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The class of Cyclotomic Aperiodic Substitution Tilings (CASTs) is introduced. Its vertices are supported on the 2nth cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers of CASTs are discussed as well as practical use cases to identify specimen with individual dihedral symmetry Dn or D2n, i.e., the tiling contains an infinite number of patches of any size with dihedral symmetry Dn or D2n only by iteration of substitution rules on a single tile.
Tilings have been the subject of wide research. Many of their properties are investigated and discussed in view of their application in physics and chemistry, e.g., in detail in the research of crystals and quasicrystals, such as D. Shechtman et al.’s renowned AlMnalloy with icosahedral point group symmetry [1]. Tilings are also of mathematical interest on their own [2,3].
Without any doubt, they have great aesthetic qualities as well. Some of the most impressive examples are the tilings in M. C. Escher’s art works [4], H. Voderberg’s spiral tiling [5,6] and the pentagonal tilings of A. Dürer, J. Kepler and R. Penrose [2,7], just to name a few.
In contrast to a scientist, a designer may have different requirements. Nevertheless the result may have interesting mathematical properties. Due to the lack of a general criteria for subjective matters of taste, we consider the following properties as preferable:
The tiling shall be aperiodic and repetitive (locally indistinguishable) to have an interesting (psychedelic) appearance.
The tiling shall have a small inflation multiplier for reasons of economy. Large inflation multipliers either require large areas to be covered or many tiles of a small size to be used.
The tiling shall yield “individual dihedral symmetry” Dn or D2n with n≥4. In other words, it shall contain an infinite number of patches of any size with dihedral symmetry only by iteration of substitution rules on a single tile.
Similar to G. Maloney we demand symmetry of individual tilings and not only symmetry of tiling spaces [8].
The most common methods to generate aperiodic tilings are:
Matching rules, as introduced in the very first publication of an aperiodic tiling, the Wangtiling by R. Berger [9]. See [10,11,12] for more details.
Cutandproject scheme, first described by de Bruijn for the Penrose tiling in [13], later extended to a general method, see [14,15] and references therein for more details, notably the earlier general method described in [16].
Duals of multi grids as introduced by de Bruijn [17] with equidistant spacings are equivalent to the cutandproject scheme [18].
Duals of multi grids with aperiodic spacings as introduced by Ingalls [19,20] are close related to Ammann bars, see [2,10,21] and [22] (Chapter 5) for details and examples.
The idea of substitution rules or substitutions in general is a rather old concept, e.g., Koch’s snowflake [23,24,25] or RepTilings [26]. However, it seems its first consequent application to tile the whole Euclidean plane aperiodically appeared with the Penrose tiling [2,3,27,28,29].
Among those methods, substitution rules may be the easiest approach to construct aperiodic tilings. Additionally, they have some other advantages:
The inflation multipliers of tilings obtained by the cutandproject scheme are limited to PVnumbers. According to [14,15] this was first noted by [16]. This limitation does not apply to substitution tilings.
Matching rules tend to be more complex than substitution rules. See [30,31] for examples.
In view of the preferable properties we will introduce the class of Cyclotomic Aperiodic Substitution Tilings (CAST). Its vertices are supported on the 2nth cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Its properties, in detail: substitution matrices, minimal inflation multiplier and aperiodicity, are discussed in Section 2. CASTs with minimal or at least small inflation multiplier are presented in Section 3 and Section 4, which also includes a generalization of the LançonBillard tiling. Section 5 focuses on several cases of rhombic CASTs and their minimal inflation multiplier. In Section 6 the “Gaps to Prototiles” algorithm is introduced, which allows to identify large numbers of new CASTs. Finally, examples of Girih CASTs with n∈4,5,7 are presented in Section 7. Except the generalized LançonBillard tiling all CASTs in this article yield local dihedral symmetry Dn or D2n.
For terms and definitions we stay close to [3,32]:
A “tile” in Rd is defined as a nonempty compact subset of Rd which is the closure of its interior.
A “tiling” in Rd is a countable set of tiles, which is a covering as well as a packing of Rd. The union of all tiles is Rd. The intersection of the interior of two different tiles is empty.
A “patch” is a finite subset of a tiling.
A tiling is called “aperiodic” if no translation maps the tiling to itself.
“Prototiles” serve as building blocks for a tiling.
Within this article the term “substitution” means, that a tile is expanded with a linear map—the “inflation multiplier”—and dissected into copies of prototiles in original size—the “substitution rule”.
A “supertile” is the result of one or more substitutions, applied to a single tile. Within this article we use the term for one substitutions only.
We use ζnk to denote the nth roots of unity so that ζnk=e2ikπnand its complex conjugate ζnk¯=e−2ikπn.
Qζn denotes the nth cyclotomic field. Please note that Qζn=Qζ2n for oddn.
The maximal real subfield of Qζn is Qζn+ζn¯.
Zζn denotes the the ring of algebraic integers in Qζn.
Zζn+ζn¯ denotes the the ring of algebraic integers (which are real numbers) in Qζn+ζn¯.
We use μn,k to denote the kth diagonal of a regular ngon with side length μn,1=μn,n−1=1.
Zμn=Zμn,1,μn,2,μn,3…μn,n/2 denotes the ring of the diagonals of a regular ngon.
2. Properties of Cyclotomic Aperiodic Substitution Tilings
We define Cyclotomic Aperiodic Substitution Tilings (CASTs) similar to the concept of Cyclotomic Model Sets as described in [3] (Chapter 7.3).
A (substitution) tilingTin the complex plane is cyclotomic if the coordinates of all vertices are algebraic integers inZζ2n, i.e., an integer sum of the2nth roots of unity. As a result all vertices of all substituted prototiles and the inflation multiplier are algebraic integers as well.
For the case that all prototilesPkof a CASTThave areas equal toAk=A(Pk)=R·sinkπn,R∈R>0,k,n∈N,0<k<nwe can use a given inflation multiplier η to calculate the substitution matrix ifη2can be written asη2=∑k=1n/2ckμn,k,ck∈N0,maxck>0),μn,k=sinkπnsinπnand the conditionsη2∉N and min((max(ck),oddk),(max(ck),evenk))≥1,(evenn>4)are met. (For simplification, all prototiles with the same area are combined.)
The inflation multiplier η of such a substitution tiling can be written as a sum of 2nth roots of unity:
η=∑k=0n−1akζ2nk(ak∈Z,max(ak)>0)
Please note that there are multiple ways to describe η. We assume ak to be chosen so that the sum is irreducible, i.e., ∑k=0n−1ak is minimal.
A substitution tiling with l≥2, l∈N prototiles and substitution rules is partially characterized by its substitution matrix M∈N0l×l with an eigenvalue λ and an left eigenvector xA.
λxAT=xATM
The elements of the left eigenvector xA contain the areas of the prototiles Ak=A(Pk). Since M∈N0l×l, the elements of xA generate a ring of algebraic integers which are real numbers.
