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TOND to TOND

Self-Similarity of Persian TOND Patterns, Through the Logic of the X-Tiles
  • Jean-Marc CasteraEmail author
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Abstract

Looking at the heritage of traditional Persian pentagonal patterns (patterns made from tiles derived from the pentagon), one can suppose that the artists, very early on, have set targets for their creation. One is the search for self-similarity; another is the search for methods of connection between the two main families of patterns. It is strange and intriguing that the historic artists did not fully achieve these targets.

This paper, following a previous publication (Castera, Nexus Netw J 18:223, 2016), proposes solutions and new developments.

There are no Penrose patterns in that story; only binary tiling and the X-Tiles.

Keywords

Persian pentagonal patterns Multilevel patterns X-Tiles Self-similarity Binary tiling 

Introduction: The Two Traditional Persian Families of Pentagonal Patterns

The impossibility to pattern the plan (without any gap or overlap) with only regular pentagons is well known. Nevertheless, Persian artists trying to overcome that impossibility have produced many patterns with local five-fold symmetry. These “pentagonal patterns” are always included in a periodic network. If it happens that they introduce some light variations which break the periodicity that does not make the pattern an aperiodic one (in other words, a 2D quasicrystal). For geometric analyse of pentagonal patterns, see for example Lee (1987) ; About connection with quasicrystals, one of the first contributions is in Makovicky (1992).

There are two main families of pentagonal Persian patterns. Let [A] be the first one, which the Iranian tradition call “Kond + Sholl,” (Mofid and Raieszadeh, 1995) and [B] the second one, “Tond.” Let [A1] be the subset of [A] made of only Kond tiles. Those families are defined by the different kinds of tiles they are composed of (Figs. 1 and 6). The vertices of these patterns are always of degree 4 (intersection of two lines). They can therefore be colored using two colors, as in a chessboard. As a result, a remarkable property emerges: In such a coloring, any two tiles that are of the same kind will also be of the same color (this is not the case in the Arabic style). That is why we can distinguish two kind of tiles. Let them be the “positive” and the “negative” tiles.
Fig. 1

The “Tond + Sholl” family set of tiles. While it can be slightly different in the Iranian tradition, this is the necessary and sufficient set for any pattern in that family to be fully self-similar. The tile N1, a stellated decagon, is “the mother” of each tile. It is never used in that way, but always decorated with one of its variations (Fig.~2). One could say that the mother is always pregnant. The Kond tiles define the subset [A1]. (© Jean-Marc Castera 2015–2018)

Note that a pattern can be drawn using only one kind of tile (positive or negative), connected by vertices that respect the continuity of the lines. The shapes of the void between the tiles are the shapes of the other kind of tiles. The following figures show the set of the tiles, the variations, and some examples.

The Kond + Sholl Family

See Figs. 2, 3, 4, and 5.
Fig. 2

The mother (Stellated decagon [10/3]). Proportions, angles, and variations. Only N11 (Sun) keeps the symmetries; N12 is oriented and N13 uses a new tile. N41 is the unique variation of the tile N4. (© Jean-Marc Castera 2015–2018)

Fig. 3

The simplest Kond pattern: Nothing but 4 stars holding hands, with variation N11. (© Jean-Marc Castera 2015–2018)

Fig. 4

Left, the simplest pattern made of Kond and Sholl tiles, with stars on each vertex of a rhombus. On the right, the stars are on the vertices of a rectangle. (© Jean-Marc Castera 2015–2018)

Fig. 5

A Turkish pattern. Note the variation N41 and flowers from the Tond family at the bottom corners. On the right, the pattern reduced to tiles from only the Kond + Sholl system (What I want to say is that to do this reduction is a good exercise). Note the tile P5, which is rarely used, and two slightly forced connections at the edges (showing also the small deviation from the original). (© Jean-Marc Castera 2015–2018)

The Tond Family

See Figs. 6 and 7.
Fig. 6

The set of Tondtiles (system [B]). Although the tradition is sometimes to use more tiles, these ones are the most common. Moreover, they are X-Tiles compatible (see section “X-Tiles”), thus they can form self-similar systems. (© Jean-Marc Castera 2015–2018)

Fig. 7

Top: the most common Tond patterns (stars are set up along a rhombus). Bottom: some slight variations. (© Jean-Marc Castera 2015–2018)

Multilevel Patterns. Reminders, and a New Case

A first-level pattern is made of large-scale tiles, each one cut into small-scale tiles that fit perfectly with the adjacent ones. That is the second-level pattern. Iranian artists call this process “the pattern into the pattern.” Would it be possible to make a third level, and so on ad infinitum, using always the same substitution rules? In this case, the pattern would be self-similar.

Of course, there is no way to make such a pattern in real mosaic; even with only three levels, the material difficulties are obvious. Nevertheless, if all the shapes of the tiles being used at the second level are used at the first level (large scale), then all the substitution rules needed are defined, the process can go on (virtually) ad infinitum, and therefore the pattern is fully self-similar.

