TTessellations and Positional Representation
Howard L. Resnikoff ∗ Abstract
The main goal of this paper is to define a 1-1 correspondence between between sub-stitution tilings constructed by inflation and the arithmetic of positional representationin the underlying real vector space.It introduces a generalization of inflationary tessellations to equivalence classes oftiles. Two tiles belong to the same class if they share a defined geometric property, suchas equivalence under a group of isometries, having the same measure, or having the same‘decoration’. Some properties of ordinary tessellations for which the equivalence relationis congruence with respect to the full group of isometries are already determined bythe weaker relation of equivalence with respect to equal measure. In particular, themultiplier for an inflationary tiling (such as a Penrose aperiodic tiling) is an algebraicnumber.Equivalence of tiles under measure facilitates the investigation of properties oftilings that are independent of dimension, and provides a method for transferring tilingsfrom one dimension to another.Three well-known aperiodic tilings illustrate aspects of the correspondence: a tilingof Ammann, a Penrose tiling, and the monotiling of Taylor and Socolar-Taylor.
Keywords:
Ammann tilings, aperiodic tilings, Fibonacci numbers, golden number, infla-tion, measure-preserving maps, multi-radix, Penrose tilings, positional notation, positionalrepresentation, non integral radix, remainder sets, silver numbers, substitution tilings, Tay-lor monotile, tessellations, 2-dimensional positional representations.
Mathematics in Civilization [18] argued that there are only two problems in mathematics: improving the ability to calculate and understanding the geometrical nature of space . Asknowledge increases, these fundamental problems are reformulated in a more sophisticated ∗ Resnikoff Innovations LLC; howard@resnikoff.com. a r X i v : . [ m a t h . DG ] M a y ay, and investigated anew. The goal of this paper is to highlight the connection betweenpositional representation for numbers and geometrical tilings of the plane. Although positional representation for real numbers is one of the most ancient andgreatest intellectual inventions and certainly amongst the most important in numericalpractice, the concept seems not to have had much influence on the internal developmentof mathematics. Other good, old, mathematical ideas, such as the unique factorization ofintegers into primes and the theorem of Pythagoras, have led to innumerable deep andimportant abstract generalizations. Not so positional representation. Positional represen-tation is still the occasional subject of mathematical papers, most often in the recreationalcategory.Tessellations have an even more ancient history but they have generally been considereddecorative rather than profound. Although vast numbers of papers and online exampleshave been devoted to tilings, the mathematical theory is recent and unstructured – largelya collection of interesting and sometimes beautiful patterns.Positional representation and tessellations have traditionally been considered indepen-dent domains of recreational mathematics. Connections between the two have not receivedmuch notice. Potential relationships could be of particular interest for tilings because theymight provide natural constructions as well as an arithmetic representation for the geo-metric relationships, and an alternative but familiar language for constructing and talkingabout tilings. In the opposite direction, tilings provide insight into new forms of positionalrepresentation – particularly those that depend on more than one radix. The paper explores this interplay. The main result is a 1-1 correspondence between aclass of positional representations and a class of tilings that includes those constructed bythe process of inflation used by Roger Penrose [15] in 1974.
It is crucial to distinguish between a positional notation – a notation that employs a finiteinventory of symbols to identify an arbitrary real number – and a positional representation ,which is a positional notation whose symbols and structure have meanings – interpretations– that are linked to the structure of arithmetic so that the notation can be used forcalculation.This distinction may be worth elaborating. Suppose that a 1-1 correspondence betweenthe field of real numbers R and a set S without any structure is given. Suppose furtherthat a notation for the elements of S that employs sequences of symbols drawn from afinite inventory is used to set up a correspondence with the elements of S . The set of suchsequences can be thought of as a ‘positional’ method for labeling the elements of S ; call Tilings are also called tessellations , from the Latin tessera for the small pieces of stone, glass or ceramictile used in mosaics. ‘Tessera’ is derived from the Greek for ‘four’ referring to the four sides of rectangularmosaic stones. We use ‘tessellation’ and ‘tiling’ interchangeably. The term ‘radix’ refers to the base of a system of positional representation.
2t a positional notation for S . These sequences contain no hint of arithmetical properties.The 1-1 correspondence could be used to map the real numbers onto the sequences of thenotation, and then to transfer the field properties of R to S but there is no assurance thatthis would result in a practical or efficient method for calculating , that is, for performingthe field operations of addition and multiplication and other operations derived from themA positional representation for R (or for C ) is a special kind of correspondence betweenreal numbers and sequences of symbols drawn from a finite inventory such that the symbolsand the sequences have meanings that explicitly relate them to the numbers and theirproperties in a way that facilitates the expression of properties of R (or C ) that are notnecessarily properties of other sets.For instance, the set of potential sentences of English can be coded as sequences ofletters from a finite alphabet. This set has the same power as R . Although the letters ofthe alphabet (together with the interword space symbol) and sequences of letters are alsoendowed with a linear order (used, for instance, to organize dictionaries), this positionalnotation for sentences does not imply an a priori ‘arithmetic’ of sentences.The first positional notation that also was an efficient positional representation wasintroduced by Akkadian mathematicians more than four thousand years ago. It originallywas limited to positive integers but was easily extended to positive numbers smaller than1. In its earliest realization it lacked a symbol for the zero, which was unreliably denotedby a gap between neighboring digits. The radix was 60 – probably the largest integer eversystematically and extensively employed as a base for hand calculation. Although 60 hasthe advantage of many divisors and leads to short expressions for the practical quantitiesthat were of interest in early times, the addition and multiplication tables are too largeto be memorized, or even used, by anyone other than a specialist. Remnants of radix 60representation are found in our notation for angles and time.Radix 10, often referred to as Arabic notation and less often but more correctly asHindu-Arabic notation, is said to have been invented between the 1st and 4th centuries byIndian mathematicians. It was adopted by Arab mathematicians many centuries later andmade its way to western Europe during the Middle Ages. Leonardo Pisano (Leonardo ofPisa), generally known as Fibonacci, brought radix 10 calculation into mainstream euro-pean thought in his book
Liber abaci – The Book of Calculation [6, 20] – first publishedin 1202. Today few remember Fibonacci’s role in the transmission of Hindu-Arabic radix10 representation of numbers to Europe, but many have heard of the sequence of numbershis book introduced as the solution to a homework problem – the ‘Fibonacci numbers’.This sequence, which begins 1 , , , , , , . . . (the n -th term is the sum of the previoustwo), plays a role in many unexpected places, from the idealized reproduction of rabbits,which was the subject of the exercise, to the growth of petals on flowers and seeds onpine cones, and not least of all, in many of the examples in this paper. The analytical See [18] for the details of early systems of numeration, and [13], section 4.1, for a brief but excellentoverview of the history. φ = (1 + √ / ∼ .
61 – the‘golden number’, defined by Euclid as the “extreme and mean ratio”, that is, the number x satisfying x = 1 + 1 /x . The larger solution of x = x + 1 is φ . The golden number wasbelieved to express aesthetically pleasing proportions for rectangles and for that reason itis often seen embodied in commercial designs and logos.No history of positional representation, no matter how brief, should omit DonaldKnuth’s introduction of 2-dimensional radices [12, 13], several variants of which appearbelow. These ideas were followed up by many others, in particular Gilbert, who studiedarithmetic in complex bases in a series of interesting papers [7, 8, 9].The theory of wavelets, a concept introduced about thirty years ago, is closely relatedto positional representation although the connection has not been emphasized. ‘Wavelets’are collections of compactly supported orthonormal functions that, in general, overlapand are bases for a broad variety of function spaces [19]. One can think of compactlysupported wavelets as a kind of positional representation for functions. Their supportsprogressively decrease in size as they ‘home in’ on the neighborhood of an arbitrary pointon the line. We shall not examine wavelets here, but the reader should be aware of theintimate and unexplored connection of that circle of ideas with positional representationsand tessellations. There have not been many fundamental applications of positional representation inpure mathematics but there are a few.Positional notation was first used to prove significant theorems by the inventor of settheory, Georg Cantor, who recognized that the properties of positional notation alone –no need for the additional arithmetical implications of positional representation – weresufficient to prove that the set of real numbers is not countable [3]. Cantor’s ‘diagonalmethod’ has become a foundation stone in the education of mathematicians. He did notneglect positional representation: Cantor’s construction of a nowhere decreasing continuousfunction that increases from 0 to 1 but is constant except on a set of measure zero (‘Cantor’sfunction’, cp. [4]) used the essential device of passing from positional representation withradix 3 to positional representation with radix 2.It has long been known that positional representation provides a way to map subsetsof R m to subsets of R n . The general method is made clear from the simplest example:mapping the unit interval onto the unit square. Having fixed the radix, say ρ = 2, fromthe representation of u ∈ R as u = (cid:80) k ≥ u k − k , construct the pair ( x, y ) as x = (cid:88) k ≥ u k − − k , y = (cid:88) k ≥ u k − k After suitable normalization, u → ( x, y ), and similar maps constructed from positionalrepresentations, is measure-preserving. Norbert Wiener made essential use of this in his Cp. [14, 26]. Tessellations of the plane are no doubt older than the first developments of positionalrepresentations of numbers, and developed examples can already be found in the ancientFertile Crescent. An example from the Sumerian city Uruk IV, circa -3100, now in thePergamon Museum in Berlin, shows triangular and diamond periodic mosaic patterns.Tessellations were taken up from a mathematical viewpoint in 1619 by Kepler, whowrote in his
Harmonices Mundi – Harmony of the World – about coverings of the planeby regular polygons.Periodic tilings of the plane can be classified by their symmetries into 17 groups, some-times called “wallpaper groups” or, more formally, “plane crystallographic groups.” Whilethe mathematical classification was the work of Evgraf Fedorov [5] in 1891, the classifi-cation has been intuitively understood for millennia by artists and craftsmen around theglobe who decorated almost every surface they could find with complex repeating patterns.Fedorov also recognized that crystals were physical realizations of periodic tiling of 3-dimensional space. This led to his classification of the 230 space groups – the symmetrygroups of crystallographic tessellations – which is among the earliest and most significantmathematical results in this field.It was not until 1974 that the mathematician and mathematical physicist Roger Penrose[15] discovered the aperiodic tessellations that bear his name. Since then it has becomea parlor game for amateur and professional mathematicians to find new and interestingexamples of aperiodic tessellations, but the subject has not stimulated much work norfound resonance in other departments of mathematics.In the past, it seems to have been generally believed that tessellation of the planewithout periodic symmetry is impossible. The concepts underlying symmetry and theirappearance in art and nature as well as their applications in mathematics and science weretraced in a beautiful book written in 1952 by Hermann Weyl [29], who was one of the mostpowerful mathematical minds of his time. That Weyl made no mention of the possibilityof aperiodic tilings demonstrates how improbable they were thought to be.