The elements of the right eigenvector xf represent the frequencies of the prototiles fk=f(Pk), so that:
λxf=Mxf
The eigenvalue λ can be interpreted as inflation multiplier regarding the areas during a substitution. In the complex plane we can conclude:
λ=η2=η·η¯(λ∈R)
With (1) and (4) the eigenvalue λ can also be written as a sum of 2nth roots of unity. In other words, the elements of the left eigenvector xA span a ring of algebraic integers which are real numbers. This ring is isomorphic to the ring of algebraic integers Zζ2n+ζ2n¯ in Qζ2n+ζ2n¯, which is the maximal real subfield of the cyclotomic field Qζ2n.
λ=b0+∑k=1(n−1)/2bkζ2nk+ζ2nk¯(b0,bk∈Z)b0=∑k=0n−1ak2
Because of the conditions regarding ak we ensure that no combination of roots of unity in the right part of the Equation (5) can sum up to an integer which is a real number. Since we need at least two roots of unity to have an inflation multiplier η>1, it follows that
b0≥2.
The length of the kth diagonal μn,k of a regular ngon with side length 1 can be written as absolute value of a sum of 2nth roots of unity. Note that μn,1=1 because it refers to a single side of the regular ngon. An example for n=11 is shown in Figure 1.
μn,1=ζ2n0=1μn,2=ζ2n1+ζ2n−1=ζ2n1+ζ2n1¯μn,3=ζ2n−2+ζ2n0+ζ2n2=ζ2n0+ζ2n2+ζ2n2¯(n>4)μn,4=ζ2n3+ζ2n1+ζ2n−1+ζ2n−3=ζ2n1+ζ2n1¯+ζ2n3+ζ2n3¯(n>5)μn,k=∑i=0k−1ζ2n2i−k+1(n>k≥1)
This also works vice versa for ζ2nk+ζ2nk¯:
ζ2nk+ζ2nk¯=μn,k+1−μn,k−1(n/2≥k>1)
As a result, the eigenvalue λ can also be written as a sum of diagonals μn,k:
λ=∑k=1n/2ckμn,k(ck∈Z;max(ck)>0)
We recall the areas An,k of the prototiles are written as:
An,k=Rsinkπn
In most cases we will find R=1 (e.g., for tilings based on rhombs spanned by roots of unity which enclose an inner angle equal kπn) or R=12 (for tilings based on isosceles triangles which are spanned by roots of unity which enclose a vertex angle equal kπn).
We furthermore recall that the length of the kth diagonal μn,k of a regular ngon with side length 1 can also be written as:
μn,k=sinkπnsinπn
We recall the Diagonal Product Formula (DPF) as described in [33,34] with some small adaptions:
μn,1μn,k=μn,kμn,2μn,k=μn,k−1+μn,k+1(1<k≤n/2)μn,3μn,k=μn,k−2+μn,k+μn,k+2(2<k≤n/2)μn,4μn,k=μn,k−3+μn,k−1+μn,k+1+μn,k+3(3<k≤n/2)
Or, more generally:
μn,hμn,k=∑i=1hμn,k−h−1+2i(1≤h≤k≤n/2)
The other diagonals are defined by:
μn,n−k=μn,k(1≤k≤n−1)
and
sinn−kπn=sinkπn.
As a result the diagonals of a ngon span a ring of diagonals Zμn. With Equations (8)–(10) can be shown that:
Zμn=Zζ2n+ζ2n¯λ∈Zμn
Because of Equations (11)–(14), we choose the substitution matrix as M∈N0n−1×n−1 and the left eigenvector as:
xA=μn,n−1⋮μn,2μn,1=1An,1An,n−1⋮An,2An,1
With Equations (14) and (19), we can find a matrix Mn,k,n/2≥k≥1 with eigenvalue μn,k for the given left eigenvector xA:
Mn,1=E=10⋯⋯⋯⋯0010⋮⋮01⋱⋮⋮⋱⋱⋱⋮⋮⋱10⋮⋮0100⋯⋯⋯⋯01Mn,2=010⋯⋯⋯01010⋮010⋱⋱⋮⋮0⋱⋱⋱0⋮⋮⋱⋱010⋮01010⋯⋯⋯010Mn,3=0010⋯⋯00101⋱⋮101⋱⋱⋱⋮01⋱⋱⋱10⋮⋱⋱⋱101⋮⋱10100⋯⋯0100⋯
To get the substitution matrix for a given eigenvalue λ as defined in Equation (11) we just need to sum up the matrices Mn,k with the coefficients ck:
λxAT=xAT∑k=1n/2ckμn,k=xAT∑k=1n/2ckMn,k=xATMnMn=∑k=1n/2ckMn,k
We recall that a substitution matrix Mn must be primitive and real positive. To ensure the latter property its eigenvalue λ∈R must be a positive sum of elements of the left eigenvector xA. For this reason, we have to modify Equation (11) accordingly:
λ=∑k=1n/2ckμn,k(ck∈N0;max(ck)>0)
With Equation (14) it can be shown that the λ with positive coefficients ck in Equation (23) span a commutative semiring N0μn which is a subset of Zμn:
λ∈N0μn⊂Zμn
Since all Mn,k are symmetric, this is also true for Mn. In this case, left and right eigenvector of Mn are equal so that xA=xf and xA represents the frequencies of prototiles as well as their areas.
Because of μn−k=μk as noted in Equations (15) and (16), the matrices and the left eigenvector can be reduced so that Mn;Mn,k∈N0n/2×n/2 and xA∈Rn/2. In detail we omit the redundant elements of xA and adjust Mn accordingly. This also enforces n/2≥k≥1. The results of the reduction are shown in Equations (26) and (28).
For xA we can write:
xA=μn,n/2⋮μn,2μn,1=1An,1An,n/2⋮An,2An,1
For oddn the substitution matrices Mn can be described by the following scheme:
M5=c21110+c11001M7=c3111110100+c2110101010+c1100010001M9=c41111111011001000+c31110110110100100+c21100101001010010+c11000010000100001M11=c51111111110111001100010000+c41111011101110101010001000+c31110011010101010101000100+c21100010100010100010100010+c11000001000001000001000001
The scheme can be continued for any oddn. The matrices are still symmetric, so that
xf=xA(oddn)
and xA still represents the frequencies of prototiles.
For evenn, the substitution matrices Mn be described by the following scheme:
M4=c20120+c11001M6=c3101020200+c2010201010+c1100010001M8=c40101202002002000+c31010020120100100+c20100201001010010+c11000010000100001M10=c51010102020202000200020000+c40101020201020102010001000+c31010002010201010101000100+c20100020100010100010100010+c11000001000001000001000001
The scheme can be continued for any evenn, but the matrices are not symmetric anymore. With Equations (3), (20) and (28), we can derive:
xf=μn,n−12μn,n−2⋮2μn,22μn,1=1An,1An,n−12An,n−2⋮2An,22An,1(evenn)
For a given matrixMnof a CASTTas defined above the eigenvalue λ can be calculated easily with Equation (23). From Equations (26) and (28) follows that:Mn=⋯cn/2⋮⋮c2cn/2⋯c2c1
In other words, the coefficientsckcan be read directly from the bottom row or the right column of matrixMn.
We also can find a vector notation for Equation (23):λ=xATcwithc=cn/2⋮c2c1(ck∈N0;max(ck)>0)
For n∈2,3 the Mn=c1Mn,1∈N;λ∈N and xA;xF∈R>0. As a result Mn is always primitive.