Surprisingly, it looks like historic Persian artists achieved only two-level patterns: there is at least one tile at the second level that does not exist at the first. So, one substitution rule is missing, which is necessary to get to a third level. In the Kond + Sholl patterns, that tile is the Sormedan (Fig.~8, and N3 in Fig.~1). Even so one can find some two-level pattern with that tile at the first level, but never in good connection with its adjacent tiles and/or never without introducing some extra tiles that do not admit a compatible substitution rule. Examples of traditional multilevel patterns design can be seen in Mofid and Raieszadeh (1995), Necipogglu (1995), Shaarbaf (1982). For western scholars investigations, see for example Bonner (2017), Cromwell (2009), Pelletier (2013).
Fig. 8

The problematic Sormedan. (© Jean-Marc Castera 2015–2018)

Two Kond Self-Similar Systems

We want to complete the traditional two-level Kond + Sholl patterns in order to make them fully self-similar. That means to pattern each tile with tiles of the same family at a reduced scale. The two first solutions come from a previous article (Castera 2016), while the third one presented here is new.

The general method consists in first searching for a covering of the different kinds of edges (three for the family A; only two for the subfamily A1), then in patterning the interior of each tile in continuity. The first solution (Fig.~9) works for the whole Kond + Sholl family. A similar solution is shown on http://www.quadibloc.com/math/pen05.htm. The second solution uses smaller second-level tiles but works only for the Kond subfamily (Fig.~12).
Fig. 9

A Kond + Sholl fully self-similar system. Bottom left, the covering of the three different kinds of edges in that family. To the right, alternative options for the tiles P1 and N3. Bottom, the covering of the three types of edges. (© Jean-Marc Castera 2015–2018)

There is actually a slight difficulty with the Sormedan (tile N3). The resolution requires that the “Golden rule” be broken, which asserts that each first-level vertex be the center of a second-level star.

System 1: This works for the whole Kond + Sholl family (Figs. 10 and 11).
Fig. 10

A two-level pattern in Shiraz. Note (highlighted to the right) the non-symmetric arrangement of the pentagons (Tiles P3). (© Jean-Marc Castera 2015–2018)

Fig. 11

Starting from the Sormedan tile, four levels are necessary for all the substitutions rules to be used. (© Jean-Marc Castera 2015–2018)

System 2: This works only for the Kond subfamily (Figs. 12 and 13).
Fig. 12

The second solution for a Kond self-similar system. To the right, alternative solutions for the tiles P1, P2, and N3. Bottom, the covering of the two types of edges. (© Jean-Marc Castera 2015–2018)

Fig. 13

This famous Darb-el-Iman pattern (Isfahan) belongs to the second system of self-similarity. Again, note that the orientations of the pentagons are not symmetric. And this is NOT an error! (© Jean-Marc Castera 2015–2018)

A Third Type of Kond Self-Similar System

In the meantime, we have found a third solution, which is the generalization of a two-level pattern seen in Iran (Fig.~14).
Fig. 14

Third type of two-level Kond pattern. Left, (Isfahan, Chahar Bagh Madrasa), only the positive tiles have a second level. To the right, the pattern is made of mirrors and all the tiles have a second level (Shiraz, Shah Cheragh). Superimposed, the covering of the two types of edges (long and short) at the first level. (© Jean-Marc Castera 2015–2018)

In these two examples, the Sormedan is still missing at the first level. Figure~15 shows a complete solution, with two options for the Sormedan. The two levels belong to the subfamily [A1], (Kond family). Applying endlessly the substitution rules from any Kond tile generates a 2D quasicrystal-like pattern, fully self-similar (Fig.~16).
Fig. 15

The five Kond tiles, along with their rules of substitution into tiles of the same family at reduced scale. That set defines a third type of self-similar system. The traditional system required in addition only a substitution rule for the tile N3, the Sormedan. Note here the two options (N3a and N3b) for that tile. Note also that, as for the two previous solutions, this one is challenging the traditional “golden rule,” which requires that all the first-level vertices be the centre of a star N1 at the second level. (© Jean-Marc Castera 2015–2018)

Fig. 16

An original third type self-similar pattern, with the Sormedan tile at the first level. The variation N11 is systematically applied in place of the N1 star. The substitution rules are defined from the first level onwards, thus this pattern would be fully self-similar… if the first level was not periodic. (© Jean-Marc Castera 2015–2018)

Transitions Between Different Families

In a two-level pattern, both levels can belong to the same families or to different ones. The historic Persian artists have explored some possibilities but, surprisingly, not all of them.

Let [A] and [B] be the two families (Kond + Sholl and Tond). There are then four transition possibilities:
  • [A] = > [A] (from Kond + Sholl at the first level to Kond + Sholl at the second level).

  • [A] = > [B] (from Kond + Sholl to Tond)

  • [B] = > [A] (from Tond to Kond + Sholl)

  • [B] = > [B] (from Tond to Tond).

