## 7. Classification

Let us sum up the relations between recurrent double sequences defined here so far. From finer to coarser we have: class (mirrored class of conjugation), the isomorphism, the same skeleton, and the same geometric content (geometric type).

Our classification of linear double sequences over ${M}_{2}\left({\mathbb{F}}_{2}\right)$ with constant border $=I$ clears the relationships between the different double sequences according to the criteria explained above. A general conclusion can be summed up in the following statement:

**Theorem** **7.1** The following statements hold:

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.

Now we are ready to present the classification with details.

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

The number n runs from 1 to 90 and is sufficient to determine the type.

The prefix P runs from A to F and encodes only some geometric information, which is defined as follows:

Consider a plane square $\alpha \beta \gamma \delta $. Let $\u03f5$ denote the middle of the segment $\beta \gamma $ and let $\eta $ denote the middle of the segment $\gamma \delta $. The point $\alpha $ should be figured as left-upper vertex. Recall that ${S}_{n}$ is the ${s}^{n}\times {s}^{n}$ initial square of the recurrent double sequence S and ${\pi}_{n}$ is the canonical projection of ${S}_{n}$ on the unit square. We consider all the half-lines $\left[\alpha {\pi}_{n}\left(a(i,j)\right)\right)$, where $a(i,j)$ does not belong to a periodic domain of S.

Let ${R}_{n}$ be the union of those half-lines and let $R=R\left(S\right)$ be the topological closure of the Hausdorff limit $lim{R}_{n}$ in the unit square.

Then the prefixes are:

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)

The author did not overcome the tentation to give names for the types. The names are a very unessential information: the readers who do not like geometric types to have their own names should just skip them.

Other syntactic rules and conventions are the following:

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:

2 → 4, 4 rules: Skeleton 1: aaa 6, aab 12, aae 12, aba 6, aea 6, oao 6, oaz 12, ooa 12, ooo 1, oou 6, oua 12, oud 12, ouo 3, ouu 6, uaz 12, udc 12, udu 6, uoa 12, uod 12, uou 3, uua 12, uud 12, uuu 3. Skeleton 2: auK 12.

2 → 4, 2 rules: Skeleton 1: aaJ 12, abI 12, aoc 12, aud 12, azK 12, oac 12, ooJ 6, ouI 6, uda 12, udK 12, uoI 6, uuL 6. Skeleton 2: JXK 12.

1 → 2, 2 rules: aaL 12, aeI 12, aIe 6, aLa 6, IIJ 6, IJI 3, JJJ 3.

3 → 6, 3 rules: abX 12, azY 12.

2 → 4, 6 rules: Skeleton 1: aeX 12, auY 12, ooX 4, uaX 12, udY 12. Skeleton 2: JKX 12.

2 → 4, 3 rules: Skeleton 1: JKY 12. Skeleton 2: uaK 12.

3 → 6, 5 rules: uab 12.

2 → 4, 5 rules: uau 6.

2 → 4, 9 rules: XIY 2, XYX 2.

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

2 → 4, 5 rules: Skeleton 1: aaK 12, azI 12. Skeleton 2: aoa 6.

2 → 4, 7 rules: abf 12, aob 6, aoe 6, aua 6, aue 6, aza 6, azb 6.

1 → 2, 2 rules: ada 6, ade 12, afa 6, aoI 12, aoL 12, auI 12, auL 12, IaI 6, IaJ 12, IoI 1, JaJ 6.

2 → 4, 6 rules: aoJ 12.

3 → 6, 48 rules: azJ 12.

2 → 4, 4 rules: IaL 12, IoJ 6, IuL 6.

2 → 4, 16 rules: IoX 4.

1 → 2, 3 rules: IuI 3.

2 → 4, 8 rules: JcJ 6, JoJ 3, JzJ 3.

6 → 12, 128 rules: JeY 12.

2 → 4, 28 rules: XaX 6.

2 → 4, 51 rules: XoX 2.

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

2 → 4, 11 rules: aca 6.

3 → 6, 181 rules: acb 12.

2 → 4, 19 rules: avd 6.

**A4 Hour Glasses**,

${d}_{1}$,

${d}_{2}$, 48 double sequences, 5 classes. See [

1].

2 → 4, 34 rules: aIc 12, aJc 12, aJd 12, aYa 6.

2 → 4, 11 rules: aId 6.

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

2 → 4, 7 rules: JeJ 6.

2 → 4, 10 rules: JoK 6.

**A6 Diamond**,

${d}_{1}$, 18 double sequences, 2 classes. See [

1].

2 → 4, 42 rules: JaL 12, JuJ 6.

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

2 → 4, 56 rules: XuX 6.

**A8 Twin Peaks**,

${d}_{1}$, 18 double sequences, 2 classes. See [

1].

2 → 4, 51 rules: acf 12, ava 6.

**A9 Open Peano**,

${d}_{1}$ 12 double sequences, 2 classes. See [

6].

