#
Linear Recurrent Double Sequences with Constant Border in M_{2}(F_{2}) are Classified According to Their Geometric Content

## Abstract

**:**

_{2}($\mathbb{F}$

_{2}); geometric content

## 1. Introduction

**Definition**

**1.1**

**Definition**

**1.2**

**Definition**

**1.3**

**Definition**

**1.4**

**Definition**

**1.5**

**Lemma**

**1.6**

**Proof:**

- A recurrent double sequence remains constant if and only if it satisfies the relation $f({x}_{0},{x}_{0},{x}_{0})={x}_{0}$ for ${x}_{0}=(1,0)$. Exactly 1024 sequences, i.e., a quarter of 4096, are constant.
- All homomorphic recurrent double sequences over $\mathbb{K}$ can be also generated by systems of substitutions. Moreover, most of them can be easily associated a geometric content, and there are only 90 such contents occurring in the 4096 homomorphisms.
- Recurrent double sequences produced by homomorphisms lying in the same class of conjugation (tend to) have the same geometric content. This fact is at the first sight a surprise, and was not foreseen by Lemma 1.6. Moreover there are two different kinds of exceptions:
- A
- A normal exception is the situation when two homomorphisms in a class of conjugation of six homomorphisms or one homomorphism in a class of conjugation of three homomorphisms produce a constant recurrent double sequence, although the other homomorphisms in the class of conjugation remain in the same (non-uniform) type.
- B
- A sporadic exception is the situation in which some recurrent double sequences with different geometric content are produced by conjugated homomorphisms. Only some few classes of conjugation (only 33 classes in 736) split in different geometric contents. Pairs of contents occurring in sporadic exceptions are from sets that were called later type 1 and type 2, type 2 and type 19, type 19 and 74, type 2 and type 74. See the Classification for the definition of those geometric types.

**Definition**

**1.7**

- A linear recurrent double sequence over ${M}_{2}\left({\mathbb{F}}_{2}\right)$ remains constant if and only if it satisfies the relation $f(I,I,I)=I$. Exactly 256 sequences, i.e., $1/16$ of 4096, are constant. Moreover, the set of matrix triples producing constant sequences is a union of classes of conjugation of triples, because the given condition is compatible with the conjugation. Otherwise said, the normal exception noticed in $\mathbb{K}$ for homomorphic double sequences does not take place anymore.
- All linear recurrent double sequences over $\mathbb{K}$ can be also generated by systems of substitutions. Moreover, most of them can be easily associated with a geometric content, and there are only 90 such contents occurring in the 4096 homomorphisms. The geometric contents are very similar with those met in the homomorphic double sequences of $\mathbb{K}$. More on this is discussed in the third section.
- Recurrent double sequences produced by matrix triples lying in the same class of conjugation
**always have**the same geometric content. More exactly, we will prove in the next section that homomorphisms lying in the same class of conjugation produce isomorphic recurrent double sequences over ${M}_{2}\left({\mathbb{F}}_{2}\right)$. Consequently, the fact that the geometric types are unions of classes of conjugation is no more surprising. Both the normal and the sporadic exceptions noticed in homomorphic double sequences over $\mathbb{K}$ completely disappear in ${M}_{2}\left({\mathbb{F}}_{2}\right)$. The sporadic exception was produced by the fact that the projection ${\pi}_{1}$ is sometimes bad and can map neighboring different elements from a double sequence onto neighboring equal elements in the other one.

## 2. Linear Recurrent Double Sequences Over Rings

**Definition**

**2.1**

**Definition**

**2.2**

**Lemma**

**2.3**

**Definition**

**2.4**

**Corollary**

**2.5**

- 1.
- Let $y\in R$ be an element. Then the recurrent double sequence defined by $(ABC,y)$ is the projection of the recurrent double sequence $(ABC,1)$ realized by the application $x\u21ddxy$. In particular, the recurrent double sequence defined by $(ABC,y)$ lives in the principal ideal $Ry$.
- 2.
- Let $\phi ,\psi \in {R}^{\times}$ be units. Then the linear double sequences defined by $(ABC,1)$, $(ABC,\phi )$ and $({\left[ABC\right]}^{\phi},1)$ are isomorphic. They are also isomorphic with the linear double sequence defined by $({\left[ABC\right]}^{\phi},\psi )$.
- 3.
- All recurrent double sequences $(DEF,1)$ produced by triples $DEF$ in some class of conjugation $\widehat{ABC}$ are isomorphic.