The last case does not exist in the traditional heritage, which is why we are going to propose solutions to it later in this article. Moreover, those solutions will be fully self-similar. First, here are some examples seen in Iran.

[A] = > [A] (from Kond + Sholl to Kond + Sholl)

This case has been examined in the above-mentioned previous publication and in the previous section of this article (A third type of self-similar system). More common examples are reduced to the subfamily [A1], except that the tile N3 does not exist at the first level (or rarely, with incompatible substitution rules). Here are more examples, which cannot generate a self-similar system. However, we give here a substitution rule for the Sormedan that, once again, is conspicuously absent at the first level (Fig.~17).
Fig. 17

These two mosaics, found in Isfahan, use mostly the same pattern. The first level belongs to the Kond family while the second includes also some Sholl tiles. Note that the positive tiles admit a geometric second level, while the others are decorated with floral elements. The superimposed design shows a generalization of this system, by setting a substitution rule for every tile, including the Sormedan. However, this is only a two level system, not self-similar, because the other tiles (bottom) of the second level do not admit any compatible substitution rules. Note the two variations for the tiles N1 and P2. (© Jean-Marc Castera 2015–2018)

[A] = > [B]. From Kond + Sholl to Tond or, More Often, from [A1] to [B]

Here is an example (Fig.~18), and its generalization to all the Tond tiles:
Fig. 18

Once again, the first level includes only Kond tiles. It is the simplest pattern in that family, made from 4 stars holding hands, arranged in a diamond shape (see Fig.~3). Superimposed is an original solution for the missing tile at the first level, the Sormedan… as usual. (© Jean-Marc Castera 2015–2018)

[B] = > [A1]. From Tond to Kond

Figure~19 shows two solutions. The first level pattern is the same (the simplest Tond pattern), which uses a limited set of tiles from that family (see Fig.~7).
Fig. 19

In these two mosaics, there are only Kond tiles at the second level, while the first level is nothing but the simplest Tond pattern. Only five Tond tiles are used (P1, P2, P3, N1 and N6). Is it possible to complete the system? (© Jean-Marc Castera 2015–2018)

Generalization of the First Example

The tile N3 admits two chiral options (N3a and N3b). The tile N5 admits a more convoluted solution, which requires the use of two variations for the tiles P1 and P2, each one with two chiral options. The star N1 can be replaced with one of its variations, for example, N1a. This proliferation of the variations is not very elegant. Its advantage is to complete the system. If not perfect, it may be the perfect level of imperfection… (Fig.~20).
Fig. 20

Generalization of the simplest transition from Tond to Kond. The tile N5 needs two variations of the tiles P1 and P2. (© Jean-Marc Castera 2015–2018)

Generalization of the Second Example

After the second generalization (Fig.~21), we have two systems of connection from any Tond pattern at the first level to a Kond pattern at the second level.
Fig. 21

Generalization of the second solution of the two-level system with first [B] level and second [A] level. Here, contrary to the previous solution, each tile has a unique substitution rule. Top left, the covering used for each of the three different edge lengths in that set. (© Jean-Marc Castera 2015–2018)

Many two-level patterns can be seen in Iran. Not all of them have a second level with all the tiles fitting correctly together or with severe local deformations. Visually, these little arrangements can go unnoticed.

This kind of things deserves further study… in a future article.

[B] = > [B]. From Tond to Tond

As far as the author knows, there is not such two-level pattern in the traditional heritage (at least in Iran). If one exists, it is certainly well hidden.

In short: We now have transitions, fully self-similar, from [A] to [A], from [A1] to [A1], from [A] to [B], from [B] to [A1]. So, we can imagine cycles like [A] = > [B] = > [A1] = > [A] = > [B] = > [A1], etc. Now we are going to propose transition solutions from [B] to [B]. Before, we have to quickly describe the “X-Tiles” system (Castera 2011).

X-Tiles

As the above-mentioned publication shows, any Tond pattern can be made from a couple of rhombi decorated with two simple lines. Let them be the “X-Tiles,” while the lines are the “X-Lines.” Those two rhombi are well known from the famous Penrose patterns, which use the same rhombi, along with constraints (matching rules) that force the pattern to be non-periodic. But the X-Tiles matching rules are different. In fact, they are the same as those of the binary pattern, which can be found at http://www.quadibloc.com/math/pen02.htm. They allow both nonperiodic and periodic patterns.