For the following discussion of cases with n≥4 we define some sets of eigenvalues (and its powers) based on Equation (23):
Seven=∑k=1n/2ckμn,k∣ck∈N0;ck=0(oddk);max(ck)>0Sodd=∑k=1n/2ckμn,k∣ck∈N0;ck=0(evenk);max(ck)>0Smixed=Sodd∪SevenC∖0Sfull=∑k=1n/2ckμn,k∣ck∈N;ck>0Sfull⊂SmixedSeven∩Sodd=Sodd∩Smixed=Smixed∩Seven=∅
For any n≥4 Equations (26) and (28) show that substitution matrix Mn is strictly positive Mn≫0 and so also primitive if the corresponding eigenvalue λ∈Sfull. Substitution matrix Mn is also primitive if some finite power of an eigenvalue λp∈Sfull(p∈N).
For oddn≥5 Equations (14) and (15) show that some finite power of diagonals of a regular ngon exist so that:
∃pμn,kp∈Sfullk≥2;
It is also possible to show that some finite power of an eigenvalue λ∈N0μn∖N0 exist so that:
∃pλp∈Sfullλ∈N0μn∖N0
As a result any substitution matrix Mn in Equation (26) with an eigenvalue λ∈N0μn∖N0 is primitive. That follows that the coefficients ck have to be chosen so that:
maxck>0(oddn≥5,k≥2)
For evenn≥4 Equations (14) and (15) show that the powers of diagonals of a regular ngon have the following porperties:
μn,kp∈Soddoddkμn,kp∈Soddevenk;evenpμn,kp∈Sevenevenk;oddpμn,k+μn,hp∈Smixedevenk;oddh∃pμn,k+μn,hp∈Sfullevenk;oddh
With the same equations it is also possible to show that:
λp∈Soddλ∈Soddλp∈Soddλ∈Seven;evenpλp∈Sevenλ∈Seven;oddpλp∈Smixedλ∈Smixed∖N0
It is also possible to show that some finite power of an eigenvalue λ∈Smixed∖N0 exist so that:
∃pλp∈Sfullλ∈Smixed∖N0
As a result any substitution matrix Mn in Equation (28) with an eigenvalue λ∈Smixed∖N0 is primitive. That follows that the coefficients ck have to be chosen so that:
minmaxck,oddk,maxck,evenk≥1(evenn≥4)
Every substitution tiling defines a substitution matrix. But not for every substitution matrixMnexists a substitution tiling.
CASTsTin Theorem 1 withn≥4are aperiodic under the condition that at least one substituion rule contains a pair of copies of a prototile whose rotational orientations differs inkπn;n>k>0;k∈N.
We recall that any substitution matrix Mn of CASTs T in Theorem 1 is primitive and that every CAST T yields finite rotations.
Under this conditions [35] (Theorem 2.3) applies, so that all orientations of a prototile appear in the same frequency.
We furthermore recall areas and frequencies of prototiles of CASTs T in Theorem 1 are given by left and right eigenvector xA and xf.
With Equations (27) and (29) the ratio between the frequencies of prototiles P2 and P1 is given by:
f(P2)f(P1)=μn,22μn,1=sin2πn2∉Q(n=4)f(P2)f(P1)=μn,2μn,1=sin2πn∉Q(n≥4)
Since f(P2)f(P1)∉Q and all orientations of a prototile appear in the same frequency, no periodic configuration of prototiles P1 and P2 is possible.
As a result the CASTs in Theorem 1 with n≥4 are aperiodic. ☐
The inflation multiplier makes no statement whether a CAST meets the condition in Theorem 2 or not. As a consequence possible solutions have to be checked case by case. However, all examples within this article fulfill the condition in Theorem 2.
All prototiles of a CASTsTin Theorem 1 withn∈2,3have identical areas. For these CASTs aperiodicity has to be proven with other methods.
The smallest possible inflation multipliers for CASTsTin Theorem 1 withn≥4are given byηmin=ζ2n1+ζ2n1¯=μn,2,oddnandηmin=1+ζ2n1=μn,2+2,evenn.
With Equation (1) we can describe the minimal eigenvalues for the smallest possible inflation multipliers:
λmin=ηmin·ηmin¯=ηmin2=μn,3+1(oddn)λmin=ηmin·ηmin¯=ηmin2=μn,2+2(evenn)
From Equation (1) we can also derive:
η1>η2⇒λ1>λ2
In other words, to prove Theorem 3, it is sufficient to show that for a substitution matrix of a CAST no smaller eigenvalues than λmin exist which fulfill the Equations (7), (23), (41) and (52). We recall that Equations (41) and (52) imply that λmin∉N.
With the following inequalities
k>μn,k(n/2≥k>1)μn,k+1>μn,k(n/2≥k+1>k≥1)μn,k+μn,h>μn,k+1+μn,h−1(n/2≥k+1>h≥2)μn,(n/2−1)+μn,1>μn,n/2
We can identify all eigenvalues which are smaller than λmin and check them individually.
For oddn we identified the eigenvalues in Table 1:
None of them fulfills the conditions in Equations (7), (23) and (41) which completes the proof for the oddn case.
For evenn we identified the eigenvalues in Table 2:
None of them fulfills the conditions in Equations (7), (23) and (52) which completes the proof for the evenn case. ☐
The definition of CASTs withn≥4can be extended to all CASTs with other prototiles than in Theorem 1.
Let a CAST T* exist with l≥lmin prototiles Pk*,l≥k≥1 with a real positive, primitive substitution matrix Mn*∈N0l×l and a real positive left eigenvector xA*∈Rl whose elements represent the relative areas A(Pk*)A(P1*) of the prototiles:
xA*=1A(P1*)A(Pl*)⋮A(P3*)A(P2*)A(P1*)=A(Pl*)A(P1*)⋮A(P3*)A(P1*)A(P2*)A(P1*)1
with
A(Pl*)≥A(Pl−1*)≥⋯≥A(P2*)≥A(P1*)>0
and
A(Pl*)A(P1*)≥A(Pl−1*)A(P1*)≥⋯≥A(P2*)A(P1*)≥A(P1*)A(P1*)=1
Since T* is a CAST by definition and all coordinates of all vertices are algebraic integers in Zζ2n the areas of all pototiles are real numbers so that
A(Pk*)∈R∩Zζ2n=Zζ2n+ζ2n¯=Zμn
and so also
A(Pk*)A(P1*)∈Zμn.
As a consequence a transformation matrix T∈Zn/2×l and its inverse T−1∈Ql×n/2 exist such that:
xA*T=xATTMn*=T−1MnTλ=xATc=xA*Tc*=xATTc*
In other words, if there is a CAST T* then also a corresponding CAST T (as defined in Theorem 1) with the same eigenvalue λ (and so the same inflation multiplier η) does exist. That includes but may not be limited to all cases where the CASTs T and T* are mutually locally derivable. ☐
A similar approach is possible for the frequencies of the corresponding CASTs T and T*. As a result Theorem 2 also applies to CAST T*.