This paper explores the relationship between tessellations and positional representation,primarily in the real vector spaces R and R equipped with the euclidean metric and themeasure m derived from it, although some concepts are formulated for R d . Two measurablesubsets of R d are essentially disjoint if the measure of their intersection is 0, and they are [30], esp. p.81. This property plays a role in constructing measures on spaces of functions in the theoryof brownian motion. ssentially identical if the measure of their intersection is equal to the measure of each ofthe sets. Thus m ( A ∪ B ) = m ( A ) + m ( B ) if and only if A and B are essentially disjoint.The objects of interest are ‘tiles’ and the tessellations made from them. A tile is a subsetof R d that has positive measure. Tiles are usually, but need not be, connected. Each tilebelongs to one of a finite collection of equivalence classes. Tiles are of the same type if theybelong to the same equivalence class. If two tiles are of the same type, each is a copy of theother. We suppose that every type of tile has infinitely many members, and that there isat least one type. A tessellation of R d , also called a tiling , is the pair consisting of a finitecollection of types of tiles of dimension d and a covering of R d by essentially disjoint tileseach of which belongs to one of the types. An overtiling is the pair consisting of a finitecollection of types of tiles of dimension d and a covering of R d by essentially identical tileseach of which belongs to one of the types. In an overtiling, tiles can be stacked on top ofone another.Here are some examples of equivalence classes of tiles: Congruence under the group ofisometries of R d is an equivalence relation. Congruent tiles are of the same type relative tothis relation. Tiles that are congruent under some subgroup of the group of isometries canalso be said to constitute a type. Congruent tiles have the same measure. Tiles that areconnected and simply connected and have the same measure but not necessarily the sameshape form an equivalence class. Tiles might be distinguished by ‘decorations’. Suppose ameasurable subset of R d is given and copies of it are colored in a finite number of distinctcolors. Say that two tiles are equivalent if they have the same color. There are as manytypes of tiles as colors used to color them. Tiles may have more general decorations, thatis, have markings on them. Those that have the same decoration form an equivalence classand constitute a type. The decorations may be used to limit how adjacent tiles may beplaced. For instance, if the decorations are curves drawn on the tiles, an allowed tessellationmight be one for which the curves are continuous across tile boundaries.Suppose that the dimension is d and that there are N types of tiles. Let { R j : 1 ≤ j ≤ N } be a set of representatives of the types of tiles. A tessellation is said to be inflationary if there is a real number ρ > O such that eachmagnified representative ρ O ( R i ) is the union of essentially disjoint tiles. This processof magnifying each R i by the same factor and then tiling it with copies of the R j is theprocess called inflation ; the factor ρ O is called the multiplier ; sometimes we shall refer to ρ itself as the multiplier. The measure of ρ O ( R i ) is ρ d m ( R i ) so repetition of the inflationprocess covers increasing volumes of R d . If the origin of the magnification lies in the interiorof R i , infinite repetition of the inflationary process results in a tessellation of R d that issaid to have been constructed by inflation . If there is an S ⊂ R d and a lattice Λ ⊂ R d (that is, a discrete additive subgroup of R d of rank d ) such that R d = (cid:83) λ ∈ Λ ( λ + S ) andthe translations S → λ + S are essentially disjoint, then the tiling is said to be periodic ; This concept is used only in a footnote on page 15. One could generalize this definition to expansive matrices but for our purposes that would only com-plicate the details. not periodic .A collection of types of tiles for which at least one tessellation is possible but no tessel-lation consistent with the constraints is periodic is said to be aperiodic . In an importantpaper that connected tessellations to undecidability problems, Wang [28] conjectured thataperiodic tilings are impossible. Five years later, Berger [2] showed the existence of aperi-odic tilings of the plane by creating a correspondence between tilings and Turing machinesand applying the undecidability of the halting problem. He constructed a set of 20,426distinct types of tiles for which an associated tessellation exists and is aperiodic. But itwas Penrose’s [15] explicit construction of aperiodic tilings using two types of tiles in 1974that captured the imagination. In both cases the ‘types’ are defined by the equivalencerelation of geometrical congruence up to sets of measure zero.***We shall see that inflationary tessellations are intimately related to algebraic numbers,and that aperiodicity is the generic situation. But first, consider tessellations that are bothperiodic and inflationary.
Theorem 1
Suppose that a tessellation of R d is both periodic with period lattice Λ andinflationary with multiplier ρ O . Then ρ O (Λ) ⊂ Λ and ρ is an algebraic number of degree d . After suitable normalizations, if d = 1 then ρ ∈ Z ;if d = 2 then ρ is an imaginary quadratic integer.Proof: If { ω k : 1 ≤ k ≤ d } is an integral basis for Λ there is a matrix A with integerentries such that ρ O ( ω j ) = A ω j . Then ρ d = | det A | so ρ is an algebraic number of degree d . If d = 1 then ρ ω = A ω with A (cid:54) = 0 a rational integer. If d = 2, then R can beidentified with C in the usual way, ( ρ O ) with a complex number temporarily denoted ρ ,and ρ ω j = (cid:80) j A ij ω j with A = (cid:18) a bc d (cid:19) a matrix of integers. A is invertible because Λ isnon-degenerate. If the tessellation is inflationary with a non zero rational integer multiplier,say n , then for any lattice Λ the periods of n Λ are nω , nω so n Λ ⊂ Λ.Are there special lattices for which other multipliers exist? Set τ = ω /ω . Withoutloss of generality, suppose that Im ( τ ) >
0; then ρ = cτ + d and τ = aτ + bcτ + d . Thus τ is aquadratic algebraic number. Since the lattice has rank 2, the quadratic is irreducible andthe number is imaginary quadratic. The isomorphism is x + iy ↔ (cid:18) x y − y x (cid:19) .
7o complete the proof, we show that ρ satisfies ρ − ( a + d ) ρ + det A = 0 by calculating τ . The discriminant of the quadratic for τ is ( a − d ) + 4 bc = ( a + d ) − A . Thus ρ = d + cτ = d + c (cid:32) a − d ± (cid:112) ( a + d ) − A c (cid:33) = a + d ± (cid:112) ( a + d ) − A ρ − ( a + d ) ρ + det A = 0 . Since 0 (cid:54) = det A ∈ Z and ρ = cτ + d , it follows that ρ is animaginary quadratic integer. (cid:3) A consequence of this theorem is that periodic inflationary tessellations of the planecorrespond to complex multiplication. Since the the Penrose tilings are inflationary andthe multiplier is ρ = √ – not an imaginary quadratic number – it follows that noparticular Penrose tiling can be periodic, whence Penrose tilings are aperiodic.The theorem has a partial converse. Theorem 2 If d ∈ { , } and Λ ⊂ R d is a lattice such that ρ O (Λ) ⊂ Λ for some ρ O ∈ Λ then there is an inflationary tiling with multiplier ρ O .Proof: If d = 1 then Λ ⊂ R can be normalized so that Λ = Z . Choose a fundamentaldomain F = Z / Λ. For example, the interval F = [0 ,
1] is essentially identical to a fun-damental domain. Then the sets in (cid:83) n ∈ Z ( n + F ) are essentially disjoint and the unionis a periodic tessellation of R . According to the hypothesis, there is a multiplier ρ ∈ Z ∗ .Evidently ρ Z ⊂ Z and ρ Z = (cid:83) k ∈ ∆ ( k + F ) where ∆ is a set of representatives of Z /ρ Z , forinstance { , . . . , | ρ | − } . This is the inflationary decomposition of the tile F .If d = 2 then identify R with C . Λ has two generators which can be normalized to { , τ } where τ ∈ C has positive imaginary part. According to the hypothesis, Λ has amultiplier ρ which is an integer in an imaginary quadratic number field Q ( √ D ) where D isa negative square-free integer. Let Z [ √ D ] be the ring of integers in this field and let ∆ be aset of representatives for Z [ √ D ] /ρ Z [ √ D ] which we call the set of digits for the radix ρ . Thenumber of elements in ∆ is ρρ = | ρ | (The ratio of the area of ρF to the area of F is thenumber of congruent copies of F that tile ρF ). If F is essentially identical to a fundamentaldomain C / Λ, then the sets in (cid:83) λ ∈ Λ ( λ + F ) are essentially disjoint and the union is a periodictessellation of C . The tessellation is inflationary because ρ F = (cid:83) δ ∈ ∆ ( δ + F ). This is nothingmore than saying that each λ ∈ Λ can be written as λ = ρλ (cid:48) + δ . (cid:3) Denote the ring of integers in Q ( √ D ) by Λ = Z [ √ D ]. It is a lattice generated over Z Including the rational integers.