This system, which is extremely simple, appeared on the occasion of a competition with the British architect Norman Foster (Fig.~22). The X-tiles naturally lead to all the Tond patterns (as defined in this article). There is no reason to believe that the traditional artists used, or were aware of, this concept.
Fig. 22

This pattern on the sphere of the new EAU council is at the root of the emergence of the X-Tiles. (© Foster+Partners)

Definition

The X-Tiles are the two “penta-rhombi” decorated with only two lines crossing at the centre of each rhombus, with angles of 36 (fat rhombus) and 108 (slim rhombus) degrees. The matching rules are given by the constraint of continuity of the X-Lines. The X-Tiles constitute an extremely simple system (Figs. 23 and 24).
Fig. 23

The two “penta-rhombi,” coming from the decagon. One fat and one slim, known as Laurel and Hardy in a certain circle of crystallographers. (© Jean-Marc Castera 2015–2018)

Fig. 24

The embellishment of the penta-rhombus with the X-Lines defines the X-Tiles. Right, two partitions of the decagon with penta-rhombi. Only the configuration “Rose 2” enables line continuity. (© Jean-Marc Castera 2015–2018)

The X-Tiles and the Tond Traditional Family of Pentagonal Patterns

The figure below (Fig.~25) shows all the different arrangements of the X-Tiles around a common vertex. After exclusion of the cases which break the X-Lines continuity, nine configurations remain. In each one, the X-Lines form, around the common vertex, a special shape, which are exactly the Tond tiles!
Fig. 25

All the valid arrangements of X-Tiles around a common vertex lead to all the Tond tiles. Bottom left, the forbidden configurations. (© Jean-Marc Castera 2015–2018)

Transition from Kond to Tond with the X-Tiles

The previous publication demonstrates two possibilities for the decomposition of Kond tiles into penta-rhombi, in configurations compatible with the X-Tiles. This leads not only to transitions from Kond to Tond patterns (Fig.~26) but also to transitions from Tond to Tond patterns, which this article is now going to explore.
Fig. 26

A transition from Kond to Tond through the X-Tiles. (© Jean-Marc Castera 2015–2018)

Tond to Tond Transition Through the X-Tiles

A binary pattern can be made tile-to-tile, using local rules. However, it is also possible to use an inflation process, which means to replace any basic tile by a set of the same kind of tiles at reduced scale. With the X-Tiles, this process leads to an infinite series of Tond patterns at decreasing scales. This solves the problem of Tond patterns’ self-similarity. However, that is certainly not the unique way to produce self-similar Tond patterns.

Self-Similarity of TOND Patterns Through the X-Tiles

Principle

Since we know that it is possible to apply inflation processes to the binary tiling, thus to the X-Tiles, we have a general method of Tond to Tond transition, with self-similarity. An illustration of the process may be the simplest explanation (Figs. 27, 28, and 29).
Fig. 27

X-Tiles inflation, applied to the N4 tile environment, leads to its partition into the same kind of tiles, with reduced scale. This process can be applied to any Tond tile (Fig.~28), and then the system becomes fully self-similar. (© Jean-Marc Castera 2015–2018)

Fig. 28

It becomes possible to forget the X-Tiles and compose multi-level patterns directly with these two-level tiles. The second level lines of any adjacent tiles will automatically match perfectly. Most of the tiles are oriented. Note that the tile N4 can exist only at the first level because, like the others, it is missing at its second level. Apart from that, the system is self-similar. (© Jean-Marc Castera 2015–2018)

Fig. 29

An original pattern from that system. The pattern uses all kinds of Tond tiles, except N2 and N5. It is original, but the originality lies more in the system itself than in any particular pattern made from this system. (© Jean-Marc Castera 2015–2018)

One can say that, in this example, the inflation rule is not the simplest (see section “Fourth Inflation Rule: System V4”). However, it leads to a relationship between the two levels, which is more elegant than others coming from simpler rules.

We now are going to explore systematically the use of different X-Tile’s inflation rules.

First Inflation Rule: System V1

The Inflation Rule

With the first application of the rule (Fig.~30), the tiles lose one symmetry axis, thus the invariance under 180° rotation. That is why we have to care about the orientation of the rhombi during the inflation process (Figs. 31 and 32).
Fig. 30

The simplest inflation rule for the binary tiling. (© Jean-Marc Castera 2015–2018)

Fig. 31

To the left, the diamonds transformed according to the inflation rule, but arranged without regard to their orientation. To the right, the unique valid arrangement in the case of that inflation rule. No gap, no overlap. (© Jean-Marc Castera 2015–2018)

Fig. 32

In addition to the inflation rule, we use this coding to fix the relative orientation of the rhombi. (© Jean-Marc Castera 2015–2018)

A visual coding fixes the right orientation. Now we have an inflation rule for each rhombus, and a rule fixing their relative orientation: we now have everything needed to continue the process. In this case, the solution is unique. We will see later that in the case of different inflation rules, there are many solutions.