The corresponding CASTs T and T* have the same eigenvalue, so Theorem 3 also applies to CAST T*.
lminis given by the algebraic degree of λ which depends on n. In other words, λ is a solution of an irreducible polynomial with integer coefficients, and λ is of at leastlminth degree.lmincan be described by Euler’s totient function denoted by Φ, in detaillmin=Φn2,oddnandlmin=Φn,evenn.
The eigenvalue λ as noted in Equations (23) and (31) is unique iflmin=n/2. This is the case if n is a prime. If n is not a prime, λ may have more than one corresponding vector c. e.g., forn=9we can show thatμ9,4=μ9,2+μ9,1. An eigenvalueλ=μ9,4−μ9,1=μ9,2implies that the corresponding vectorc∈100−1,0010. Only the latter one is real positive. As a consequence we will considerλ∈N0μnas noted in Equation (24) if at least one real positive c exist.
The results in this chapter may apply to other substitution tilings as well, e.g., substitution tilings with dense tile orientations and individual cyclic symmetryCnas described in [36]. In detail the inflation multipliers and areas of prototiles can be described with Equations (1) and (25).
3. CASTs with Minimal Inflation Multiplier
In the this section, we discuss CASTs with minimal inflation multiplier as noted in Theorem 3.
3.1. The Odd n Case
The substitution matrix Mn,min for CASTs with minimal eigenvalue λmin=μn,3+1,oddn is given by the following scheme:
M5,min=2111M7,min=211120101M9,min=2110120110200101Mn,min=2110⋯0120⋱⋱⋮10⋱⋱⋱00⋱⋱⋱01⋮⋱⋱0200⋯0101(oddn)
The case n=5 describes the Penrose tiling with rhombs or Robinson triangles and individual dihedral symmetry D5. The cases with n>5 are more difficult and require additional prototiles. For n=7 we give an example of a CAST T* with individual dihedral symmetry D7 as shown in Figure 2 and the following properties. A(Pk) stands for the areas of a prototile Pk.
xA*=μn,3μn,2μn,3+μn,1μn,1=1A(P1)A(P3)A(P2)A(P1′)A(P1)T=101001000011M7*=1130121010010020
A further example with n=9, minimal inflation multiplier but without individual dihedral symmetry D9 has been found but is not included here.
Another CAST with n=7, individual dihedral symmetry D7 and the same minimal inflation multiplier is given in [33] (Figure 1 and Section 3, 2nd matrix) and shown in Figure 3.
CASTs with minimal inflation multiplierηmin=ζ2n1+ζ2n1¯=μn,2and individual dihedral symmetryDnexist for everyoddn≥5.
The status of Conjecture 1 is subject to further research.
Formally all preferable conditions are met. However, for large n, this type of CAST tends to be complex and the density of patches with individual dihedral symmetry Dn tends to be small.
The substitution rules of the CASTs in Section 3.1 have no dihedral symmetry. The minimal inner angle of a prototile isπn. For this reason the CASTs in Section 3.1 may have individual dihedral symmetryDnbut not D2n.
3.2. The Even n Case
The substitution matrices Mn,min for CASTs with minimal eigenvalue λA,min=μn,2+2 is given by the following scheme:
M4,min=2122M6,min=210221012M8,min=2100221001210012Mn,min=2100⋯0221⋱⋱⋮01⋱⋱⋱00⋱⋱⋱10⋮⋱⋱1210⋯0012(evenn)
The LançonBillard tiling (also known as binary tiling) [3,37,38] is a rhomb tiling with inflation multiplier η=1+ζ2n1 for the case n=5. It is possible to generalize it for all n≥4. The approach is very similar to the approach in [39]. The generalized substitution rules are shown in Figure 4.
The left eigenvector xA and substitution matrix Mn* are given by:
xA*=μn−1+μ2μn−1μn−1⋮μ1=1A(P1)A(PC)A(PR)A(Pn−1)⋮A(P1)M4*=1110011100100102002110012M6*=1110000111000010000100002101100121020001200001002M8*=111000000111000000100000010000200101100021010000012100000101200200010020000100002
Foroddn, the scheme and the substitution matrixMn*can be separated into two independent parts. We choose the part which relays on prototilesPk,n>k≥1,oddkonly. The left eigenvector is given by the areas of the rhombic prototiles with side length 1 and areaA(Pk)=sinkπn. With Equations (15) and (16) we can write:1A(P1)A(P2n/4+1)⋮A(Pn−4)A(P3)A(Pn−2)A(P1)=1A(P1)A(Pn/2)⋮A(P4)A(P3)A(P2)A(P1)=μn,n/2⋮μn,4μn,3μn,2μn,1=xA(oddn)
As a result, we can use Equation (26) to describe the substitution matrix:Mn=Mn,2+2E(oddn)
Regarding generalized LançonBillard tilings, we can confirm the results sketched in [40].
The generalized LançonBillard tiling does not contain any patches with individual dihedral symmetry Dn or D2n. However, for n=4 and n=5, other CASTs with individual symmetry D8 and D10 have been derived and submitted to [32]. An example for n=4 is mentioned in Section 6.
4. CASTs with Inflation Multiplier Equal to the Longest Diagonal of a Regular Odd <italic>n</italic>Gon
Another interesting approach to identify CASTs with preferred properties is to choose a CAST as described in Theorem 1 and a relative small inflation multiplier, in detail the longest diagonal of a regular ngon with oddn:
η=μn,n/2(oddn)
Because of Equations (4), (14) and (26), we can write:
Mn=Mn,n/22=∑i=1n/2Mn,i=nn−1⋯21n−1n−1⋯21⋮⋮⋱⋮⋮22⋯2111⋯11
For the case n=5, we have the Penrose tiling, as described in [3] (Ex. 6.1) and [2] (Figure 10.3.14). For the case n=7, we have different tilings in [32] (“Danzer’s 7fold variant”), [33] (Figure 11) and [41]. More examples for all cases n≤15 can be found in [42] (“Half Rhombs”). The matrix in Equation (80) is equivalent to those mentioned in [42] (“Half Rhombs”). However, it is much older. To the knowledge of the author, it appeared first in [43].
CASTs as described in Theorem 1 with inflation multiplierη=μn,n/2, substitution matrixMn=Mn,n/22and exactlyn/2prototiles andn/2corresponding substitution rules exist for every oddn≥5.
The substitution rules have no dihedral symmetry, so different combinations of the prototiles chirality within the substitution rules define different CASTs with identical inflation multiplier. For the cases 7≤n≤11, solutions with individual dihedral symmetry Dn have been found by trial and error. For the case n=7, see Figure 5. For the case n=11, see Figure 6. Because of the complexity of that case, just one vertex star within prototile P4 has been chosen to illustrate the individual dihedral symmetry D11.
For everyoddn≥5, the substitution rules of a CAST as described in Conjecture 2 can be modified so that the CAST yields individual dihedral symmetry Dn.
Formally all preferable conditions are met. However, for CASTs with large n as described in Conjecture 3 the density of patches with individual dihedral symmetry Dn tend to be small.