8y 1 and ω = (cid:40) √ D, D ≡ , √ D , D ≡ ρ = n + n ω with n , n ∈ Z .These tilings are primarily of interest because of their connection with the theory ofelliptic functions and number theory. Another reason is that they yield positional represen-tations for complex numbers with respect to the radix ρ , and thereby establish a connectionbetween a class of tessellations of the plane and positional representations with an imag-inary quadratic integer radix. Indeed, suppose that ρ ∈ Λ and | ρ | >
1, and write theinflation relation in the form F = 1 ρ (cid:91) δ ∈ ∆ ( δ + F )where ∆ is a complete set of representatives of Λ /ρ Λ. Then F is just the remainder setand the positional representation for an arbitrary remainder is z = ∞ (cid:88) k =1 z k ρ − k , z k ∈ ∆ (1)When the cardinality of ∆ – the number of digits in the positional representation –is 2, there are three distinct imaginary quadratic fields and three possible multipliers upto multiplication by a unit of the field. Listed in order of increasing trace of the complexgenerator of the field, they are ρ = i √ i √ , and 1 + i . In each case a convenient choiceof digits is { , } so these can be considered as generalizations to C of the conventionalbinary representation for R . Pictures of the three remainder sets and their decompositionsare shown on pages 168-9 of reference [19]. ***Here is a simple example of the correspondence between a tiling and a positional repre-sentation with ρ = 2. There will be | ρ | = 4 digits. The simplest tiling of the plane by unitsquares – repetition of one type of square as in a cartesian coordinate grid – is periodicwith lattice generated by z (cid:55)→ z + 1 , z (cid:55)→ z + i . The inflationary multiplier is the rationalinteger 2. The tiling has an algebraic realization by the inflation equation2 R = R ∪ (1 + R ) ∪ (1 + i + R ) ∪ ( i + R ) (2)which is satisfied by R = { z = x + iy ∈ C : 0 ≤ x, y ≤ } . With this solution, eq(2) is anessentially disjoint union. In this case multipliers that map the lattice into itself have theform m + ni, m, n ∈ Z , which are imaginary quadratic (“Gaussian”) integers.Equation (2) is also a realization of a system of positional representation for the fieldof complex numbers. It realizes the arithmetic of C within the framework of a positional9epresentation in the sense that R can be considered a remainder and every z ∈ C can bewritten in the form z = [ z ] + { z } where { z } = ∞ (cid:88) k =1 z k k ∈ R, z k ∈ ∆ = { , , i, i } and [ z ] is a Gaussian integer. The elements of ∆ are the digits of the representation. Thisrepresentation is not unique, just as the representation of real numbers by decimals is notunique. Nevertheless, each z has a representation relative to the remainder set.Let us call a number with a positional representation that has only non negative powersof the radix a positional representation integer . In the example above, the set of positionalrepresentation integers coincides with the ring of Gaussian integers; hence it is a lattice.But it is not always the case that the integers of a number field and the positional repre-sentation integers of an associated positional representation are the same. This differencecan sometimes be exploited to prove aperiodicity of a tessellation.***A checkerboard is also a tessellation of the plane by unit squares but now the squaresare of two types, distinguished not by a geometrical property but by a decoration – theircolor, say black and white. This tiling can also be described algebraically. If B , resp. W ,denotes a black, resp. white, square then2 B = B ∪ (1 + i + B ) ∪ (1 + W ) ∪ ( i + W )2 W = B ∪ (1 + i + B ) ∪ (1 + W ) ∪ ( i + W ) (3)In this example each set B and W is geometrically similar to a scaled-up version of itself,and can be decomposed into an essentially disjoint union of copies of tiles.The colors are not geometrical properties of the tiles, but their role can be replaced bymanifestly geometrical properties by deforming the tiles. One way would be to deform theboundaries of the squares: for the black, cut out a triangular notch from two adjacent sidesand add semicircular pips to the opposite sides so that the area of the square is conserved;for the white, cut out semicircular notches and adjoin triangular pips so that the tilescan be joined as in a jigsaw puzzle. These modifications force certain relationships in thetessellation. These relationships are also expressed by eq(3).Yet another way to think about the checkerboard might be to consider the tiles as 2-faced and to ‘color’, or otherwise distinguish, opposite faces. Reflection in the plane of thecheckerboard in an ambient R would represent the mapping from one face to the other.The literature does not seem to have a general theorem claiming that an arbitrary equiv-alence relation could be replaced by differences in the shapes of the tiles although somethinglike that must be true for a limited category of equivalence relations. The modified hexagonof the Taylor monotile shown in fig. 14 (page 36; cp. [24]) is the most complicated exampleof this process known to the author; in this case the geometrical alteration results in adisconnected tile. 10**The next example is a version of the famous Penrose aperiodic tessellation. Here thereare also two types of tiles: isosceles triangles denoted R and R ; the subscripts are themultiples of π/ u = exp( iπ/
5) be the generator of thegroup of rotations of order 10. The inflation factor – the radix ρ – is the golden number φ = √ . The inflation equations are φ R = (cid:0) u R (cid:1) ∪ ( R ) φ R = (cid:0) u R (cid:1) ∪ (cid:0) u R (cid:1) ∪ (cid:0) u R (cid:1) (4)The tiles that correspond to the remainder sets are isosceles triangles. Their decomposition,implied by the inflation equation eq(4), is shown in fig. 6 on page 24.These equations lead to generalized positional representations, at first for points in theremainder sets, and then, by inflation and finally rotation of the wedge-shaped sectors, forarbitrary points of the complex plane. The translations appearing in eq(4) are the digits:∆ = { , } . There are several remainder sets that are reflected and rotated as the positionalrepresentation advances from digit to digit.The positional representations have the form z = (cid:88) k z k φ k u k , z k ∈ ∆ , u k ∈ { u n : 0 ≤ n < } (5)and each remainder set is the set of all representations of the kind specified by the equations.Figure 1, resp. fig. 2, displays the expansions in R , resp. R , through 8 digits. The colorcoding and diameter of the disk that represents a number are arranged so that the pointscorresponding to a given number of digits can easily be seen. The large red disks correspondto { /φ, /φ } ; the smaller orange disks correspond to 2-digit expansions, etc.Inflation by φ extends this to a sector in the plane, and rotation by powers of u extendsthe sector to the entire plane. Observe that the triangular remainder sets R and R –which were originally thought of as the tiles – are fully determined by the equations, andthe partition of each remainder set is essentially disjoint.The point that corresponds to a finite expansion is a vertex of a deflated copy of aremainder set. In this sense, it labels the deflated remainder set. In each remainder set thepositional representation for a number defines a polygonal path – call it simply a path –from the origin to the point representing the number and labeling the deflated remainderset: just add the complex numbers – the vectors – corresponding to successive digits. Thepaths are a microscope that opens up a universe of detail in the progressively deflatedremainder sets. These equations were used to generate the Penrose pinwheel shown in fig. 17 on page 39. φ ∼ .
61. Penrose R φ + u φ + u φ + u φ + u φ + 1 φ + u φ + u φ = −
352 + 8 √ i (cid:113) − √ u express the changes ofdirection along the progressively smaller segments of the path. Note that each turn is bythe angle 4 π/
5. ***It has been said that a remarkable property of Penrose tilings is that every finitepattern of tiles is repeated infinitely often somewhere else in the tiling. An unremarkableproperty of positional representations is that every pattern – every sequence – of digitsin the representation of a number is repeated in the sequence of digits of infinitely manynumbers. For inflationary tilings with positional representations, the discussion above showsthat these facts are the same.Let us elaborate this observation. Any finite portion T of the tessellation of the planecan be deflated until it becomes a subset of each of the remainder sets. Within the remainderset each tile has a distinguished vertex – the point in a subtile that corresponds to 0 inthe reminder set – labelled by a finite expansion in the positional representation. Thus thecollection of tiles is in a correspondence with a collection of positional representations, andhence with the corresponding paths. If T is a subset of a deeply nested remainder set R ,then the initial segments of the paths will coincide until the vertex specifying R is reached,after which they may diverge to the tiles they represent within R . Exactly the same is truefor the corresponding numbers and their positional representations. So all of these numberswill share an initial sequence of digits D initial until some place in the notation, at whichpoint a vertex of the appropriate deflated tile has been specified, and different sequencesof digits { D k } will follow the initial common segment D initial , one for each tile in T .12igure 2: Radix φ ∼ .