The relative frequency of the two rhombi converges to the golden number, which confirms that a pattern coming from that process is not periodic. It is easy to demonstrate this property (see case V4) (Fig.~33).
Fig. 33

Considering all the possible rhombuses configurations around a common vertex, after inflation, leads to all the tiles that can emerge from the V3 system. (© Jean-Marc Castera 2015–2018)

Figure~34 shows the set of all Tond tiles that can emerge from this V1 system.
Fig. 34

The set of tiles that can emerge from the V3 inflation system. The tiles N1 and N2 are missing. (© Jean-Marc Castera 2015–2018)

Order of Appearance of the Tiles

See Fig.~35.
Fig. 35

Successive applications of the inflation process. The tiles appear gradually. At the fourth application, all the possible tiles in this system have emerged. Level 1 (first X-Tiles inflation): Tiles P1 and P2. Level 2: Tile N6: Level 3. Tiles N4 and N5. Level 3: Tiles P3 and N3. (© Jean-Marc Castera 2015–2018)

The Two-Level Tiles

Could the superimposition of successive levels leads to a self-similar system defined directly by Tond tiles inflation? Figure~36 shows examples of superimposition of two successive levels (Levels 3 + 4, and 3 + 5). In the two cases, the cutting of the second level tiles by the edges of the first level is not really elegant, giving a feeling of disorder. However, that disorder may have some aesthetic value. Figure~37 shows the set of all the tiles with three levels of inflation. Each one, except the tile N5, requires two options, which are used in the double-sided tiles of the “Abyme puzzle” (Fig.~38). That pattern has also been used for the album cover of “Different Tessellations” by the British pianist, composer and improviser Veryan Weston.
Fig. 36

Left, Levels 2 and 3 superimposition. Right, levels 3 and 5. (© Jean-Marc Castera 2015–2018)

Fig. 37

The Tond tiles with 3 superimposed levels of inflation. Each one leads to two different configurations, except the most symmetric, N5. (© Jean-Marc Castera 2015–2018)

Fig. 38

The “Abyme puzzle.” Each tile, except N5, is printed recto-verso with the two variations of a three-level inflation. Right, the album cover of “Different Tessellations” (Veryan Weston, Leo Svirsky and the Vociferous Choir, 2011). (© Jean-Marc Castera 2015–2018)

Second Inflation Rule: System V2

The Inflation Rule

See Figs. 39, 40, and 41.
Fig. 39

The second X-Tiles inflation rule. (© Jean-Marc Castera 2015–2018)

Fig. 40

Due to the dissymmetry of the second rhombus after inflation, it is necessary to takes in account the orientation. That is why a coding is added to the figure. Contrary to the previous case V1, there are many different valid choices. (© Jean-Marc Castera 2015–2018)

Fig. 41

Here is the point of view of decorated rhombuses. Again, the slim rhombus loses one symmetry axis after the first inflation. See further (section “Working with Decorated Rhombuses”) on this article. (© Jean-Marc Castera 2015–2018)

Here again, it is not surprising that the relative frequency of the two kind of rhombuses converges to the golden number.

The Set of All the Tond Tiles that Can Emerge from the V3 System

The six negative Tond tiles have been generated by the different arrangements of the X-Tiles around a common vertex (Fig.~25). In each of those arrangements, the X-Tiles are connected to this common vertex by their 36° angles for the slim rhombus, and by their 72° angles for the fat one. It so happen that, after inflation, these rhombuses have the same immediate environment. Therefore, their arrangements will lead again to the same six negative tiles. The three positives tiles as well emerge (Fig.~42).
Fig. 42

All the Tond tiles emerge from the system V2. (© Jean-Marc Castera 2015–2018)

Order of Appearance of the Tiles

See Fig.~43.
Fig. 43

All the Tond tiles emerge after 3 inflation levels: P1, P2, N6 since the first level, P3, N2, N4 at the second, and eventually N1 and N3 at the third. (© Jean-Marc Castera 2015–2018)

The Two-Level Tiles

See Figs. 44 and 45.
Fig. 44

Superimposition of two successive levels (levels 3 and 4). Here comes the idea of an inflation defined directly on the tiles. (© Jean-Marc Castera 2015–2018)

Fig. 45

Unfortunately, several different inflations are necessary for some kind of tiles. Moreover, the edges of the first level cut the second level tiles in ways that are not always very elegant. (© Jean-Marc Castera 2015–2018)

Looking at two successive inflation levels, we got the idea to define an inflation process directly on the Tond tiles, without going through the X-Tiles. The result is much more… calm than with the first system, V1, but the decomposition of some tiles is not unique, and the second level tiles are badly cut by the first level edges.

Remark: Other Valid Orientation Options in the V2 System

Figure~46 shows two orientation systems, which are lightly different from the one we have used here. The first differs by the orientation of one only rhombus, the second by more. Visually, the differences between the resulting Tond patterns may not be dramatic. However, are the relative amount of the different tiles affected? Does every tile of the system will emerge?
Fig. 46

Two valid alternative options for the rhombuses orientation, among others. (© Jean-Marc Castera 2015–2018)

We are going to discuss those questions with the next inflation rule.

Third Inflation Rule: System V3

The Inflation Rule

See Figs. 47 and 48.
Fig. 47

The third X-Tiles inflation rule, system V3. (© Jean-Marc Castera 2015–2018)

Fig. 48

When connected, the inflated tiles will fill the gaps at the edges always the same way. Therefore, it is possible to continue the inflation with the decorated rhombuses (to the right). (© Jean-Marc Castera 2015–2018)

Note that the covering of the edges is symmetric.