2 → 4, 27 rules: JIK 6, JXJ 6.

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

2 → 4, 15 rules: JKJ 6.

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

4 → 8, 59 rules: XJX 6.

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

2 → 4, 15 rules: aKa 6.

4 → 8, 30 rules: aXc 12.

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

2 → 4, 41 rules: aKd 6, aXa 6.

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

2 → 4, 60 rules: XbX 6.

4 → 8, 120 rules: XuY 6.

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

1 → 2, 3 rules: IXI 2.

2 → 8, 12 rules: JKL 6.

2 → 4, 21 rules: XJY 6, XXX 2.

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

2 → 4, 9 rules: IIX 4.

2 → 8, 16 rules: JXY 12.

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

2 → 4, 12 rules: IXY 4.

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

2 → 4, 36 rules: XXY 4.

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

2 → 4, 12 rules: acI 12, afI 12.

2 → 4, 16 rules: acJ 12, adJ 12, adL 12, afL 12.

2 → 4, 7 rules: adI 12.

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

2 → 4, 10 rules: IJK 12.

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

2 → 4, 9 rules: abL 12, aeJ 12, aoK 12.

2 → 4, 5 rules: IuJ 12, JcL 12.

2 → 4, 7 rules: JaK 12.

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

2 → 4, 10 rules: JoX 12.

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

4 → 8, 19 rules: abY 12.

2 → 4, 11 rules: aeY 12.

2 → 4, 19 rules: aoX 12.

2 → 4, 8 rules: IuX 12.

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

2 → 4, 25 rules: aaX 12.

2 → 4, 16 rules: aoY 12.

2 → 4, 13 rules: IaY 12, JuY 12.

2 → 4, 22 rules: JuX 12.

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

2 → 4, 24 rules: Skeleton 1: abe 12, aeb 12, auc 12. Skeleton 2: avc 12.

4 → 8, 8 rules: aJK 12.

2 → 4, 8 rules: aLK 12, aYI 12.

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

2 → 4, 9 rules: IaX 12.

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

2 → 4, 9 rules: aXI 12.

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

2 → 8, 25 rules: adK 12.

2 → 4, 14 rules: avI 12.

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

2 → 4, 14 rules: IaK 12.

2 → 8, 23 rules: JuL 12.

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

2 → 4, 16 rules: aKI 12.

2 → 8, 28 rules: aXK 12.

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

4 → 8, 30 rules: aXY 12.

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

4 → 8, 56 rules: JaX 12.

2 → 4, 28 rules: JcX 12.

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

4 → 8, 30 rules: aKJ 12,

2 → 4, 15 rules: aKL 12,

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

2 → 4, 112 rules: adX 12, avY 12.

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

2 → 4, 52 rules: JaY 12, JcY 12.

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

2 → 4, 104 rules: JzX 12.

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

2 → 4, 52 rules: aJL 12, aLJ 12.

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

2 → 4, 60 rules: aJY 12, aLY 12.

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

2 → 4, 30 rules: JIX 12.

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

2 → 4, 19 rules: aaY 12.

2 → 4, 28 rules: auX 12, azX 12.

2 → 4, 16 rules: IJX 12, JXX 12.

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

2 → 4, 27 rules: abK 12, aeK 12, auJ 12, azL 12.

2 → 4, 8 rules: Skeleton 1: IXJ 12. Skeleton 2: JJK 12.

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

2 → 4, 20 rules: JJX 12.

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

2 → 4, 51 rules: aIK 12, aYJ 12, aYL 12.

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

2 → 4, 52 rules: Skeleton 1: aJe 12, Skeleton 2: aKb 12, aXd 12,

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

4 → 8, 32 rules: acX 12.

2 → 4, 32 rules: afX 12.

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

2 → 4, 56 rules: aIX 12.

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

2 → 4, 58 rules: XaY 12.

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

2 → 4, 15 rules: acL 12.

2 → 4, 30 rules: afJ 12.

4 → 8, 30 rules: avK 12.

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

2 → 4, 116 rules: adY 12.

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

2 → 4, 112 rules: aKX 12.

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

2 → 4, 31 rules: aIY 12.

4 → 8, 62 rules: aYX 12.

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

4 → 8, 16 rules: uvI 12.

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

4 → 8, 13 rules: ubb 12, uve 12, uvf 12.

4 → 8, 10 rules: uJI 12, uXJ 12, uXK 12.

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

4 → 8, 31 rules: uba 12, uvb 12, uvc 12.

4 → 8, 17 rules: ubf 12.

4 → 8, 8 rules: uXI 12.

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

4 → 8, 17 rules: uXX 12.

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

4 → 8, 14 rules: ubI 12.

4 → 16, 25 rules: uvJ 12.

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

4 → 8, 58 rules: uvX 12, uvY 12.

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

4 → 8, 116 rules: ubX 12.

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

4 → 8, 30 rules: ubc 12, uva 12, uvd 12.