**Proof:**

## 3. Substitution

**Definition**

**3.1**

**Definition**

**3.2**

**Definition**

**3.3**

**Notation**: All matrix representations occurring in this section are indexed starting with 0.

**Definition**

**3.4**

- In other words: in every x-position of Y occurs an element of $\mathcal{X}$.
- The last symbol $\mathsf{\Sigma}$ denotes a function $\mathsf{\Sigma}:\mathcal{X}\to \mathcal{Y}$ called substitution. The natural number s must be $\ge 2$ and is called expansion factor.
- The elements of $\mathsf{\Sigma}$ seen as (finite) subset of $\mathcal{X}\times \mathcal{Y}$ are called substitution rules.
- The cardinality r of $\mathcal{X}$ is the number of substitution rules.

**Definition**

**3.5**

**Definition**

**3.6**

**Lemma**

**3.7**

**Definition**

**3.8**

**Definition**

**3.9**

**Definition**

**3.10**

**Definition**

**3.11**

**Theorem**

**3.12**

- 1.
- $\forall \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}i\ge 0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}b(i,0)=\overrightarrow{\lambda}(i\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}n)$.
- 2.
- $\forall \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}j\ge 0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}b(0,j)=\overrightarrow{\mu}(j\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}m)$.
- 3.
- There exists $Q\in \mathbb{N}$ such that $R\left(Q\right)=S\left(Q\right)$ and ${\mathcal{N}}_{x}\left(S\left(Q\right)\right)={\mathcal{N}}_{x}\left(S(Q-1)\right)$.

**Notation**: All systems of substitution used in this article are expansive systems of (context-free) substitutions. As a shorthand we call them systems of substitution, or only substitutions. A double sequence produced by such systems is said to be of substitution.

**Lemma**

**3.13**

## 4. Skeleton

**Definition**

**4.1**

**Definition**

**4.2**

**Definition**

**4.3**

**Definition**

**4.4**

- Let $s\left(i\right):=a(i,0)$. There exists a system of word substitutions generating the sequence s.
- Let $t\left(j\right):=a(0,j)$. There exists a system of word substitutions generating the sequence t.
- For some $f:{A}^{3}\to A$, $a(i,j)=f\left(a\right(i,j-1),a(i-1,j-1),a(i-1,j\left)\right)$ for all $i,j\ge 1$.

**Lemma**

**4.5**

**Proof:**

**Definition**

**4.6**

## 5. Convergence and Geometric Content

**Definition**

**5.1**

**Definition**

**5.2**

- there exist an $m\times m$ matrix M occurring in m-position in a,
- for all i, ${U}_{i}$ is a connected union of m-translations of M,
- for all i, $\mathsf{\Sigma}\left({U}_{i}\right)\subseteq {U}_{i+1}$.

**Definition**

**5.3**

**Definition**

**5.4**

**Lemma**

**5.5**

**Proof:**

**Definition**

**5.6**

**Definition**

**5.7**

**Lemma**

**5.8**

**Proof:**

**Examples**: Now we show some examples concerning the variability inside a geometric type. The images used to figure left upper minors of recurrent double sequences are pixel-wise computed using a fixed correspondence between the finite set A and a set of colors. For $A={M}_{2}\left({\mathbb{F}}_{2}\right)$ we have fixed 16 colors, with o represented by white. We do not find important to give the complete list of colors, but we recall that the borders are constant and equal with I.

## 6. Primitive Double Sequences

**Definition**

**6.1**

**Definition**

**6.2**

**convention**: the multiple ${\mathsf{\Sigma}}^{\prime}$ contains in its description only those $kx\times kx$ minors that really occur in a.