Again, the relative frequency of the two kind of rhombuses converges to the golden number.

If the process is applied ad infinitum to a rhombus, its limit becomes a fractal curve.

The Set of All the Tond Tiles that Can Emerge from the V3 System

The tiles P1, P2, P3 and N6 occur since the first level. Note that the gap at the boundary of the first level tiles is always filled in a unique way, so it becomes possible to continue the inflation process from the decorated tiles (Fig.~48, right). It so happen that the vertexes of these rhombuses are pieces of stars… except one of them, an obtuse vertex of a slim rhombus (see the arrow on the figure). Therefore, each vertex of such a pattern is the centre of a star, except if connected to a slim rhombus by such an obtuse vertex. So, it is sufficient to look at all these possibilities to find out all the others negative tiles that can emerge from a pattern made of these decorated tiles (whether by inflation or not) (Figs. 49 and 50).
Fig. 49

The tiles N2 and N4 occur around a connection with the special obtuse vertex of a slim rhombus. (© Jean-Marc Castera 2015–2018)

Fig. 50

Complete set of the tiles that can emerges from the system V3. Tiles N3 and N5 are missing. (© Jean-Marc Castera 2015–2018)

That system provide all the Tond tiles, except N3 and N5. Now, in order to continue the inflation process, it needs to fix the relative orientations of the rhombuses. There are many valid configurations, including the following (Fig.~51):
Fig. 51

Four valid options of rhombuses orientations in the V3 system. (© Jean-Marc Castera 2015–2018)

These different configurations have an incidence on the resulting patterns:
  1. 1.

    The relative frequency of the different type of tiles is variable.

     
  2. 2.

    In the case of V3.1, the tile N4 never occurs. Indeed, that tile comes from the connection of one fat rhombus and two slim in a head to tail configuration, which is impossible in that option.

     

Instead, the option V3.4 will increase the frequency of the tile N4.

In short:
  • Option V3.1: No tile N4. The others negative tiles occur since the second level.

  • Option V3.2: Few N4 tiles. They appear only at the third inflation level: six in the inflation of the fat rhombus, only one with the slim.

  • Option V3.3: Tile N4 occurs since the second level. There are two in the inflation of the fat rhombus, but none with the slim.

  • Option V3.4: Tiles N4 since the second level. There are three in the inflation of the fat rhombus and two with the slim. At the third level, there are 94 and 58.

Option V3.1

Order of Appearance of the Tiles
See Figs. 52, 53, 54, and 55.
Fig. 52

At the first inflation, the three positive tiles and the negative N6 are here. (© Jean-Marc Castera 2015–2018)

Fig. 53

Second level of inflation. Here emerge the tiles N1 and N2. (© Jean-Marc Castera 2015–2018)

Fig. 54

Third level, detail. Emphasized is the boundary between the second level components. Obviously, there is no new kind of tiles: all of them existed already at the previous level. (© Jean-Marc Castera 2015–2018)

Fig. 55

The tiles emerging from the V3.1 option. In addition to N3 and N5, the tile N4 is missing. (© Jean-Marc Castera 2015–2018)

The Two-Level Tiles and the Interlacings

The superposition of two successive levels provides a mapping of the first level tiles by the next level. Unfortunately, the two levels do not fit very well, and some tiles needs two chiral options. But a good news occurs when considering two successive levels of the same parity (levels 1 and 3 on Fig.~56 right): Even though the second level tiles are badly cut, it becomes possible to design interlacings, whose lines fit nicely with the third level tiles.
Fig. 56

To the left, levels 1 and 2. In term of tiles, the two levels still do not fit nicely: many second level tiles are cut without regard to their symmetries. However, the nice thing is that we can design interlacings, which match perfectly with the third level tiles (Right). (© Jean-Marc Castera 2015–2018)

Those interlacings are the thinnest possible. We could design wider ones, but they would not fit that well.

We know that the other options provide the tile N4. Let have a look on the option V3.4.

Option V3.4

Here the orientations are set up with a maximum of slim rhombuses in a head to tail configuration, in order to allow the emergence of as many tiles N4 as possible (Fig.~57). Is it the maximum solution? Not sure.
Fig. 57

To the left, the rhombuses orientation. Same as for the previous case, all positives tiles, and N6 as well, occur since the first level. The others negative tiles occur at the second level. Including N4, highlighted on the figure. (© Jean-Marc Castera 2015–2018)

Figure~58 shows a detail of the pattern after three levels of inflation, and the previous level tiles as well. The pattern of these tiles is still imperfect: many second level tiles are cut without respect to their symmetries.
Fig. 58

The same zoom as in Fig.~54. The difference is due to the tiles N4. In light color, the previous level of negative tiles. (© Jean-Marc Castera 2015–2018)

Fourth Inflation Rule: System V4

The Inflation Rule

Here the inflation rules are a little bit tortuous. Indeed, if we use only the two rhombuses (Fig.~59a), at each inflation step, the resulting pattern leave gaps. No overlapping, but octagonal gaps, which have to be filled by the third shape.
Fig. 59

(a) That inflation rule needs the use an extra shape, the decagon. (b): The equivalent full surface of the rhombuses. (c): After the first inflation, we have the positive Tond tiles P1 and P2, and the negative tiles N5 and N6 as well. Note that the covering of the edges is not symmetric. (© Jean-Marc Castera 2015–2018)

The algorithm is not obvious, it needs some testing.