61. Penrose R D new is insertedbetween the common segment D initial and each D k . This has the geometrical effect ofdeflating the tiles T further. Looked at through the other end of the microscope, when thetessellation has been re-inflated to the level where the tiles are their original size, they willbe located somewhere else, depending on the path for the sequence D new .***This is the model we will generalize. It suggests that the idea of a positional representationbe extended to include a remainder set for each type of tile. Our attention will generallybe restricted to ambient spaces R or R and R will be identified with C . Let R i be a finitecollection of tiles for an inflationary tessellation with radius ρ . The inflationary hypothesisis equivalent to ρ R i = (cid:91) j ( δ ij + u ij ( R j )) , δ ij ∈ ∆ , u ij ∈ O (6)where O is a finite subgroup of the orthogonal group. In the plane, the action R → u ij ( R )is either R → u R or R → u R where u is a root of unity. Iteration of this system of relationsshows that for each i and z ∈ R i there are digits z k such that z = ∞ (cid:88) k =1 z k ρ k u k (7)This is a positional representation for z that corresponds to the tessellation.Thus 13igure 3: 8-digit path in R from 0 to φ + u φ + u φ + u φ + u φ + φ + u φ + u φ . Theorem 3
If an inflationary tiling corresponds to a positional representation, then everycompact part of the tiling is contained infinitely often in other parts of the tiling.Proof:
Let us recapitulate what has already been said for the Penrose tiling. After apossible rotation to bring the region into concordance with the remainder sets, every finitecollection T of tiles is contained in an inflated version of a remainder set, say ρ n R i . Thetiles in the collection have a positional integer representation. These leading digits coincidein the deflated set ρ − n T ⊂ R i . Select an arbitrary finite sequence of digits and rotationsand follow it by the given one. The concatenation is the distinguished vertex of a deflatedremainder set after some number of iterations. Inflating that set produces a region of thetiling in which T reappears. (cid:3) ***Let us restrict our attention to classes of tiles that have the same measure. Consideran inflationary tessellation of R d formed from N types of tiles. An N -rowed matrix U ,called the partition matrix , can be constructed from this information. The element U ij isthe number of tiles of type j required in the tessellation of the inflated tile ρ O ( R i ). Thisnumber is a non negative rational integer. The vector v whose j -th component is m ( R j ) –the measure of R j – is an eigenvector of U . Indeed, from the definitions, U v = ρ d v (8)The components of the eigenvector lie in the field generated by the eigenvalue ρ d . Con-versely, given the eigenvector v , the eigenvalue can be expressed as ρ d = v t U vv t v For rotations O the determinant is 1; for orthogonal transformations with det O = −
1, package the signwith ρ . ρ d lies in the field generated by the measures of the different types oftiles, and ρ lies in that field with d -th roots adjoined. Theorem 4
A multiplier ρ for an inflationary tessellation is an algebraic integer of degreeat most N d .Proof: ρ d is a root of the characteristic polynomial of U , which has rational integers ascoefficients and leading coefficient 1. Therefore ρ d is an algebraic integer of degree at most N . (cid:3) These simple remarks already tell us that not every real number greater than 1 inabsolute value can be the multiplier of an inflationary tessellation. This limitation, whichinterweaves algebraic number theory with geometry, is part of what makes the class ofinflationary tessellations interesting.If there is only one type of tile then U is a matrix of order 1 whose sole entry is thepositive integer n > R are needed in an essentiallydisjoint tiling of ρR . Hence the eigenvalue equation is ρ d = n which determines ρ up to aroot of unity. This is the case that applies to periodic tilings using a single type of tile, likethe white marble hexagons that were used by interior designers of an earlier generation totile the bathroom floors. Here the tiles are congruent but the result applies more generallyto tiles that have equal area. In particular, if there were a monotile – a single tile thatcovers the plane aperiodically in an inflationary tessellation – this equation would apply.A tiling of the plane by a process similar to inflation that uses just one type of tile wasdiscovered by Joan Taylor [27]; it will be discussed below. The partition matrix U contains information that is sometimes sufficient to prove thatan associated tessellation is aperiodic. The basic idea goes back to Penrose [15] but hereit appears in the more general setting of equivalence relations not necessarily limited togeometrical congruence.The multiplier ρ d is an eigenvalue of the N × N matrix U . The entry U ij is the number ofcopies of tiles of type j required to partition the inflated tile ( ρ O ) R i of type i . In particular, (cid:80) Nj =1 U ij is the total number of tiles needed to tile ρR i . The partition matrix of the n -thiterated inflation is U n ; the entries on its i -th row are the number of tiles of each typerequired to tile ( ρ O ) n R i and σ ni := (cid:80) Nj =1 ( U n ) ij is the total number of tiles required to tile( ρ O ) n R i . Theorem 5
Let U be the partition matrix of a tessellation. If there is a j such that lim n →∞ ( U n ) ij /σ ni is not rational, then the tessellation of R i with matrix U is aperiodic.Proof: Suppose the tessellation were periodic. Then there would exist some set S whosetranslates form an essentially disjoint cover of R d . This set can be covered by an integral It is inflationary only in the limit. Inflation by the multiplier yields an overtiling of ρR . U n ) ij /σ ni are all rational. Take an in-creasing sequence of essentially disjoint translates: each corresponding ratio for the unionis constant, so each limit is also rational. Thus no tiling with partition matrix U is periodic,so the tiling is aperiodic. (cid:3) This method of showing aperiodicity will be referred to as the proof by irrationality .The theorem provides a way to prove that some tilings consisting of tile types that differonly in color are aperiodic. The critical ingredient is a number-theoretic property of thepartition matrix. Note that this approach to proving aperiodicity cannot be used if thereis only one type of tile. ***Thus far it has been shown that a multiplier for an inflationary tessellation must bean algebraic integer. We have seen an example of an aperiodic tessellation and made aconnection between it and a generalization of positional representation where the twotypes of tiles correspond to two types of remainder sets.Now we will indicate how this generalization of positional representation arises naturally– why more than one type of remainder set occurs – and show that it further constrainsthe multiplier.The simplest way to see this is to try to construct 1-dimensional inflationary tessella-tions – tilings of R – with multiplier 1 < ρ ≤
2. Suppose the tile R is an interval. Moreover,consider a cover of ρR by two copies of R : ρR = R ∪ ( v + R ) with v ∈ R . Iteration of thisequation yields a positional representation for elements of R as R = x : x = (cid:88) k ≥ δ k ρ k v , δ k ∈ { , } (9)and it follows that the endpoints of R are 0 and (cid:80) k ≥ ρ − k v = v/ ( ρ − R by setting v = 1. Then R = [0 , / ( ρ − R ∩ (1 + R ) = [1 , / ( ρ − R will be essentiallydisjoint only when ρ = 2 and only then will it produce a tiling. This motivates us tointroduce new types of tiles to make the cover essentially disjoint for other 1 < ρ < ρR can be expressed as the essentially disjoint union ρR = [0 , ∪ (1 + R ), whichintroduces the new tile R = [0 , R is the essentially disjoint union ρR = [0 , ρ ] = R ∪ (1 + [0 , ρ − R = [0 , ρ −
1] that inflates as ρR = R ∪ (1 + [0 , ρ ( ρ − − yielding a sequence of tiles R n = [0 , x n ] with x n = ρ ( ρ ( . . . ( ρ − − . . . − − ρ n − n − (cid:88) k =0 ρ k The number of types of tiles can be limited to n by requiring ρ n = n − (cid:88) k =0 ρ k This constraint makes the space spanned by the ρ k over Q n -dimensional; forces the mul-tiplier to be an algebraic integer of a special type; and replaces the original overlapped setunion for R by a collection of interwoven essentially disjoint set unions for the R k .This roughly indicates how positional representations with many remainder sets arise.***We will fill in the details of this procedure to produce an infinite collection of aperiodictessellations, starting with aperiodic tessellations of R and use them to construct aperiodictessellations of higher dimensional spaces, concentrating on R .The complexity of both results and presentation increases rapidly as the radix grows.In order to keep the calculations comparatively simple and the discussion informative itwill be helpful to concentrate on positional representations that, like the binary system,only use the digits 0 and 1. Constructing these positional representations is the task ofsection 3. If 1 < ρ ≤ ρ can be used as the radix of a positional representation with digits { , } , which may be thought of as a kind of ‘generalized binary representation’. Every x in the remainder set R has a positional representation x = ∞ (cid:88) k =1 x k ρ k , x k ∈ { , } (10)from which it follows that R = { x : 0 ≤ x ≤ / ( ρ − } The series for x can be conveniently expressed in positional notation as x = (0 · x x . . . x n . . . ) ρ (11) Subject to the obvious inequalities on ρ that insure the number x n is not negative.
17f the radix ρ is fixed, write x = 0 · x x . . . x n . . . We make the convention that an infinite repetition of the sequence x k x k +1 . . . x k + l isabbreviated x k x k +1 . . . x k + l and that an infinite sequence of trailing zeros may be omitted.***Among these generalized binary representations are a family with particularly interestingproperties. Let N ∈ Z + and select a sequence of N ‘bits’ b j ∈ { , } with b N = 1. Put b = (0 · b . . . b N ) , and assume that 2 N b is odd and greater than 1. The integer 2 N b satisfies1 < N b < N . The silver number of index b , denoted s b , is the largest real root of thepolynomial of degree N P b ( x ) := x N − N (cid:88) j =1 b j x − j (12) Lemma 1
The silver number s b exists and < s b < .Proof: By definition, 2 N b > x ≥ P b ( x ) > P b (2) = 2 N − N (cid:88) j =1 b j − j ≥ N − N (cid:88) j =1 − j > , while P b (1) = 1 − (cid:80) Nj =1 b j ≤
0. Since P b ( x ) is continuous, it has a real root, and hence alargest real root, between 1 and 2. Since P b ( x ) > P b (2) for x >
2, it has no larger real root. (cid:3)
For N = 2 there is one silver number: s / = φ = √ , the golden number. Note thatlim N →∞ s b = 2. Theorem 6
The silver numbers are algebraic integers of degree greater than 1.Proof:
A silver polynomial is monic with rational integer coefficients. Thus its roots arealgebraic integers. The proof reduces to showing that P b ( x ) is irreducible over Q . Supposeotherwise. Then there exist distinct co-prime integers p, q such that P b ( p/q ) = 0. Since b N = 1, this assumption implies p N − N − (cid:88) j =1 b j p N − j q j = q N The left side is divisible by p (recall that N > (cid:3) P b generates identities that produce distinct finite positional represen-tations for the same number. This comes about as follows. Let the radix be ρ = s b . Then ρ satisfies 1 = (0 · b . . . b N ) ρ (13)This implies that there are infinitely many finite positional representations that denotethe same number. The golden number provides the simplest example. With b = 3 / ρ shows that the following representationsare equal: 0 · ·
011 = 0 · · P b ( x ) in the form U := ( u ij ) , u ij = b j if i = 11 if j = i − < i ≤ N U = b b b . . . b N . . .