**Lemma**

**6.3**

**Proof:**

**Definition**

**6.4**

**Lemma**

**6.5**

**Proof:**

**Lemma**

**6.6**

**Proof:**

**Theorem**

**6.7**

## 7. Classification

**Theorem**

**7.1**

- Two linear recurrent double sequences over ${M}_{2}\left({\mathbb{F}}_{2}\right)$ with constant border $=I$ are isomorphic if and only if they belong to the same class (mirrored class of conjugation) or are both constant.
- There exist non-isomorphic recurrent double sequences having the same skeleton.
- There exist recurrent double sequences in same geometric type, whose skeletons have the same substitution type $x\to sx$ and the same number of rules, but who have however different skeletons.
- Many of the 90 geometric types contain double sequences with different types of substitution.

**Conventions and notations**: The notation $Pn$ for the geometric types consist of a prefix P and a number n.

- A, if $R\left(S\right)\in \{\varnothing ,\alpha \beta ,\alpha \gamma ,\alpha \beta \cup \alpha \gamma ,\alpha \beta \gamma \delta \}$. (51 types)
- B, if $R\left(S\right)\in \{\alpha \u03f5\gamma \delta ,\alpha \eta \gamma \beta \}$. (17 types)
- C, if $R\left(S\right)=\alpha \u03f5\gamma \eta $. (4 types)
- D, if $R\left(S\right)\in \{\alpha \beta \gamma ,\alpha \delta \gamma \}$. (11 types)
- E, if $R\left(S\right)\in \{\alpha \u03f5,\alpha \eta ,\alpha \beta \u03f5,\alpha \delta \eta \}$. (4 types)
- F, if $R\left(S\right)\in \{\alpha \gamma \eta ,\alpha \gamma \u03f5\}$. (3 types)

- We denote with ${d}_{1}$ the symmetry around the diagonal $\alpha \gamma $, with ${d}_{2}$ the symmetry around the diagonal $\beta \delta $ and with m the median symmetry. When we say that a geometric type fulfills some symmetry, we mean that the geometric content, as Hausdorff limit, does it.
- We denote by t the fact that a half limit is isometric with the other half by translation. The absence of symmetry is abbreviated $ns$.
- We describe a geometric type in the following way:
**number name**, symmetry, number of elements, number of classes. - This line is always followed by the list of skeletons occurring in the type, each of them given together with the list of its own classes.
- If there are different skeletons in the same geometric type and if they have the same type of substitution and the same number of rules, then those skeletons are numbered like “Skeleton 1”, “Skeleton 2”, etc.
- Every name of a class is followed by its number of elements.

**A1 Homogenous**, ${d}_{1}$, 522 double sequences + 256 constant double sequences, 58 classes + 34 classes producing constant double sequences. The non-constant double sequences are given here explicitly:

**A2 Pascal Triangle**, ${d}_{1}$, 262 double sequences, 34 classes.

**A3 Pascal Triangle and Diagonal**, ${d}_{1}$, 24 double sequences, 3 classes.

**A5 Double Wave Pascal**, ${d}_{1}$, 12 double sequences, 2 classes.

**A7 Brilliant**, ${d}_{1}$, 6 double sequences, 1 class.

**A10 Swallow**, ${d}_{1}$, 6 double sequences, 1 class.

**A11 Squares**, ${d}_{1}$, 6 double sequences, 1 class.

**A12 Angel**, ${d}_{1}$, 18 double sequences, 2 classes.

**A13 Butterfly Families**, ${d}_{1}$, 12 double sequences, 2 classes.

**A14 Four Stars**, ${d}_{1}$, 12 double sequences, 2 classes.

**A15 Trace 1**, ${d}_{1}$, 16 double sequences, 4 classes. These double sequences are all primitive.

**A16 Trace 2**, ${d}_{2}$, 16 double sequences, 2 classes. These double sequences are all primitive.

**A17 Trace Median**, $ns$, 4 double sequences, 1 class. These double sequences are all primitive.

**A18 Trace Rectangular**, $ns$, 4 double sequences, 1 class. These double sequences are all primitive.

**A19 Mirrored Triangle**, m, 84 double sequences, 7 classes.

**A20 Mirrored Rectangles**, m, 12 double sequences, 1 class.

**A21 Long Triangles I**, t, 72 double sequences, 6 classes.

**A22 Shifted Triangles**, $ns$, 12 double sequences, 1 class.

**A23 Shifted Long Triangles I**, $ns$, 48 double sequences, 4 classes.

**A24 Left Meteorites**, $ns$, 60 double sequences, 5 classes.

**A25 Right Meteorites**, $ns$, 84 double sequences, 7 classes.

**A26 Left Comets**, $ns$, 12 double sequences, 1 class.

**A27 Right Comets**, $ns$, 12 double sequences, 1 class.

**A28 Pythagoras vs. Pascal**, $ns$, 24 double sequences, 2 classes.

**A29 Left Broken Arrows**, $ns$, 24 double sequences, 2 classes.

**A30 Right Broken Arrows**, $ns$, 24 double sequences, 2 classes.

**A31 Interrupted I**, $ns$, 12 double sequences, 1 class.

**A32 Interrupted Long**, $ns$, 24 double sequences, 2 classes.

**A33 Interrupted Short**, $ns$, 24 double sequences, 2 classes.

**A34 Lamps**, $ns$, 24 double sequences, 2 classes.

**A35 Half Lamps**, $ns$, 24 double sequences, 2 classes.

**A36 High Half Lamps I**, $ns$, 12 double sequences, 1 class.

**A37 Small Half Lamps**, $ns$, 24 double sequences, 2 classes.

**A38 Pairs Small Half Lamps**, $ns$, 24 double sequences, 2 classes.

**A39 Balks**, $ns$ 12 double sequences, 1 class.

**A40 Shifted Balks Generation**, $ns$, 60 double sequences, 5 classes.

**A41 Shifted Balks Decay**, $ns$, 72 double sequences, 6 classes.

**A42 Shifted Rectangles**, $ns$, 12 double sequences, 1 classes.

**A43 Shifted Diamonds**, $ns$, 36 double sequences, 3 classes.

**A44 Cancer**, $ns$, 36 double sequences, 3 classes.

**A45 Dragon I**, $ns$, 24 double sequences, 2 classes.

**A46 Fish**, $ns$, 12 double sequences, 1 class.

**A47 Stone Chain**, $ns$, 12 double sequences, 1 class.

**A48 Quadrilaterals**, $ns$, 36 double sequences, 3 classes.

**A49 Regatta I**, $ns$, 12 double sequences, 1 class.

**A50 Trapezes**, $ns$, 12 double sequences, 1 class.

**A51 Falling Stars**, $ns$, 24 double sequences, 2 classes.

**B52 Isosceles Triangles**, m, 12 double sequences, 1 class.

**B53 Long Triangles II**, t, 72 double sequences, 6 classes.

**B54 Shifted Long Triangles II**, $ns$, 60 double sequences, 5 classes.

**B55 Falling Comets**, $ns$, 12 double sequences, 1 class.

**B56 Pascal vs. Pythagoras**, $ns$, 24 double sequences, 2 classes.

**B57 Dragon II**, $ns$, 24 double sequences, 2 classes.