CASTs withoddnin Conjectures 2 and 3 can easily be identified by choosing the set of prototiles as set of isosceles triangles with leg length 1 and vertex angleskπn. Such isosceles triangles have areas as described in Equation (12). Because of Equation (16) we can use a similar approach as in Remark 9, in detail we only use the triangles with vertex anglekπn,oddkand a left eigenvectorxAas noted in Equation (77). As a result all inner angles of all prototiles are intgeres multiples ofπn. The approach was used in [2] (Figure 10.3.14), [3] (Ex. 6.1), [32] (“Danzer’s 7fold variant”), [42] (“Half Rhombs”) and herein, see Figure 5.
The substitution rules of the CASTs in this section have no dihedral symmetry. The minimal inner angle of a prototile isπn. For this reason the CASTs in this section may have individual dihedral symmetryDnbut not D2n.
5. Rhombic CASTs with Symmetric Edges and Substitution Rules
For many cases of CASTs with given n and rhombic prototiles, the minimal inflation multiplier can be described if some additional preconditions are met. In the following section, we will focus on cases where all edges of all substitution rules are congruent and the inner angles of the rhomb prototiles are integer multiples of πn.
“Edge” means here a segment of the boundary of a supertile including the set of tiles which are crossed by it.
Furthermore we consider only cases where all rhombs of the edge are bisected by the boundary of the supertile along one of their diagonals as shown in Figure 7.
Despite these restrictions we still have several options left:
There are two ways to place rhombs on the edge of substitution rules. We recall that the inner angles of the rhombs are integer multiples of πn. We can place all rhombs on the edge so that the inner angles either with even or odd multiples of πn are bisected by the boundary of the supertile. We will call these two cases “even” and “odd edge configuration”, for details see Figure 7. A “mixed” configuration is not allowed, because it would force the existence of rhombs with inner angle equal to k+12πn.
We can choose the symmetry of the substitution rules and their edges. Possible choices are dihedral symmetry D1 and D2. Edges with dihedral symmetry D1 can have the boundary of the supertile or its perpendicular bisector as line of symmetry. The smallest nontrivial solution for the latter case is the generalized GoodmanStrauss tiling [39]. Since this example does not provide individual dihedral symmetry Dn or D2n in general, we will focus on the other case.
Substitution rules of rhombs which appear on the edge of a substitution rule are forced to have the appropriate dihedral symmetry D1 as well. This is also true for substitution rules of prototiles which lie on the diagonal, i.e., a line of symmetry of a substitution rule. The orientations of the edges have to be considered as well. These three conditions may force the introduction of additional rhomb prototiles and substitution rules. Additionally, the existence of edges with orientations may require additional preconditions.
To avoid this problem, a general dihedral symmetry D2 can be chosen for the substitution rules and their edges.
Parity of the chosen n may require different approaches in some cases, similar to the example of the generalized LançonBillard tiling in Section 3.2 and Figure 4.
The options above can be combined without restrictions, so in total we have eight cases as shown in Table 3.
Within this section, we will denote a rhomb with side length d0≡1 and an inner angle kπn, n>k>0 which is bisected by the boundary of a supertile as Rk and its diagonal which lies on the boundary of a supertile as dk. The line segments which appear on the edges of substitution rules in the cases with even edge configuration are denoted by R0 and their length by d0. Under those conditions, we can write the inflation multiplier η as sum of the dk:
η=∑i=0(n−1)/2α2id2i(αk∈N0,cases1and3)η=∑i=0n/2−1α2i+1d2i+1(αk∈N0,cases2and4)
The inflation multiplier η can also be written as a sum of 2nth roots of unity, where the roots occur in pairs ζ2ni+ζ2ni¯ so that η∈R:
η=α0+∑i=0(n−1)/2α2iζ2ni+ζ2ni¯(αk∈N0,η∈R,cases1and3)η=2+ζ2n1+ζ2n1¯α1+∑i=0n/2−1α2i+1ζ2ni+ζ2ni¯(αk∈N0,η∈R,cases2and4)
We borrow a remark regarding worms from [44]: “By definition, a parallelogram has two pairs of parallel edges. In a parallelogram tiling therefore there are natural lines of tiles linked each sharing a parallel edge with the next. These lines are called worms and were used by Conway in studying the Penrose tilings [2]. We will follow Conway to call these lines worms.”
Every rhomb Rk with an inner angle kπn, n>k>0 which is bisected by the boundary of a supertile work as entryexitnode of two worms. We also can say that such a rhomb on the edge “reflects” a worm back into the substitution rule. The line segments R0 work as entryexitnode of one worm only.
The KannanSorokerKenyon (KSK) criterion as defined in [45,46] decides whether a polygon can be tiled by parallelograms. We use a simplified phrasing for the criterion: All line segments R0 and all inner line segments of rhombs Rk, n>k>0 on the edge of the substitution rule serve as nodes. All corresponding nodes of the substitution rules are connected by lines. The KSK criterion is fulfilled if for every intersection of lines the inner angles between the corresponding nodes are larger than 0 and smaller than π.
We start with observations of the substitution rule for rhomb prototile R1 or Rn−1. In detail we will focus on those both edges which enclose the inner angle equal πn and which we will denote as “corresponding edges”. We will denote the area where the both edges meet as the “tip” of the substitution rule.
The rhombs on the edges must not overlap. For this reason, for the tip of the substitution rule, only three configurations are possible as shown in Figure 8. Obviously, a tip as shown in Figure 8c is compliant to the cases 2 and 4 with odd edge configuration and Figure 8b to cases 1 and 3 with even edge configuration. Figure 8a requires the even edge configuration as well. Since all edges are congruent, it must be the start and the end of the same edge, which meet on that vertex. Since start and end of the edge are different, it can not have dihedral symmetry D2. For this reason, the tip in Figure 8a is not compliant to case 3.
Any rhomb Rk with n−3≥k≥3 on one edge implies the existence of a rhomb Rk−2 on the corresponding edge. In turn, rhomb Rk on one edge is implied by a rhomb Rk+2 on the corresponding edge or a rhomb Rk on the opposite edge. An example is shown in Figure 9.
Any rhomb R2 on one edge implies the existence of a line segment R0 on the corresponding edge. In turn, rhomb R2 on one edge is implied by a rhomb R4 on the corresponding edge or a rhomb R2 on the opposite edge.
Any rhomb R1 on one edge implies the existence of a rhomb R1 on the corresponding edge. In turn, rhomb R1 on one edge is implied by a rhomb R3 on the corresponding edge or a rhomb R1 on the opposite edge.
Any line segment R0 on one edge is implied by a rhomb R2 on the corresponding edge or a line segment R0 on the opposite edge.
Any rhomb Rn−2 on one edge implies the existence of a rhomb Rn−4 on the corresponding edge. In turn, rhomb Rn−2 on one edge is implied by a rhomb Rn−2 on the opposite edge. (Rhomb Rn does not exist, because the inner angle would be zero.)
Any rhomb Rn−1 on one edge implies the existence of a rhomb Rn−3 on the corresponding edge. In turn, rhomb Rn−1 on one edge is implied by a rhomb Rn−1 on the opposite edge. (Rhomb Rn+1 does not exist, it would have an inner angle greater than π or smaller the 0.)
If for a rhomb Rk,n>k>0 on one edge two related elements (rhomb or line segment) Rk−2 and Rk+2 exist on the corresponding edge, Rk−2 is closer to the tip than Rk+2.