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

4 → 8, 50 rules: ubJ 12, uvL 12.

4 → 8, 25 rules: ubK 12.

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

4 → 8, 104 rules: uJL 12.

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

4 → 8, 28 rules: ubL 12, uvK 12.

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

4 → 8, 51 rules: uJa 12, uJd 12, uXb 12.

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

4 → 8, 58 rules: uXY 12.

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

4 → 8, 112 rules: uXa 12, uXf 12.

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

4 → 8, 106 rules: uJX 12.

4 → 8, 53 rules: uJY 12.

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

4 → 8, 37 rules: uJJ 12, uXL 12.

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

4 → 8, 108 rules: ubd 12, ube 12.

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

3 → 6, 32 rules: uvu 6.

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

4 → 8, 32 rules: ubu 6.

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

4 → 8, 43 rules: uJu 6.

4 → 8, 86 rules: uXv 12.

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

4 → 8, 41 rules: uJv 6, uXu 6.

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

2 → 4, 3 rules: aIb 6, aJa 6, oIu 6, oJo 3, uad 12, uLu 3, oaf 12.

2 → 4, 6 rules: Skeleton 1: aYc 12. Skeleton 2: oau 12, uov 6.

2 → 4, 9 rules: oXo 2, uIv 6.

2 → 4, 5 rules: uub 12.

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

2 → 4, 6 rules: Skeleton 1: aac 12. Skeleton 2: ave 12.

2 → 4, 4 rules: aad 12, aaf 12, acd 12, adc 12, aII 12, aIJ 12, aJI 12, aJJ 12, aLI 12, oaI 12, oaL 12, oIa 12, oII 2, oJc 12, oJI 6, uaI 12, uaL 12, uIa 12, uII 6, uLa 12, uLI 6.

4 → 8, 6 rules: abc 12, aub 12.

2 → 4, 10 rules: Skeleton 1: abd 12. Skeleton 2: oaJ 12, udL 12.

2 → 4, 8 rules: Skeleton 1: aIL 12, aLL 12, oIJ 6, oJJ 6, uIL 6, uLL 6. Skeleton 2: oJa 12.

2 → 4, 7 rules: oaa 12, oab 12, oae 12, uaa 12, uae 12, udd 12, udf 12, uob 12.

2 → 4, 51 rules: oIX 4.

2 → 4, 16 rules: Skeleton 1: oXI 4. Skeleton 2: uIX 12.

2 → 4, 40 rules: oXX 4, uLX 12.

2 → 4, 5 rules: udI 12, uuJ 12.

2 → 4, 11 rules: uId 12.

3 → 6, 51 rules: uLd 12.

6 → 12, 128 rules: uLJ 12.

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

4 → 8, 12 rules: acK 12.

2 → 4, 12 rules: afK 12, avJ 12, avL 12.

2 → 4, 22 rules: oaK 12.

2 → 4, 14 rules: uoJ 12.

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

2 → 4, 14 rules: acY 12, afY 12.

2 → 4, 19 rules: oaX 12.

2 → 4, 11 rules: uoX 12.

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

4 → 8, 20 rules: oJX 12,

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

4 → 8, 10 rules: oJK 12.

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

2 → 4, 16 rules: oXa 12.

4 → 8, 16 rules: uLb 12.

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

2 → 4, 20 rules: oXJ 12.

2 → 4, 10 rules: uIJ 12.

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

4 → 8, 13 rules: aJX 12.

2 → 4, 13 rules: aLX 12, aYY 12.

2 → 4, 16 rules: oaY 12, uuX 12.

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

2 → 4, 23 rules: oXb 12, uIb 12.

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

4 → 8, 28 rules: oJe 12.

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

3 → 6, 8 rules: ouv 12.

4 → 8, 8 rules: uav 12.

3 → 6, 9 rules: ubz 12.

3 → 6, 10 rules: uJe 12.

3 → 6, 14 rules: uJf 12.

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

2 → 4, 5 rules: aKK 12, aXL 12.

4 → 8, 5 rules: aXJ 12.

2 → 4, 7 rules: ouJ 12.

2 → 4, 9 rules: uaJ 12, udJ 12.

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

2 → 4, 16 rules: aXX 12.

2 → 4, 19 rules: ouX 12.

2 → 4, 17 rules: uaY 12.

2 → 4, 31 rules: udX 12.

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

4 → 8, 16 rules: Skeleton 1: aJf 12. Skeleton 2: aXb 12.

2 → 4, 16 rules: aKf 12, aXe 12.

2 → 4, 18 rules: oub 12.

2 → 4, 34 rules: uac 12, udb 12.

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

4 → 8, 35 rules: oav 12, uuv 12.

4 → 8, 32 rules: uJb 12, uJc 12, uXd 12, uXe 12.

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

4 → 8, 28 rules: oJu 12.

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

4 → 8, 28 rules: Skeleton 1: oXu 12. Skeleton 2: uJz 12.