00 1 0 . . . . . . (15)where b j ∈ { , } . The characteristic polynomial of U is P b ( x ) so s b is the largest realeigenvalue of U .Fix the degree N and the index b and drop them from the notation. Lemma 2
The silver number s is an eigenvalue of U . If v is an eigenvector of U thatbelongs to s , then v is proportional to the vector whose components are v k = s − k .Proof: The components of an eigenvector v for s satisfy s v k = v k − for 2 ≤ k ≤ N and s v = (cid:80) Nk =1 b k v k = 1. Then v k = c s − k , c (cid:54) = 0. Since v (cid:54) = 0 it can be normalized so that c = 1, i.e. (cid:80) k v k = 1. (cid:3) That is, expansions such as eq(10) that have finitely many non-zero digits. Tessellations of R Silver numbers are connected to tessellations by identifying the companion matrix U withthe partition matrix for an inflationary tessellation. Properties of the companion matrixthereby become properties of tessellations and, conversely, properties of the partition matrixfor a tessellation become properties of the corresponding positional representation.Every ρ ∈ R such that 1 < ρ ≤ { , } . The remainder set is R = (cid:40) x : x = ∞ (cid:88) k =1 x k ρ k (cid:41) , x k ∈ { , } (16) R is an interval one of whose endpoints is 0. The other is (cid:80) ∞ k =1 1 ρ k = ρ − , a number thatis greater than 1 unless ρ = 2. Separating the first digit from the sum in eq(16) shows that R satisfies the set theoretic identity ρR = R ∪ (1 + R ) (17)As was mentioned above, unless ρ = 2, the two sets on the right are not essentially disjointbecause their intersection contains the interval [1 , / ( ρ − R by a collection of remainder sets such that the decomposition corresponding toeq(17) will be a union of essentially disjoint subsets. Here is where the special propertiesof silver numbers come in.Suppose that U is the companion matrix of a silver polynomial and ρ is the largestreal eigenvalue of U – the associated silver number. The eigenvector belonging to to ρ isproportional to v = ( v k ) where v k = ρ − k , 1 ≤ k ≤ N .Equation (16) implies that ρR = [0 , ρ/ ( ρ − ρ implies ρρ − ρρ − N (cid:88) k =1 b k ρ − k which tells us that the length of ρR can be written as the sum of lengths of N essentiallydisjoint intervals. So introduce sets R k = ρρ − , ρ − k ] , ≤ k ≤ N (18)The intervals b k R k can be arranged in any order as essentially disjoint subsets thatcover R . We shall say that the remainder set b k R k is essential if b k (cid:54) = 0, and that theorder R , . . . , R N is the natural order ; note that the lengths of the intervals are decreasingalthough only the essential ones contribute to the essentially disjoint covering of R .Introduce constants c k = (cid:26) k = 1 (cid:80) k − j =1 b j ρ − j if 1 < k ≤ N R k arranged in the standard order satisfy the system of equations ρ R = (cid:83) Nk =1 ( c k + b k R k ) ρ R k = R k − , ≤ k ≤ N (19) Theorem 7
The decomposition eq(19) is an essentially disjoint cover of the original re-mainder set R and R = ρR . Moreover, the decomposition implies the measure relationshipsexpressed by the companion matrix.Proof: For the first part it is enough to observe that the right hand endpoint of R k isthe left hand endpoint of R k +1 (so the intervals are essentially disjoint) and that the sumof their lengths is ρρ − (cid:80) Nj =1 ρ − j = ρρ − , which is the length of ρR .For the second, the measure is invariant under translations and reflection so m ( c k + R k ) = m ( R k ) and the decomposition implies U v = ρv . (cid:3) This procedure produces an infinite number of ‘tilings’ of R similar to barcode or astrange piano keyboard. They may be thought uninteresting. However, although they donot look like much to the eye, it turns out that they are, in general, aperiodic. Theyalso provide a foundation for constructing higher dimensional tessellations that are alsoaperiodic.For instance, if ρ = s / = φ then the companion matrix is given by eq(21). Figure 4shows the 10-digit decomposition of R constructed as above. It turns out that U is thematrix for dimension 2 that describes the aperiodic plane tessellations of Ammann andPenrose. Figure 4: Radix φ : 10-digit decomposition of remainder set R .***These tilings are aperiodic. We shall sketch two methods of proof. The first, followingPenrose, shows that the ratio of the number of tiles of the two tile types is irrational. Thesecond proves the non existence of a period lattice. Theorem 8 If ρ is a silver number whose companion matrix U defined by eq(15), then U is the partition matrix for a inflationary tessellation of R and the tessellation is aperiodic. roof: The proof can be reduced to a tessellation in standard order because the calcula-tions that will be made are order-independent.Since ρ is a silver number, ρ is an algebraic irrational and the digits in the associatedpositional representation are 0 and 1.The partition matrix elements count the number of tiles belonging to each class thatare required to cover each tile representative ρR k . ρR is partitioned into a total of (cid:80) Nk =1 b k tiles whereas every other ρR k is covered by R k − , hence by 1 tile. After n inflations thenumber of tiles of type j required to partition ρ n R i is the matrix element ( U n ) ij . U determines a linear recurrence sequence and the matrix entries are the numbers inthe sequence. Let us make this explicit. Let denote the vector whose entries are 1. Thenthe j -th entry of U n is the number of tiles required to partition ρ n R j . Denote this numberby a n + j . This sequence is generated by the relation a n = (cid:80) Nj =1 b j a n − j , or equivalently, by( a n +1 , . . . a n +1+ j , . . . , a n +1+ N ) t = U ( a n , . . . a n + j , . . . , a n + N ) t It is well-known that the solution has the form a n = N (cid:88) j =1 c j λ nj where c j are constants and the λ j are the eigenvalues of U , i.e. the roots of P b ( x ). Theproduct of the roots is 1 since b N = 1. Recall that ρ >
1. An application of Rouch´e’s theoremshows that each root has absolute value bounded by ρ . Hence the eigenvalue ρ dominatesand therefore lim n →∞ a n +1 /a n = ρ , which is irrational because P b ( x ) is irreducible of degreegreater than 1. This implies the 1-dimensional tessellation defined by U is aperiodic. (cid:3) ***For the golden number, the recurrence defined by the partition matrix U generates theFibonacci sequence. It follows that if e j := ( δ jk ) t where δ jk is the Dirac delta, then (cid:0) U d (cid:1) n e j counts the number of tiles of measure v j of type k in the n -th iteration of the inflationprocess. Thus, for the Penrose tessellation with matrix U given above, an inflation of thelarger tile consists of 1 copy of the smaller and 2 copies of the larger, and an inflation ofthe smaller tile is the union of 1 copy of each tile type: U (cid:18) (cid:19) = (cid:18) (cid:19) (cid:18) (cid:19) = (cid:18) (cid:19) , U (cid:18) (cid:19) = (cid:18) (cid:19) and so forth for the powers of U d . ***22nother way to prove theorem 8 depends directly on the positional representation. Let R (cid:48) denote a fixed remainder set R k . Since any x ∈ R (cid:48) has a representation of the form x = (cid:88) k ≥ x k ρ k , x k ∈ { , } each number in the inflated remainder set ρ n R (cid:48) can be written y := [ y ] + { y } where[ y ] = n (cid:88) k =1 x k ρ n − k , { y } = ∞ (cid:88) k = n +1 x k ρ n − k The number { y } is just another element of the remainder set. Regarding [ y ], if the radixwere an integer, [ y ] would naturally be called the “integer part” of y . If ρ is not an integer,the set of polynomials in ρ with coefficients in { , } does not have most of the propertiesof a ring of integers so we call these numbers positional representation integers . Lemma 3
Let < ρ < be a silver number and let Z [ ρ ] denote the set of positionalrepresentation integers, i.e. the polynomials of finite degree with coefficients in { , } . Then (cid:54)∈ Z [ ρ ] .Proof: Suppose the contrary. Then 2 is a polynomial in ρ with coefficients in { , } . Since ρ >
1, this polynomial cannot have more than 1 term. Hence 2 = ρ n for some n ∈ Z + . Butthis shows ρ is not a silver number. (cid:3) The lemma tells us that the positional representation for 1 + 1 is the sum of 1 and aremainder; indeed, the silver equation asserts 2 = 1 · b . . . b N .Now we can complete the second proof of theorem 8. If an inflationary tessellation isperiodic, then the period lattice is the ring of rational integers. In particular, the sum ofany two elements is an element of the lattice. Since 2 does not belong to the lattice, everytessellation with these tiles is not periodic. Hence the tessellation aperiodic. (cid:3) This way of proving aperiodicity extends to higher dimensions and, in principle, alsoto some tessellations that have just one class of tiles.
Suppose that n ∈ Z + . A consequence of U v = sv is U n v = s n v (20) We must adjoin the negative expansions as well.