Since all edges are congruent we can derive the following inequalities for αk:
α0≥α2≥α4≥…α2(n−1)/2≥0(αk∈N0,cases1and3)α1≥α3≥α5≥…α2n/2−1≥0(αk∈N0,cases2and4)
Because of Equations (8) and (9), the inflation multiplier can also be written as a sum of diagonals μn,k:
η=∑k=1n/2βkμn,k(βk∈N0,cases1and3)η=μn,2+2∑k=1n/2βkμn,k(βk∈N0,cases2and4)
Equations (81)–(84) give no hint on which αk=0, so that a rhombic CAST still can exist. For this reason, we have to extend our observations to the substitution rules of rhomb Rn/2.
For case 1b and 3b:
Any line segment R0 on the edge implies the existence of a rhomb Rn−1 on the correspondent edge or a line segment R0 on the opposite edge. As shown in Figure 10a,b, the existence of rhomb Rn−1 on the edge is not required to meet the KSK criterion.
Any rhomb R2 on the edge implies the existence of a rhomb Rn−3 on the correspondent edge or a rhomb R2 on the opposite edge. As shown in Figure 10c,d, the KSK criterion is only met if at least one Rn−3 exists on the edge.
αn−3>αn−1≥0(cases1band3b)
For case 1a and 3a:
The line segment R0 on the edge implies the existence of a line segment R0 on the opposite edge only (rhomb Rn does not exist).
Any rhomb R2 on the edge implies the existence of a rhomb Rn−2 on the correspondent edge or a rhomb R2 on the opposite edge. So the KSK criterion is only met if at least one Rn−2 exists on the edge.
αn−2≥1(cases1aand3a)
For case 2b and 4b:
Any rhomb R1 on the edge implies the existence of a rhomb Rn−2 on the correspondent edge or a rhomb R1 on the opposite edge. So the KSK criterion is only met if at least one Rn−2 exists on the edge.
αn−2≥1(cases2band4b)
For case 2a and 4a:
Any rhomb R1 on the edge implies the existence of a rhomb Rn−1 on the correspondent edge or a rhomb R1 on the opposite edge. So the KSK criterion is only met if at least one Rn−1 exists on the edge.
αn−1≥1(cases2aand4a)
In the next part we will determine the minimal inflation multiplier ηmin of a rhombic CAST. We will discuss only case 1 in detail to show the general concept.
We continue our observations of the substitution rule R1 or Rn−1. There are four possible combinations of the corresponding edge orientations as shown in Figure 11.
We recall Equations (85) and (89). To get a small inflation multiplier, the αk with evenk have to be chosen as small as possible. We use the KSK criterion to check whether substitution rules can exist for different orientations of the corresponding edges and equal amounts of rhombs Rk and Rk+2 so that αk=αk+2. The checks are illustrated in Figure 12. Actually, the figure just shows the case αk=αk+2=1, while the result is true for αk=αk+2>1 as well.
The results for α0=α2 in Figure 12a–c show that the KSK criterion is fulfilled for more than one combination.
The results for αk=αk+2;k>0 in Figure 12d–f show that the KSK criterion is fulfilled for one orientation only.
At this point we have to introduce an additional condition. For cases 1 and 2, we consider only rhombic CASTs whose prototiles R1 or Rn−1 yield at least two different orientations of the corresponding edges.
The author is only aware of one example of a rhombic CAST whose prototilesR1orRn−1show only one orientation of the corresponding edges, namely the AmmannBeenker tiling withn=4. Forn>4, the existence of such rhombic CASTs seem to be unlikely. However, a proof to rule out this possibility is not available yet.
With the results and conditions above we can conclude:
α0≥α2(case1)α2k>α2k+2;k>0(case1)
With Equations (9), (83), (85) and (87), we can describe the minimal inflation multiplier:
ηmin=∑i=1n/2−1μn,i+μn,i+1(case1)
For case 1b with oddn, we can use the same concept of orientations as shown in [33] (Figure 2) which leads to rhombs with orientations as shown in Figure 13. As a result, the orientations of the edges are globally defined. In detail, the edges which enclose an inner angle πn have orientations as shown in Figure 11c,d and a tip as shown in Figure 8a.
For case 1a with evenn this simplification can not be used. The edges which enclose an inner angle πn have not only orientations as shown in Figure 11c,d. Depending on how exactly the orientations are defined one of the orientations in Figure 11a or Figure 11b may appear.
Similar approaches are possible for cases 2, 3 and 4. For cases 3 and 4 the symmetry conditions of the substitution rules and its edges enforce αkeven,k>1, just α0 is not required to be even.
With the minimal inflation multiplier ηmin as shown in Table 4, Table 5, Table 6 and Table 7, we can start the search of rhombic CASTs by trial and error. The minimal rhomb edge sequences therein were also obtained in this way, so additional solutions with varied sequences obtained by permutation may exist. Examples for all eight cases are shown in Figure 14, Figure 15, Figure 16 and Figure 17:
For all cases as listed in Table 3, there exist rhombic CASTs for any n with edge configuration and inflation multiplierηminas shown in Table 4, Table 5, Table 6 and Table 7 which yield individual dihedral symmetryDnor D2n.
A general proof of Conjecture 4 is subject to further research. However, a proof for case 4b already exists. The results in [47] for oddn match very well with our results in case 4b. The case evenn in the same publication is almost identical to our case 3a, up to a difference of one line segment R0 in the center of the edge of the substitution rules.
Case 1a withn=4is equivalent to the AmmannBeenker tiling [10,32,48,49].
Socolar’s 7fold tiling [32] (credited to J. Socolar) can be derived from case 1b withn=7with a modified minimal rhomb edge sequence.
Two substitution steps of the Penrose rhomb tiling are equivalent to one substitution step in case 1b withn=5.
6. Gaps to Prototiles Algorithm
In this section, we sketch a “Gaps to Prototiles” algorithm to identify CASTs for a given n and a selected edge of a substitution rule (as in Definition 2).
Conditions:
All prototiles have inner angles equal kπn.
All edges of all substitution rules are congruent and have dihedral symmetry D2.
As discussed in Section 5 the tiles on the edge have to be placed, so that the inner angles either with even or odd multiples of πn are bisected by the boundary of the supertile.
The tiles on the edge are bisected by one or two lines of symmetry of the edge. This implies dihedral symmetry D1 or D2 of the corresponding substitution rules.
The inflation multiplier η must fulfill the conditions in Theorem 1.
The inflation multiplier η is defined by the sequence of tiles which are part of the edge.
Algorithm:
We start with the prototiles which appear on the edge of the substitution rule.
We start the construction of the substitution rules by placing the prototiles on the edge.
If the edge prototiles overlap the algorithm has failed. In this case, we may adjust the sequence of rhombs or other equilateral polygons on the edge and start another attempt.
We try to “fill up” the substitution rules with existing prototiles under consideration of the appropriate dihedral symmetry D1 or D2. If gaps remain, they are defined as new prototiles and we go back to step (2). Please note, if a gap lies on one or two lines of symmetry, the substitution rule of the new prototile must also have the appropriate dihedral symmetry D1 or D2.