**B58 Regatta II**, $ns$, 12 double sequences, 1 class.

**B59 Descendant**, $ns$, 36 double sequences, 3 classes.

**B60 Ascendant**, $ns$, 36 double sequences, 3 classes.

**B61 High Half Lamps II**, $ns$, 12 double sequences, 1 class.

**B62 Skyrockets**, $ns$, 24 double sequences, 2 classes.

**B63 Cancer Lamps**, $ns$, 36 double sequences, 3 classes.

**B64 Cut Lamps**, $ns$, 12 double sequences, 1 class.

**B65 Parallelograms**, $ns$, 24 double sequences, 2 classes.

**B66 Interrupted II**, $ns$, 24 double sequences, 2 classes.

**B67 Broken Arrows**, $ns$, 24 double sequences, 2 classes.

**B68 Slim Triangles**, $ns$, 24 double sequences, 2 classes.

**C69 Pascal Slim Cut**, ${d}_{1}$, 6 double sequences, 1 class.

**C70 Pascal Slim Vertex**, ${d}_{1}$, 6 double sequences, 1 class.

**C71 Single Butterflies**, ${d}_{1}$, 18 double sequences, 2 classes.

**C72 Pairs of Butterflies**, ${d}_{1}$, 12 double sequences, 2 classes.

**D73 Diagonal**, ${d}_{1}$, 98 double sequences, 13 classes.

**D74-Pascal Rotated**, ${d}_{2}$, 560 double sequences, 53 classes.

**D75 Shifted Pascal**, $ns$, 72 double sequences, 6 classes.

**D76 Shifted Double Pascal**, $ns$, 48 double sequences, 4 classes.

**D77 Half Shifted Pascal**, $ns$, 12 double sequences, 1 class.

**D78 Shifted Triangles Rotated**, $ns$, 12 double sequences, 1 class.

**D79 Splits**, $ns$, 24 double sequences, 2 classes.

**D80 Shifted Double Pascal**, $ns$, 24 double sequences, 2 classes.

**D81 Comets Up**, $ns$, 60 double sequences, 5 classes.

**D82 Comets Down**, $ns$, 24 double sequences, 2 classes.

**D83 UFOs I**, $ns$, 12 double sequences, 1 class.

**E84 Median to Vertex**, $ns$, 60 double sequences, 5 classes.

**E85 Single Long Triangle**, $ns$, 72 double sequences, 6 classes.

**E86 Shifted Single Long triangle**, $ns$, 48 double sequences, 4 classes.

**E87 UFOs II**, $ns$, 84 double sequences, 7 classes.

**F88 UFOs III**, $ns$, 72 double sequences, 6 classes.

**F89 Shadows**, $ns$, 12 double sequences, 1 class.

**F90 Double Shadows**, $ns$, 24 double sequences, 2 classes.

## 8. Affine Recurrence Rules and Two Secret Types

- All these double sequences are generated by systems of substitutions.
- For non-constant double sequences ABC, adding a constant D in the recurrence does not change the geometric type.
- For most of the triples ABC generating constant double sequences, the various types ABCD generate double sequences in one of the 90 geometric types given above. The geometric type depend only on the triple ABC, and not on the constant $D\in {M}_{2}\left({\mathbb{F}}_{2}\right)\backslash \left\{0\right\}$ that have been added in the recurrence formula.
- There are only two new geometric types revealed in the (formerly) constant double sequences, as follows:
**A91 Byzantine Mosaics**, ${d}_{1}$, got for example by all XIXD with $D\in {M}_{2}\left({\mathbb{F}}_{2}\right)\backslash \left\{0\right\}$. These sequences are primitive. All XIXD are of type $2\to 4$ with 12 rules and with the same skeleton for all $D\in {M}_{2}\left({\mathbb{F}}_{2}\right)\backslash \left\{0\right\}$.**B92 Double Triangles**, $ns$, got for example by ubYD$D\in {M}_{2}\left({\mathbb{F}}_{2}\right)\backslash \left\{0\right\}$. These sequences have periodic domains and a conventional geometric content. All ubYD are of type $4\to 8$ with 12 rules, and have again the same skeleton for all $D\in {M}_{2}\left({\mathbb{F}}_{2}\right)\backslash \left\{0\right\}$.