23e can give this eigenvalue equation a different geometrical interpretation. It asserts thatmagnification by the factor s of the linear dimensions of a subset of R dn of volume sv j yieldsa set whose volume is the linear combination (cid:80) k U jk v k of the dn -volumes represented bythe v k .The most accessible cases of this interpretation are d = 1 ,
2; the most interesting is d = 2.Assume that ρ is the golden number φ = √ ∼ .
61. The Frobenius matrix – thepartition matrix – is U = (cid:18) (cid:19) (21)Thus U = (cid:18) (cid:19) Identify R with the complex field C and introduce the notation (cid:104) A (cid:105) for the equivalenceclass of subsets of the plane whose area is A .The eigenvector of U is v = ( (cid:104) (cid:105) , (cid:104) φ (cid:105) ) t . The relations expressed by U v = φ v areprecisely those between the areas of the darts and kites of Penrose’s aperiodic tiling of theplane. But not every realization of these relations as areas arises from a conventional tilingbecause the ‘tiles’ that have the same area need not have the same shape. Figures 5 and6 contrast the remainder set decompositions for the aperiodic tilings of Ammann [1] andPenrose [15]; both have radix φ and the same area condition matrix U (cp. eq(21)).Figure 5: Radix φ : Remainder set decompositions based on a non periodic tiling of Am-mann.Figure 6: Radix φ : Remainder set decompositions for Penrose remainder sets.24**In this context the reader may have observed that the matrix U itself corresponds to thetessellation known as ‘Ammann’s Chair’ with radix √ φ . How does this fit in to the generalstructure? According to what has been said, U corresponds to a 1-dimensional tessellation.But observe that √ φ is also a silver number because it is (the largest real) eigenvalue of U = (22)and U = (23)The characteristic polynomial of the first matrix is the silver polynomial x = x + 1 so ofcourse the roots include the square roots of the golden polynomial. Division by x yieldsthe positional representation 1 = (0 · √ φ Thus Amman’s Chair is a 2-dimensional example corresponding to the 4-rowed matrix U which decouples to describe two tilings, the first and third rows describing the decompo-sition of two remainder sets; the third and fourth, the (same) decomposition of the othertwo. ***Consider the group of isometries of R more closely. In the language of complex geometrythe orthogonal group O (2) is the extension of the group of rotations by complex conju-gation. Thus one could posit tile equivalence relative to O (2), including conjugation, ormerely relative to the proper subgroup of rotations. Assume the latter, and the positionalrepresentation with two remainder sets R k and radix ρ = φ . Abbreviate u = exp( iπ/ ρ R = (1 + u R ) ∪ R ρ R = ( u/ρ + u R ) ∪ (1 + u R ) ∪ (1 + u R ) (24)The digits for this positional representation are ∆ = (cid:110) , , ρ e iπ/ (cid:111) . The geometry is 2-dimensional so the number of the digits is the least integer greatest than or equal to | ρ | . See fig. 10 on page 27 for the 1-digit decompositions. Indeed, the n -th root of a silver number is also a silver number. φ = φ + 1 ∼ .
61, 3 digits are expected and that is what we have found. The digitsare the coordinates of the vertices of a triangle similar to R . The partition matrix forthis system is the Ammann-Penrose matrix, i.e. the square of U given by eq(21). Thisimplies that the tiling associated with the recursion eq(24) describes an aperiodic tiling. Isit the Penrose tiling? The answer is “No”. Although the two remainder sets are the sameas the Penrose triangles shown in fig. 6 and their decompositions also appear to be thesame, the tiling described by eq(24) is constrained by equivalence under the rotation grouprather than the full orthogonal group, so fewer geometrical options are possible. In fact,the tessellation associated with eq(24) is not edge-to-edge. The Penrose edge-to-edge tilingsrequire complex conjugation – geometrically, reflection – for their presentation. Figures 7and 8 show 6-digit approximations to the remainder sets without complex conjugationdefined by eq (21).Figure 7: Remainder set R for the Penrose-like tessellation without complex conjugation.Not edge-to-edge. 6 digits.Figure 8: Remainder set R for the Penrose-like tessellation without complex conjugation.Not edge-to-edge. 6 digits.Continuing to explore this example, suppose tiles are equivalent if and only if they arecongruent under translation or reflection. Tiles with different orientations belong to distinctclasses, taking into account that orientations and orientations reflected in the real axis areequivalent. Since the rotations are elements of the cyclic group generated by z → e iπ/ z ,there will be a total of 10 classes for each tile shape: a total of 20. If the tile classes aredifferently colored, the result is the edge-to-edge pseudo-Penrose tiling shown in figure 926igure 9: An edge-to-edge Penrose tiling of the triangular remainder set (tile) R withangles { , , } π/ φ . Tiles that are translations or reflections in the base of theisosceles triangle (complex conjugates) of each other belong to the same type. There are20 distinct orientation types. 6 digits.for the triangular tile with angles { , , } π/
5. This tessellation is aperiodic for the samereason that the usual Penrose tessellation is: the ratios of the number of tiles of differenttypes is irrational. The partition matrix is a 20 ×
20 array whose largest eigenvalue is, asexpected, φ .Carrying this line of thinking further, suppose that two tiles are equivalent if theyare similar hexagons. Then the ‘Ammann’s Chair’ tessellation encountered above, withdecompositions shown in figure 10, is an aperiodic monotiling of the plane. The ‘two’ tilesare equivalent and hence ‘the same’ since the larger is similar to the smaller (the factor isthe radix, √ φ ; the digits are 0 and 1). Of course, this is not what is normally meant whenspeaking of a ‘monotiling’, naive intuition implicitly expects congruence under the fullgroup of isometries, but the example does emphasize the importance of clearly specifyingthe equivalence relation when discussing a tiling.Figure 10: Ammann’s Chair: Remainder set decompositions for a positional representationwith radix √ φ . The small remainder set R S is shown in red; R L in blue. R L = √ φ R S .27 An infinite class of aperiodic tessellations in R This section constructs tessellations of R by taking cartesian products of tessellations of R defined by silver numbers. We have already seen that if U is a partition matrix for R , then U d is a partition matrixfor R d . This is how, for instance, the partition matrix for the Penrose and Ammann’s Chairtilings arise. However, the partition matrix alone does not determine the geometry of thetiles, as these examples show: the former employs triangles; the latter a non-convex hexagonwhose angles are all multiples of π/ N (cid:88) k =1 b k ρ − k (25)where b k ∈ { , } and b N = 1. The silver number ρ defined by this equation is the largestreal root. The Frobenius companion matrix U for the polynomial is given by eq(14); takeit as the partition matrix. The corresponding 1-dimensional geometrical partition of theunit interval consists of the sub-intervals b k R k where R k = { x : 0 ≤ x ≤ ρ − k } , ≤ k ≤ N .If the R k are arranged edge-to-edge in any order, their union is essentially disjoint andis an interval of length 1 because of eq(25). Recall that these tessellations of R are allaperiodic. We will show that their aperiodicity extends the the d -dimensional tessellationsbuilt up from them. We shall not be concerned with the particular order of arrangement,so suppose it is R , . . . , R N which is the decreasing order of the lengths of the essentialsubintervals.Consider the d -fold cartesian product of this partition. It provides a partition of theunit hypercube in R d into n d hyper-rectangles. This partition of the hypercube induces aninflationary tessellation with multiplier ρ for which the hypercube is a remainder set of a d -dimensional positional representation with radix ρ . We shall prove that it is aperiodic.Restrict the following discussion to R for simplicity. Define a collection of rectanglesby R i,j = (cid:8) ( x , x ) : 0 ≤ x ≤ ρ − i , ≤ x ≤ ρ − j (cid:9) , ≤ i, j ≤ N (26)The area of R ij is 1 /ρ i + j . Now inflate the largest square R . ρR is a square of side 1.Recalling eq(25), the inflated square can be partitioned into an essentially disjoint unionof ( (cid:80) b k ) rectangles each of which is a translation and rotation of an R ij . Each square R ii occurs once; each R ij with i < j occurs twice. From this information the associatedpartition matrix U , which has order N ( N + 1) /
2, is easily constructed. Note that this U is not a Frobenius companion matrix. *** The method can be readily generalized to R d but nothing essentially new is gained from the increasedcomplexity of presentation. Some of the b k may be 0. N = 2 so ρ is the golden number. In the previous paragraph itwas implicitly assumed that the equivalence relation for tiles is congruence under the fullgroup of isometries. Now relax this to congruence under translations. Then the number oftile types increases to N d and the partition matrix entries are 0 or 1. Indeed, the partitionmatrix for the four tile types is U = (27)The irrational number ρ is the largest eigenvalue; it has eigenvector ( ρ , ρ, ρ, ρ R = R ∪ (1 + R ) ∪ ( i + R ) ∪ (1 + i + R ) ρ R = R ∪ ( i + R ) ρ R = R ∪ (1 + R ) ρ R = R (28)The digits are ∆ = { , , i, i } . Figure 11 illustrates the 6-digit decomposition of theremainder set R = { z : 0 ≤ Re ( z ) , Im ( z ) ≤ /ρ } .Figure 11: Radix ρ = φ ∼ .
61 for the silver polynomial x = x + 1. Six-digit aperiodicdecomposition of the remainder set R for a positional representation of radix ρ .***Consider N = 3 and tile equivalence under the full group of isometries. The irreduciblesilver number equation is x = x + x + 1 with largest real root ρ ∼ . U = (29)Its largest eigenvalue is ρ . Using these decompositions to inflate R ρ ∼ .