The second line of the figure (Fig.~59b) shows the equivalent surface of the rhombuses, thereby calculating the relative frequency of the two shapes (although this arrangement is not X-Tiles compatible):

We can see on the figure that, after inflation, each fat rhombus gives way to 20 fat and 12 thin rhombuses.

Similarly, each slim rhombus gives way to 12 fat and 8 slim.

Let un be the number of fat tiles and vn the number of slim tiles in a pattern made after n inflations.

After a new inflation, the new pattern has un + 1 fat and vn + 1 slim tiles, with the relations:
$$ {\mathrm{u}}_{\mathrm{n}+1}\ =\ 20{\mathrm{u}}_{\mathrm{n}}\ +\ 12{\mathrm{v}}_{\mathrm{n}}\ \mathrm{and}\ {\mathrm{v}}_{\mathrm{n}+1}\ =\ 12{\mathrm{u}}_{\mathrm{n}}\ +\ 8{\mathrm{v}}_{\mathrm{n}}. $$

Or, if rn  =  un/vn,  rn + 1  =  (20rn  +  12)/(12rn  +  8).

The serial {rn} converges, its limit r satisfies the equation r2 – r – 1 = 0, which solution is the golden ratio.

Not surprising, for a pattern made of that kind of tiles.

That system generates all the different Tond tiles… except N4.

Since the first inflation, we have the positives tiles P1 and P2, and the negatives tiles N5 and N6 as well (Fig.~59c). At the next inflation, we get the rest: P3, N1, N2, and N3.

Now, we can wonder: What is really interesting in that tortuous system?

The answer comes when looking at the superimposition of different levels of the resulting patterns.

Firstly, have a look on what happen in the previous cases, when superimposing two successive inflations (Fig.~60). In these cases, the second level tiles do not fit nicely with the first level. It is better with the system V2 than with the system V1, but still not satisfying. But with the system V4, everything looks very good (Fig.~61). It is not absolutely perfect with two successive levels: one edge of the little tile P1 is still badly broken by the first level edges. But if we consider two levels of the same parity (levels 2 and 4 on Fig.~61 right), there is only a little, quite unnoticeable imperfection.
Fig. 60

Superposition of two successive levels in the cases of the systems V1 (left) and V2 (right). (© Jean-Marc Castera 2015–2018)

Fig. 61

Case of the system V4. To the left: two successive levels (levels 2 and 3). To the right: two successive levels of same parity (levels 2 and 4). (© Jean-Marc Castera 2015–2018)

The complete set of the two level tiles is shown in Fig.~28. It includes all the Tond tiles. Even though the tile N4 do not emerges from the inflation system, it can be designed apart. However, it can only exist at the first level. After that, it is lost in inflation.

The other nice thing here is that the system provides a natural, easy way to draw interlacings (Figs. 62 and 63). Moreover, their width is the correct size, in an aesthetic sense (which was not the case in Fig.~56).
Fig. 62

The interlacings, drawn at the first level (left) and at the second level as well (right). (© Jean-Marc Castera 2015–2018)

Fig. 63

Design of the tiles at three successive levels, with the interlacings on the two first levels. All the lines fit nicely together. (© Jean-Marc Castera 2015–2018)

Working with Decorated Rhombuses

In some cases of inflation systems, it is possible, since the first level, to replace the new rhombuses by a decoration of the initial ones with the resulting second level tiles. That happens when the pattern of the edges of the rhombuses is symmetric. Figure~69 shows the examples from the systems V2 and V3.

The decoration of the fat rhombus in the second case leads to the most common pentagonal pattern, which can be seen in Persian, Arabic, or Indian traditional styles. However, there are simpler ways to generate it (Castera 1996; Castera and Jolis 1991). It happens that the artists have recognized, and used, the second (slim) rhombus (Fig.~68). Sometimes, being fascinated by symmetry, the Moroccan artists wanted that rhombus decoration to be symmetric (Fig.~65c).