If no gaps remain the algorithm was successful.
In some cases the “Gaps to Prototiles” algorithm delivers results with preferable properties as shown in Figure 18, Figure 19 and Figure 20 and inflation multipliers as shown in Table 8. However, some results contain a very large number of prototiles with different sizes and complex shapes as in Figure 19. It is not known yet whether the algorithm always delivers solutions with a finite number of prototiles.
7. Extended Girih CASTs
“Girih” is the Persian word for “knot” and stands for complex interlaced strap works of lines, which are a typical feature of Islamic architecture and design. A common definition is given in [50]: “Geometric (often starandpolygon) designs composed upon or generated from arrays of points from which construction lines radiate and at which they intersect.” The oldest known examples of star pattern date back to the 8th century AD [51]. Girih designs are known in many styles and symmetries, see [52] for examples. A variant of Girih design relies on Girih tiles and tilings as shown in the reproduction of the Topkapi Scroll in [53]. The decorations on the tiles consist of lines which run from tile to tile when they are joined together. So the borders between joined tiles seem to disappear.
According to [53] the shapes of all girih tiles are equilateral polygons with the same side length and inner angles kπ5, k∈{2,3,4,6} including:
Regular decagon with inner angles 4π5
Regular pentagon with inner angles 3π5
Rhomb with inner angles 2π5 and 3π5
Convex hexagon with inner angles 2π5,4π5,4π5,2π5,4π5,4π5
Convex hexagon with inner angles 3π5,3π5,4π5,3π5,3π5,4π5
Because of these properties, Girih tilings are cyclotomic tilings as well.
A Girih (cyclotomic) aperiodic substitution tiling was derived from a mosaic at the Darbi Imam Shrine, Isfahan, Iran which dates back from 1453. It relies on the regular decagon and two hexagons and has individual dihedral symmetry D10. It was published in [54]. However, the complete set of substitution rules can be found in [55,56]. Additional examples of Girih CASTs have been discovered and submitted to [32]. More examples have been published in [56,57,58].
That rises the question if Girih CASTs with other symmetries and relative small inflation multiplier exist. For this reasons we have to define the extended Girih CASTs. The following definition turned out to be useful for a given n≥4.
All prototiles of an extended Girih CAST are equilateral polygons with the same side length.
The inner angles of all prototiles are kπn, k∈2,3…n−1,n+1,n+2…n−2.
One of the prototiles may be a regular ngon with inner angles n−2πn.
One of the prototiles may be a regular 2ngon with inner angles n−1πn.
Please note that prototiles with inner angle πn are forbidden due to aesthetic reasons.
For the extended Girih CASTs in this section we choose the following properties:
All edges of the substitution rules are congruent and have dihedral symmetry D2.
All substitution rules except those for regular ngons with oddn have dihedral symmetry D2.
The substitution rule of the regular ngon with oddn has dihedral symmetry D1.
The substitution rule of the regular 2ngon has dihedral symmetry D2n for oddn and Dn for evenn.
Examples for extended Girih CASTs in this section with n∈4,5,7 are shown in Figure 21, Figure 22, Figure 23 and Figure 24 and its corresponding eigenvalues are shown in Table 9.
The Girih CASTs in Figure 21, Figure 22 and Figure 24 have been obtained by a trial and error method under the following conditions:
In every corner of every substitution rule a regular 2ngon is placed.
Edge and inflation multiplier have been derived from a periodic pattern of regular 2ngons and their inter space counterparts.
For n≥8 this approach might fail. As an alternative it is possible to reuse inflation multipliers from rhombic CASTs in Section 5, case 1 and 2. See Table 4 and Table 5 for details.
The Girih CAST in Figure 23 has been derived from [56] (Figures 14 and 15). In detail the nonconvex hexagons were replaced and the star shaped gap in the center of the substitution rule of the pentagon prototile [56] (Figure 15b) was eliminated. This was possible by changing the symmetry of the substitution rule from dihedral symmetry D5 to D1. The tilings in [56] (Figures 14 and 15) were derived by an analysis of patterns shown in the Topkapi Scroll, in detail [53] (Panels 28, 31, 32, 34).
Please note that the caseevennrequires special care to make sure that the regular2ngons with dihedral symmetryDnalways match. It seems that the existence of a substitution rule of the regular2ngons with dihedral symmetryD2nrequires the existence of prototiles with inner angleπnwhich are forbidden due to our preconditions. As a result, additional substitution rules might be necessary for prototiles with the same shape but different orientations.
The decorations at the prototiles are related but not necessarily equivalent to Ammann bars.
8. Summary and Outlook
Although the motivation behind this article was mainly aesthetic, some significant results have been achieved. Cyclotomic Aperiodic Substitution Tilings (CASTs) cover a large number of new and well known aperiodic substitution tilings as shown in Table 10. The properties of CASTs, in detail their substitution matrices and their minimal inflation multipliers, can be used as practical starting point to identify previously unknown solutions. For many cases, such solutions yield individual dihedral symmetry Dn or D2n.
The different approaches to identify CASTs have their individual advantages and disadvantages. The preferable properties as listed in the introduction section may be complemented by a high frequency of patches with dihedral symmetry. A promising approach we do not discuss in this article may be the choice of inflation multipliers which are PV numbers.
The results in this paper focus strictly on the Euclidean plane so that Equation (4) applies. However, the methods described herein might be adapted for other cases as well.
Finally, several conjectures have been made, which require further research.
Acknowledgments
The author dedicates this paper to his daughter Xue Lili and his parents Marita and Herbert. He would like to thank Michael Baake (Bielefeld University), Dirk Frettlöh (Bielefeld University), Uwe Grimm (The Open University, Milton Keynes), Reinhard Lück and Christian Georg Mayr (Technische Universität Dresden) for their support and encouragement. The author is aware that this article might not meet everyone’s standards and expectations regarding a mathematical scientific paper. He kindly asks for the readers indulgence and hopes that the content and the sketched ideas herein are helpful for further research despite possible formal issues. The recent publications of G. Maloney [8], J. Kari and M. Rissanen [47] and T. Hibma [40,42], who found similar results, demanded a response on short notice.
Conflicts of Interest
The author declares no conflict of interest.
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The diagonals μ11,k of a regular hendecagon with side length 1 can be written as a sum of 22nd roots of unity as described in Equation (9).
Cyclotomic Aperiodic Substitution Tilings (CAST) for the case n=7 with minimal inflation multiplier. The black tips of the prototiles mark their respective chirality.
CAST for the case n=7 with minimal inflation multiplier as described in [33] (Figure 1 and Section 3, 2nd matrix).
Generalized LançonBillard tiling.
CAST for the case n=7 with inflation multiplier μ7,3.
CAST for the case n=11 with inflation multiplier μ11,5. One vertex star within prototile P4 has been chosen to illustrate the individual dihedral symmetry D11.
The ”Edges” of a substitution rules are defined as the boundaries of the supertile (dashed line) and the rhombs bisected by it along one of their diagonals. The figure illustrates how the rhombs can be placed accordingly for even edge configuration (a); and odd edge configuration (b). The inner angles of the rhombs are integer multiples of πn and are denoted by small numbers near the tips (Example n=7).