83 for the silver polynomial x = x + x + 1. Decomposition of thesix remainder sets R k for a positional representation of radix ρ . In order of increasing area,the sets are, from top to bottom and left to right: R through R (see text). Note that R and R have equal area. Rotationed and reflected tiles are equivalent. Theorem 9
The d -dimensional tessellations constructed above are aperiodic.Proof: The matrix U has ρ d as largest eigenvalue so Penrose’s proof by irrationalityapplies. (cid:3) At this point we have an extensive inventory of examples and procedures for building 1-and 2-dimensional tessellations. Now it is time to reconsider inflationary tilings for whichtiles are said to be of the same type if they have the same measure.First of all, recall the existence of measure preserving maps R → R . Identify theunit square S ⊂ R with the unit interval I ⊂ R . Divide the square into 4 essentiallydisjoint congruent squares of side 1 / I ; the order doesn’t matter. Repetition of this30igure 13: Radix ρ ∼ .
83 for the silver polynomial x = x + x + 1. 4-digit tessellation ofthe largest remainder set R for a positional representation of radix ρ . See text for details.process, subdividing each small square into 4 smaller squares of equal area and representingthem on the corresponding subintervals of equal length, leads in the limit to a mapping ofthe square onto the line (and, reciprocally, of the line onto the square) that is well definedexcept on a set of measure 0. The map can be extended to a measure-preserving map from R onto R in the same way. This idea goes back to the earliest days of Lebesgue measure.From this description it is evident that such a construction preserves essential disjoint-ness. Lemma 4
Suppose that µ : R n → R n (cid:48) is a measure preserving map. Then A, B ⊂ R n areessentially disjoint if and only if µ ( A ) and µ ( B ) are essentially disjoint.Proof: Two sets A and B are essentially disjoint if and only if the measure of theirintersection is 0, which is equivalent to m ( A ∪ B ) = m ( A )+ m ( B ). If µ is measure preservingthen m ( A ∪ B ) = m (cid:48) ( µ ( A ∪ B )), m ( A ) = m (cid:48) ( µ ( A )), and m ( B ) = m (cid:48) ( µ ( B )) so m ( A ∪ B ) − m ( A ) − m ( B ) = 0 if and only if m (cid:48) ( µ ( A ∪ B )) − m (cid:48) ( µ ( A )) − m (cid:48) ( µ ( B )) = 0. (cid:3) Inflationary tessellations – positional representations – provide us with additional ex-plicit structures that establish measure preserving mappings from one dimension to an-other. Let us start with an inflationary tessellation of the plane. Denote the collection ofremainder sets (or tile class representatives) R j . Let the partition matrix be U . Let ρ bethe largest real eigenvalue of U . Note that this eigenvalue equation acts on areas, so theeigenvalue is not the multiplier that expands the linear dimensions of a tile. That numberis √ ρ because the dimension is 2. 31 mediates between the area of an inflated remainder set m ( ρR j ) and the areas of theremainder sets that partition it. The measures m ( R k ) are non negative numbers and canbe interpreted as lengths. If a remainder set R j be selected, it can be represented on anintegral of length m ( R j ), say I j := [0 , m ( R j )], and the partition of R j satisfies m ( R j ) = 1 ρ (cid:88) k U jk m ( R k )which describes how many copies of the interval of length m ( R k ) are needed to fill out I j . At each level, corresponding to each digit in the representation, the particular order ofplacement of the sub-intervals within the larger one that contains it, does not matter. Nordo the translations and rotations and reflections that may occur in the description of the2-dimensional tessellation matter, for they do not change the measure of the partitioningtile. Note that the essentially disjoint 2-dimensional inflationary tessellation or positionalrepresentation is carried over into a measure-equivalent an essentially disjoint structure bythe measure preserving map. This process transfers the structure of the tiling, insofar as itis reflected in the area of each tile, to an interval on the line. We shall call it the projection of an inflationary tessellation onto R .As to the associated radix, note that U refers to the plane tessellation and ρ is theeigenvalue derived from it. After the measure-preserving map is applied and the situationis observed on R , areas have been converted to intervals, the matrix U is still the partitionmatrix for the nested line segments, and the eigenvalue ρ is – in this setting – the multiplierfor the inflationary n-dimensional tiling of R . Thus it also is the radix for the positionalrepresentation of R associated to the tiling.An interesting example is the Penrose tiling, whose partition matrix is U = (cid:18) (cid:19) and largest eigenvalue φ ∼ .
61. From eq(4) on page 11, there are just two 1digits’ – 0and 1. The ratio of the areas x = m ( R ) /m ( R ) satisfies x = x + 1 so x = ρ (because R ⊂ R ). Choose the unit of length so that the areas are (1 /ρ, /ρ ).Remembering that the inflated areas increase by the factor ρ , the two remaindersets can be associated with intervals of lengths 1 /ρ and 1 /ρ , and then partitioned intonested subintervals. Let the remainder intervals that represent the 2-dimensional remaindertriangles be I = [0 , /ρ ] , I = [0 , /ρ ]Then the partition equations can be taken in the form The measure-preserving maps are functors that carry the structure of inflationary tessellations or po-sitional representations from one space to another. I = (cid:16) ρ + I (cid:17) ∪ I ∪ (1 + I ) ρ I = (cid:16) ρ + I (cid:17) ∪ I (30)with digits { , , /ρ } . Recall that 1 /ρ = ρ − Theorem 10
The projection of an aperiodic inflationary tiling is an aperiodic inflationarytiling.
In the other direction, the loss of the rotations and reflections as well as arrangementof the positional digits in the plane leaves open the question of whether there is a 2-dimensional tiling in the sense of equivalence under isometry. If there is, then the characterof the original 1-dimensional tiling ‘injects’ to form an inflationary 2-dimensional tiling.Thus the 1-dimensional tilings that are projections of the Penrose tiling are also aperi-odic, and so forth.Defining tile type relative to the equivalence relation of equal measure frees us to inves-tigate a variety of questions about tessellation independent of dimension. The equivalencerelation can be constrained later to bring it accord with the conventional view.***Finally, it may be worth mentioning that there is another way to eliminate the depen-dence of the tiling equations on rotations and reflections. In a given tessellation, the groupgenerated by the orthogonal transformations that appear in the decomposition equations isfinite. Therefore, new tile types and corresponding new remainder sets can be introduced,one for each group element. The original replacement equations imply equations for the newremainder sets but now the replacement equations only contain translations of remaindersets. 33
Positional representation integers Z [ ρ ] for R This section continues the study of 1-dimensional tessellations under the assumptions that1 < ρ ≤ { , } . The positional representation integers are the elements of Z [ ρ ] = ± z : z = (cid:88) k ≥ z k ρ k , z k ∈ ∆ (31)where the sum is finite. The sum and product of elements of Z [ ρ ] are defined for only somepairs of elements, but when they are defined, both are commutative and the distributivelaw x ( y + z ) = x y + x z is satisfied for x, y, z ∈ Z [ ρ ]. For any z ∈ Z [ ρ ] and n ∈ Z ,0 + z = z, z = z, ρ n Z [ ρ ] ⊂ Z [ ρ ]so 0 is an additive unit and 1 a multiplicative unit, and inflation of Z [ ρ ] by the multiplier ρ is a subset of Z [ ρ ]. Each element has an additive inverse. If { m j } ∩ { n k } = ∅ then (cid:80) ρ m k + (cid:80) ρ n k is always defined whatever ρ . Denote the number of elements in a finite set S by S ). If { m j + n k } ) = { m j } ) ∗ { n k } ) (i.e., the exponents of the termwiseproducts are all different) then ( (cid:80) ρ m k ) ( (cid:80) ρ n k ) is defined.In general, 2 = 1 + 1 (cid:54)∈ Z [ ρ ]. Of course 2 always has a positional representation but aremainder may appear.The relationship between the sets of integers Z [ ρ ] and lattices will be considered next.Three examples in R suggest how different these sets of ‘integers’ can be depending on theradix. In all cases the set of digits is ∆. A lattice Λ ⊂ R has a single generator so Λ = λ Z for some λ > R . Unless otherwise stated, assume that 1 < ρ < { , } . Proposition 1 If ρ = 2 then Z [ ρ ] = Z is a lattice and a ring. This positional representation is just the signed standard binary representation forintegers.