See Figs. 64, 65, 66, 67, 68, and 69.
Fig. 64

Decorated rhombuses, rectangles, pentagons, or hexagons can be seen in different styles of traditional patterns. Right, the top panel is made of fat penta-rhombuses while the bottom one is made of slim penta-rhombus. Note that both are not the same scale. (© Jean-Marc Castera 2015–2018)

Fig. 65

That Zellij medaillon (a) on a spandrel of the Andalous mosque in Fes (Morocco) is made of four decorated fat penta-rhombuses (b). These rhombuses can also be seen as made from the two standard decorated rhombuses (c). See also Fig.~69, bottom. (© Jean-Marc Castera 2015–2018)

Fig. 66

The design of this plaster panel at the medersa Bou Inania in Fès (Morocco) is made of fat rhombuses decorated, and 20-pointed stars included into decagons. (© Jean-Marc Castera 2015–2018)

Fig. 67

This is the simplest and most famous pentagonal pattern. It can be seen in different styles and different variations. From left to right: India (Agra), Morocco (Rabat), Iran (Tehran), India (Agra). (© Jean-Marc Castera 2015–2018)

Fig. 68

The two decorated penta-rhombs, in a wooden door (Yazd, Iran). (© Jean-Marc Castera 2015–2018)

Fig. 69

The two decorated rhombuses coming from the V2 X-Tiles inflation system (top), and from the V3 system (bottom). Bottom left, the Moroccan style interpretation of the slim rhombus, with two symmetry axis. (© Jean-Marc Castera 2015–2018)

Note that, because the edges of these decorated rhombuses are symmetric, it becomes possible to use them as a mapping of any pattern made of the two penta-rhombuses. Including any generalized Penrose pattern, binary tiling, or periodic patterns.

The next figures show three applications of these systems (Figs. 70 and 71).
Fig. 70

Detail of a pattern made from the first system of rhombus mapping with Tond tiles. Bottom, the limits of the rhombuses are highlighted. (© Jean-Marc Castera 2015–2018)

Fig. 71

Detail of a pattern made from the second system of rhombus mapping with Tond tiles (which comes from the X-Tiles system V3). Note that this pattern is oriented. (© Jean-Marc Castera 2015–2018)

Figure~72 shows the Moroccan-style variation of the Fig.~71. Note that:
  1. 1.

    There is no more orientation.

     
  2. 2.

    In order to make it a suitable pattern, we have modified the edges. Now, the pattern is perfectly inserted into a periodic structure, the edges are the symmetry axis.

     
  3. 3.

    If the pattern is suitable for Persian artists, it is not still perfect for the Moroccans.

     
Fig. 72

A Moroccan-style variation from the pattern in Fig.~71, with modification of the edges, in order to make the pattern periodic. This pattern uses the modified slim rhombus, so it is not oriented anymore. (© Jean-Marc Castera 2015–2018)

Just a little thing is missing (although it can be easily added… at least along the horizontal edges). Indeed, the Moroccan artists and craftsmen cannot handle that the tiles are cut at the limit of a pattern, even though that limit is a symmetry axis. They want the pattern to be framed by a “river,” made of entire tiles… but this is another story.

To Go Further

Just an example of original pattern (Fig.~73):
Fig. 73

This detail of an original pattern is a mix of Tond to Tond and Tond to Kond systems. (© Jean-Marc Castera 2015–2018)

This pattern is made of a first-level Tond pattern with interlaces, with, at the second level, an alternation of Tond and Kond patterns. It can be considered a mix between a Tond to Tond and a Tond to Kond systems that fit nicely together. Of course, this is fully self-similar, even though only two levels are drawn.

The idea is to have all the different style of pentagonal patterns connected together inside a same self-similar system.

Because everything is in relation with everything, wish those relations be harmonious and peaceful.

Conclusion

The connection between the X-Tiles and the Tond patterns was very unexpected.

Not only it shows that a whole family of complex-looking patterns can be reduced to arrangements of only two kind of so simple tiles but also it gives a solution to a question that the traditional Persian artists – for mysterious reasons – had not solved. Moreover, that solution produces not only two levels but self-similar patterns.

Cross-References

References

  1. Bonner J (2017) Islamic geometric patterns. Their historical development and traditional methods of construction. Springer, New YorkCrossRefGoogle Scholar
  2. Castera JM (1996) Arabesques: Art Décoratif au Maroc, ACR Edition, Paris (English edition in 1999)Google Scholar
  3. Castera JM (2011) Flying patterns. In: Proceedings of the ISAMA/Bridges, Coimbra. Can be downloaded from Bridge’s web site, or from http://castera.net/entrelacs/public/articles/Flying_Patterns.pdf
  4. Castera JM (2016) Persian variations. Nexus Netw J 18:223.  https://doi.org/10.1007/s00004-015-0281-5CrossRefzbMATHGoogle Scholar
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  13. I strongly recommend also the reading of every publication from Antony Lee, Peter Cromwell (available on https://girih.wordpress.com/), Emil Makovicky and Craig Kaplan
  14. With a special mention to the recent brilliant work of the French mathematician Armand Jaspar, available on line: http://patterns-islamiques.fr/

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Independent ArtistParisFrance

Section editors and affiliations

  • Bharath Sriraman
    • 1
  • Kyeong-Hwa Lee
    • 2
  1. 1.Department of Mathematical SciencesThe University of MontanaMissoulaUSA
  2. 2.Department of Mathematics Education, College of EducationSeoul National UniversitySeoulSouth Korea

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