Substitution rule for rhomb R1 or Rn−1: The three possible configurations of the tip are shown in (a), (b) and (c) (Example n=7).
Substitution rule for rhomb R1 or Rn−1: Rhomb Rk on the edge and its relatives at the corresponding edge (black) and the opposite edge (blue) (Example n=7, k=4).
Substitution rule for rhomb R(n±1)/2: The KannanSorokerKenyon (KSK) criterion requires the existence of rhomb Rn−3 on the edge as shown in (c) and (d). However, this is not the case for rhomb Rn−1 as shown in (a) and (b) (Case 1b, example n=7).
Substitution rule for rhomb R1 or Rn−1: The possible configurations of the edges orientations are shown in (a), (b), (c) and (d) (Example n=7).
Substitution rule for rhomb R1 or Rn−1: KSK criterion for corresponding edges with different orientations with α0=α2 are shown in (a), (b), (c) and α2k=α2k+2;k>1 are shown in (d), (e) and (f).
Orientations of edges (and orientations of lines of symmetry) for rhomb prototiles in case 1b (Example n=11).
Rhombic CAST examples for case 1a (n=6) and case 1b (n=7).
Rhombic CAST examples for case 2a (n=6) and case 2b (n=7). The shown example for case 2b was slightly modified to reduce the number of prototiles to n/2 as in case 1b. In detail, the edges of the rhomb prototiles have orientation as shown in Figure 13.
Rhombic CAST examples for case 3a (n=6) and case 3b (n=7).
Rhombic CAST examples for case 4a (n=6) and case 4b (n=7).
CAST for the case n=7.
CAST for the case n=7, derived from the GoodmanStrauss tiling in [39].
CAST for the case n=4, derived from the generalized LançonBillard tiling in Figure 4.
Extended Girih CAST for the case n=4.
Girih CAST for the case n=5.
Another Girih CAST for the case n=5, derived from [56] (Figures 14 and 15) and patterns shown in the Topkapi Scroll, in detail [53] (Panels 28, 31, 32, 34).
Extended Girih CAST for the case n=7.
symmetry0900019t001_Table 1
Eigenvalues to be checked for the oddn case.
Sum of Diagonals
Sum of Roots of Unity
Conditions for n
μn,5
1+ζ2n2+ζ2n2¯+ζ2n4+ζ2n4¯
n=11
μn,4
ζ2n1+ζ2n1¯+ζ2n3+ζ2n3¯
n≥9
μn,3
1+ζ2n2+ζ2n2¯
n≥7
μn,2
ζ2n1+ζ2n1¯
n≥5
μn,2+1
1+ζ2n1+ζ2n1¯

3
3
n≥7
2
2

1
1

symmetry0900019t002_Table 2
Eigenvalues to be checked for the evenn case.
Sum of Diagonals
Sum of Roots of Unity
Conditions for n
μn,6
ζ2n1+ζ2n1¯+ζ2n3+ζ2n3¯+ζ2n5+ζ2n5¯
n=12
μn,5
1+ζ2n2+ζ2n2¯+ζ2n4+ζ2n4¯
n∈{10,12}
μn,4
ζ2n1+ζ2n1¯+ζ2n3+ζ2n3¯
n≥8
μn,3
1+ζ2n2+ζ2n2¯
n≥6
μn,2
ζ2n1+ζ2n1¯
n≥4
μn,4+1
1+ζ2n1+ζ2n1¯+ζ2n3+ζ2n3¯
n=8
μn,3+1
2+ζ2n1+ζ2n1¯
n≥6
μn,2+1
1+ζ2n1+ζ2n1¯

3
3

2
2

1
1

symmetry0900019t003_Table 3
Definition of cases of rhombic CASTs.
Substitution Rules and Their Edges Have at Least Dihedral Symmetry D_{1}
Substitution Rules and Their Edges Have Dihedral Symmetry D_{2}
Even edge configuration
Case 1a (evenn)Case 1b (oddn)
Case 3a (evenn)Case 3b (oddn)
Odd edge configuration
Case 2a (evenn)Case 2b (oddn)
Case 4a (evenn)Case 4b (oddn)
symmetry0900019t004_Table 4
Rhombic CAST substitution rule edge configuration for case 1.
n
Minimal Rhomb Edge Sequence
Minimal Inflation Multiplier η_{min}
4,5
0−2¯
μn,2+1
6,7
0−2¯−4−0−2¯
μn,3+2μn,2+1
8,9
0−2¯−4−0−2¯−6−4−0−2¯
μn,4+2μn,3+2μn,2+1
10,11
0−2¯−4−6−8−0−2¯−4−0−2¯−6−4−0−2¯
μn,5+2μn,4+2μn,3+2μn,2+1
…
…
…
symmetry0900019t005_Table 5
Rhombic CAST substitution rule edge configuration for case 2.
n
Minimal Rhomb Edge Sequence
Minimal Inflation Multiplier η_{min}
4,5
1−3−1
μn,2+2μn,2+2
6,7
1−3−1−5−3−1
μn,2+2μn,3+2μn,2+2
8,9
1−3−5−1−3−7−1−5−3−1
μn,2+2μn,4+2μn,3+2μn,2+2
10,11
1−3−5−7−1−3−1−5−9−3−1−7−5−3−1
μn,2+2μn,5+2μn,4+2μn,3+2μn,2+2
…
…
…
symmetry0900019t006_Table 6
Rhombic CAST substitution rule edge configuration for case 3.
n
Minimal Rhomb Edge Sequence
Minimal Inflation Multiplier η_{min}
4,5
0−2−0−2−0
2μn,2+3
6,7
0−2−4−0−2−0−2−0−4−2−0
2μn,3+4μn,2+3
8,9
0−2−4−6−0−2−4−0−2−0−2−0−4−2−0−6−4−2−0
2μn,4+4μn,3+4μn,2+3
…
…
…
symmetry0900019t007_Table 7
Rhombic CAST substitution rule edge configuration for case 4.
n
Minimal Rhomb Edge Sequence
Minimal Inflation Multiplier η_{min}
4,5
1−3−1−1−3−1
μn,2+22μn,2+4
6,7
1−3−5−1−3−1−1−3−1−5−3−1
μn,2+22μn,3+4μn,2+4
8,9
1−3−5−7−1−3−5−1−3−1−1−3−1−5−3−1−7−5−3−1
μn,2+22μn,4+4μn,3+4μn,2+4
…
…
…
symmetry0900019t008_Table 8
Inflation multipliers of CASTs identified by the “Gaps to Prototiles” algorithm.
n
Inflation Multiplier
Corresponding Figure
7
2μ7,3+2μ7,2+1
Figure 18
7
μ7,2+2
Figure 19
4
μ4,2+2
Figure 20
symmetry0900019t009_Table 9
Inflation multipliers of extended Girih CASTs.
n
Inflation Multiplier
Corresponding Figure
4
μ4,2+1
Figure 21
5
μ5,2+2(μ5,2+1)
Figure 22
5
2(μ5,2+1)
Figure 23
7
μ7,2+2μ7,1+2
Figure 24
symmetry0900019t010_Table 10
Inflation multipliers and individual symmetry of some CASTs.