Proposition 2 If < ρ < then Z [ ρ ] is not contained in a lattice.Proof: Suppose Z [ ρ ] ⊂ Λ. Then 1 ∈ Z [ ρ ] implies 1 = nλ with n ∈ Z + . Hence λ =1 /n . Similarly, ρ j = k j /n so m j = k j n j − for j ≥ n j − divides m j . Let the primefactorizations be m = (cid:89) l p d l l , n = (cid:89) l p e l l Such a structure is an example of a ringoid , but having a name for it is not of much help. k j = (cid:89) l p jd l − ( j − e l l so d l ≥ (1 − /j ) e l for j >
1. Taking j >> d j and e j are integers, itfollows that d j ≥ e j . At least one of these inequalities must be strict; otherwise m = n , acontradiction. But d j > e j implies ρ = m/n ≥ p j ≥
2, a contradiction. (cid:3)
Now consider whether a lattice can be a subset of Z [ ρ ]. Here is an interesting example. Proposition 3 If ρ = √ then Z [ ρ ] contains Z as a proper subset.Proof: Every integer n ≥ Z [ ρ ] has a representation of the form n = (cid:88) k ≥ n k k/ = (cid:88) k ≥ n k k + √ (cid:88) k ≥ n k +1 k = m m √ , m , m non negative rational integersIn particular, Z ⊂ Z [ ρ ] and √ Z ⊂ Z [ ρ ]. Both inclusions are proper because √ (cid:54)∈ Z and1 (cid:54)∈ √ Z . Moreover, Z [ ρ ] is dense in R , hence not a subset of a lattice. (cid:3) This example is an instance of an interesting class of positional representations. Theremainder set is R = [0 , √ x ∈ R can be written in the form x = x + √ x for x , x ∈ [0 , x ∈ (1 , √
2) has uncountably many different representations because x ∈ (0 ,
1) can be chosen at will whence x = ( x − x ) / √ ∈ (0 , Proposition 4 If Λ = λ Z ⊂ Z [ ρ ] then λ and ρ are of the same type, i.e. algebraic ortranscendental. If ρ is algebraic, then both belong to the number field generated by ρ andare algebraic integers.Proof: For each k ∈ Z + there is a Q k ∈ Z [ ρ ] such that kλ = Q k . Each Q k is a polynomialin ρ whose coefficients are 0 or 1. In particular, λ = Q . It follows that λ and ρ are of thesame type, i.e. if one is, respectively, algebraic or transcendental, so is the other.From k = kλλ = Q k /Q it follows that Q k − kQ = 0 and the polynomial has leadingcoefficient 1 (Note that if k > l the degree of Q k is greater than the degree of Q l ). Hence ρ is an algebraic integer. Therefore kλ = Q k lies in the field generated by ρ . (cid:3) If a plane tessellation is periodic and inflationary, then theorem 1 implies that ρ isa quadratic imaginary integer and ρ O (Λ) ⊂ Λ where O is a suitable orthogonal linear35igure 14: Radix = 2. Taylor aperiodic monotile. The line segments have zero width andare shown only for illustrative purposes. The tessellation is not inflationary.transformation. Continuing to restrict our attention to multipliers for which 1 < ρ d ≤ < ρ ≤
2, the suitable quadratic imaginary fields are given by ρ O = i √ ρ = 1 + i , and ρ = i √ and their multiplies by units of the field Q ( ρ ) eachgenerates. In each case Z [ ρ ] is the associated ring of algebraic integers, which is a lattice. Joan M. Taylor recently devised a monotile – a tile that can tessellate the plane onlyaperiodically [27], [25], [23].There are various ways to think about this tiling. From a conventional perspective, itis the standard hexagonal tiling of the plane but the tiles are ‘decorated’ and replacementrules are promulgated that insure only aperiodic tilings arise. There is only one type ofdecoration. This description – which is standard in the field – may give the impressionthat not so much has been accomplished; some might be better satisfied with a specificgeometric shape that only allowed aperiodic tilings. Socolar and Taylor have shown thatsuch a tile exists: the decorations are equivalent to a hexagon whose boundary has beensuitably deformed. The deformation introduced by Socolar and Taylor is shown in figure14. Note that this tile is not self-similar. Taylor’s monotile, shown in figure 14, is a distorted version of a regular hexagon. Thearea of the monotile is the same as the area of the hexagon from which it was constructed.Figure 15 shows 7 interlocking copies the Taylor monotile so the monotile is not self-similar. Socolar and Taylor prove that this process can be continued to an essentiallydisjoint covering the plane and that the covering is aperiodic. The figure employs color to See the next subsection for a self-similar monotile. This image agrees with the black tile in Figure 6(a) of [23] but differs from what purports to be thesame tile in figure 3 of [25]. n copies of the monotileafter which the scale of the figure is divided by 2 n , then the limit will be a regular hexagonwhose side can be taken to have length 1. This presentation is also equivalent to employingcongruent hexagonal tiles that differ only in color. There will be 4 types, say red, green,teal and magenta.The tessellation is not inflationary and therefore it is not an example of the tilingsconsidered in this paper. However, a second tiling due to Taylor, whose discovery chrono-logically preceded the tiling just described, is inflationary. We shall study this tiling inmore detail. ***The method of proving a tiling is aperiodic by showing that the limit of the ratio ofthe number of tiles of different types that occurs in a sequence of regions of increasing andunbounded measure is irrational cannot be used when there is only one type of tile.An approach which would apply to monotilings might be to decorate each tile typewith a line segment so that the partition rules for tile placement insure that the linesegments become continuous curves without endpoints. As a tile is inflated and the rulesare applied, increasingly large regions of the plane will be covered. The condition thata curve constructed this way not have endpoints means it starts and ends at infinity, oris a closed curve in a compact region of the plane. If there were a nested collection ofclosed curves of increasing length, then the tessellation could not be periodic. This is theprinciple that Taylor [27] and Socolar and Taylor [24] use to prove the existence of an37periodic monotile, which will be discussed below. But it is well to note that it could alsoapply when there is more than one type of tile.Although the result is valid, the principle as stated is not correct. Consider, for example,the inflationary tiling of the plane by translated squares defined by eq(2) on page 9. Afterinflation, the recursion fills the first quadrant. Extend the covering to the plane by rotatingthe figure by multiples of π/
2. There is only one type of tile and there are no rotations.Decorate the square with a diagonal line. The decoration of the basic tile propagates inthe tessellation, shown in fig. 16, as a sequence of concentric nested squares of unboundedsize. These curves are continuous, closed and have finite length. The emergent red patternis not periodic, but the underlying geometric tiling is periodic.Figure 16: Decorated square tessellation showing nested sequence of concentric squares ofincreasing size. See text for details.The kernel idea that nested curves of increasing length imply non periodicity is basicallycorrect but it requires some modification. If the underlying tiling were periodic with periodlattice Λ, then decorations of the tessellation would project onto the quotient torus C / Λ.It is the nature of these curves that decides the question of periodicity. If the tessellationis periodic, then the images of the decorations on the torus will be a (system of) closedcurves of finite length. Otherwise, the tessellation cannot be periodic.Returning to the decorated square, the projected curves collapse to a simple loop onthe torus.Returning to the Penrose tessellation, recall that it has an inflationary constructionbased on two remainder sets: triangles R jkk where a subscript, say j , denotes the angle jπ/ R and R are isosceles triangles. Mark R with a line seg-ment joining the midpoints of the isosceles edges, and R with a line segment joiningthe midpoint of the base to the opposite (acute) vertex. Figure 17 shows a Penrose 5-digitpinwheel tiling based on R where only the line segments are indicated. They form con-tinuous curves. It is easily seen (but it is not quite so easy to prove) that there are nestedsequences of closed curves that meander about the perimeter of a sequence of regular pen-tagons of increasing size centered on the axis of the pinwheel. It seems intuitively clear38igure 17: Penrose pinwheel based on remainder set R with line segments, showingnested continuous curves of increasing length. 5 digits. The image is generated by thepositional representation eq(4). See text for details.that the curves cannot project onto closed curves of finite length on any quotient torus,but it may not be easy to prove this. ***The high degree of interest in the recently discovered aperiodic monotile – a single tileclass that yields only aperiodic tilings – may make it worthwhile to to provide more detailsabout its connection with positional representation. Taylor’s initial discovery involved 14congruent trapezoids with decorations that determined their placement. This was latersupplemented by the equivalent (disconnected) monotile described above and illustratedin figure 14.Fourteen congruent trapezoids are suitably decorated so that appropriate tiling rulespair them to form hexagons. A hexagon inflated by the radix ρ = 2 can be tiled by fourdecorated congruent hexagon tiles and 6 congruent trapezoids. Repeated inflations coveran ever increasing internal region with hexagons and a region contiguous to the boundaryof the inflated hexagons with trapezoids. We will derive the equations that correspond toTaylor’s trapezoidal construction.Reference [17] gave a system of coupled equations for the 14 decorated Taylor trape-zoidal remainder sets as well as a single equation that encapsulates all the information.Next we present an equivalent but slightly simpler version of that equation which describes39igure 18: A portion of the trapezoid-based monotiling produced by eq(32). The field oftrapezoids is shown in the background for reference. Hexagons, related to the Socolar-Taylorhexagonal monotile, are formed by pairs of trapezoids. 5 iterations.the aperiodic Taylor tiling. Put ρ = 2 and ω = e πi/ . Denote complex conjugation by an overbar. Theorem 11
The recursion eq(32) produces the Taylor-Socolar aperiodic monotiling. Thedigits are { , , ω, ω } . ρ R = R ∪ (cid:0) ω + ω R (cid:1) ∪ (cid:0) ω + ω R (cid:1) ∪ (cid:0) ω R (cid:1) (32)Figure 18 illustrates Taylor’s aperiodicity proof. The figure was constructed from eq(32)as follows: Draw a line perpendicular to the parallel edges of the trapezoid from a vertexto the base. The resulting figures consists of an array of triangles. The idea of the proof ofaperiodicity is that the triangles have increasing size as the tessellation is inflated to covermore of the plane, and that arbitrarily large triangles are incompatible with periodicity.The alternative argument based on the fact that a fundamental domain for a periodicarray can be translated by lattice elements without changing the lattice can in principle beapplied to monotile tessellations. If the array of positional integers, i.e. the set of polyno-mials whose coefficients are digits drawn from ∆ evaluated at the radix ρ , is not a latticethen the tessellation by remainder sets cannot be periodic. This method does apply to thePenrose aperiodic tiling and the product space tilings with silver number radix, but it doesnot apply to the Taylor trapezoid monotile for in that case, the set of positional integersis the lattice { z = m + ωn } where ω = exp( iπ/ The system described below orients the trapezoid vertically and has simpler digits. eferences [1] Ammann, R., B. Gr¨unbaum, and G.C. Shephard (1992), “Aperiodic tiles,” DiscreteComp. Geom. no. 1, 1-25.[2] Berger, R. (1966), “The undecidability of the domino problem,” Mem. Amer. Math.Soc.
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