## References

- Prunescu, M. Recurrent double sequences that can be generated by context-free substitutions. Fractals
**2010**, 18, 65–73. [Google Scholar] [CrossRef] - Prunescu, M. Undecidable properties of recurrent double sequences. Notre Dame J. Formal Logic
**2008**, 49, 143–151. [Google Scholar] [CrossRef] - Prunescu, M. Self-similar carpets over finite fields. Eur. J. Combinatorics
**2009**, 30, 866–878. [Google Scholar] [CrossRef] - Passoja, D.E.; Lakhtakia, A. Carpets and rugs: An exercise in numbers. Leonardo
**1992**, 25, 69–71. [Google Scholar] [CrossRef] - Prunescu, M. Recurrent two-dimensional sequences generated by homomorphisms of finite abelian groups with periodic initial conditions. Preprint 2010. http://home.mathematik.uni-freiburg.de/prunescu/prunescu.html To appear in Fractals.
- Prunescu, M. Counterexample to context-free substitution in recurrent double sequences over finite sets. Preprint 2010. http://home.mathematik.uni-freiburg.de/prunescu/prunescu.html.
- Allouche, J.-P.; Shallit, J. Automatic Sequences-Theory, Applications, Generalizations; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Allouche, J.-P.; Shallit, J. The Ubiquitous Prouhet-Thue-Morse Sequence; Sequences and their applications (Singapore 1998), springer series discrete mathematics in theoretical computer science; Springer: London, UK, 1999; pp. 1–16. [Google Scholar]
- Muchnik, A.A.; Pritykin, Y.L.; Semenov, A.L. Sequences close to periodic. Russ. Math. Surv.
**2009**, 64, 805–871. [Google Scholar] [CrossRef] - Willson, S.J. Cellular automata can generate fractals. Discrete Appl. Math.
**1984**, 8, 91–99. [Google Scholar] [CrossRef] - Mandelbrot, B.B. The Fractal Geometry of Nature; W.H. Freeman and Company: San Francisco, CA, USA, 1982. [Google Scholar]
- Kari, J. Theory of cellular automata: A survey. Theor. Comput. Sci.
**2005**, 334, 3–33. [Google Scholar] [CrossRef] - Wolfram, S. A New Kind of Science; Wolfram Media: Champaign, IL, USA, 2002. [Google Scholar]
- Penrose, R. Pentaplexity. Math. Intell.
**1979**, 2, 32–37. [Google Scholar] [CrossRef] - Wang, H. Proving theorems by pattern recognition. Commun. ACM
**1960**, 3, 220–234. [Google Scholar] [CrossRef] - Robinson, R.M. Undecidability and nonperiodicity for tilings of the plane. Invent. Math.
**1971**, 12, 177–209. [Google Scholar] [CrossRef] - Grunbaum, B.; Shephard, G.C. Tilings and Patterns; Dover Publications: Mineola, NY, USA, 2011. [Google Scholar]
- Lindenmayer, A.; Prusinkiewicz, P. The Algorithmic Beauty of Plants; Springer Verlag: Berlin, Germany, 1996. [Google Scholar]
- Rozenberg, G.; Salomaa, A. Lindenmayer Systems: Impacts on Theoretical Computer Science, Computer Graphics, and Developmental Biology; Springer Verlag: Berlin, Germany, 1992. [Google Scholar]
- Baake, M.; Moody R., V. (Eds.) Directions in Mathematical Quasicrystals; CRM Monograph Series; AMS: Providence, RI, USA, 2000. [Google Scholar]
- Moody, R.V. The Mathematics of Aperiodic Order. Proceedings of the NATO Advanced Study Institute on Long Range Aperiodic Order; Kluwer Academic Publishers: Norwell, MA, USA, 1997. [Google Scholar]
- Frettlöh, D. Duality of model sets generated by substitution. Rom. J. Pure Appl. Math.
**2005**, 50, 619–639. [Google Scholar] - Perron, O. Zur Theorie der matrices. Math. Ann.
**1907**, 64, 248–263. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Prunescu, M.
Linear Recurrent Double Sequences with Constant Border in *M*_{2}(F_{2}) are Classified According to Their Geometric Content. *Symmetry* **2011**, *3*, 402-442.
https://doi.org/10.3390/sym3030402

**AMA Style**

Prunescu M.
Linear Recurrent Double Sequences with Constant Border in *M*_{2}(F_{2}) are Classified According to Their Geometric Content. *Symmetry*. 2011; 3(3):402-442.
https://doi.org/10.3390/sym3030402

**Chicago/Turabian Style**

Prunescu, Mihai.
2011. "Linear Recurrent Double Sequences with Constant Border in *M*_{2}(F_{2}) are Classified According to Their Geometric Content" *Symmetry* 3, no. 3: 402-442.
https://doi.org/10.3390/sym3030402