halmos called_{n}"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

nis aperiodictiling, 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

nby using aof the multigrids invented by de Bruijn for his analysis of Penrose tilings. I call these modified gridsperiodic variant. I examined not onlystar gridsdensedot packings, but alsosparseones in which the rosettes are quite widely separated. The density of the rosettes in a tiling ('density') is determined by the design of its—the basic structural unit of the star grid dual of the tiling. Star grids and lattice stars are defined below.lattice starFor 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.

Back to GEOMETRY GARRET