TOND to TOND
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 selfsimilarity; 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 XTiles.
Keywords
Persian pentagonal patterns Multilevel patterns XTiles Selfsimilarity Binary tilingIntroduction: 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 fivefold 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).
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
The Tond Family
Multilevel Patterns. Reminders, and a New Case
A firstlevel pattern is made of largescale tiles, each one cut into smallscale tiles that fit perfectly with the adjacent ones. That is the secondlevel 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 selfsimilar.
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 selfsimilar.
Two Kond SelfSimilar Systems
We want to complete the traditional twolevel Kond + Sholl patterns in order to make them fully selfsimilar. 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.
There is actually a slight difficulty with the Sormedan (tile N3). The resolution requires that the “Golden rule” be broken, which asserts that each firstlevel vertex be the center of a secondlevel star.
A Third Type of Kond SelfSimilar System
Transitions Between Different Families
In a twolevel 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.
[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 selfsimilar. First, here are some examples seen in Iran.
[A] = > [A] (from Kond + Sholl to Kond + Sholl)
[A] = > [B]. From Kond + Sholl to Tond or, More Often, from [A1] to [B]
[B] = > [A1]. From Tond to Kond
Generalization of the First Example
Generalization of the Second Example
Many twolevel 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 twolevel pattern in the traditional heritage (at least in Iran). If one exists, it is certainly well hidden.
In short: We now have transitions, fully selfsimilar, 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 “XTiles” system (Castera 2011).
XTiles
As the abovementioned publication shows, any Tond pattern can be made from a couple of rhombi decorated with two simple lines. Let them be the “XTiles,” while the lines are the “XLines.” 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 nonperiodic. But the XTiles 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.
Definition
The XTiles and the Tond Traditional Family of Pentagonal Patterns
Transition from Kond to Tond with the XTiles
Tond to Tond Transition Through the XTiles
A binary pattern can be made tiletotile, 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 XTiles, this process leads to an infinite series of Tond patterns at decreasing scales. This solves the problem of Tond patterns’ selfsimilarity. However, that is certainly not the unique way to produce selfsimilar Tond patterns.
SelfSimilarity of TOND Patterns Through the XTiles
Principle
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 XTile’s inflation rules.
First Inflation Rule: System V1
The Inflation Rule
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.
Order of Appearance of the Tiles
The TwoLevel Tiles
Second Inflation Rule: System V2
The Inflation Rule
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
Order of Appearance of the Tiles
The TwoLevel Tiles
Looking at two successive inflation levels, we got the idea to define an inflation process directly on the Tond tiles, without going through the XTiles. 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
We are going to discuss those questions with the next inflation rule.
Third Inflation Rule: System V3
The Inflation Rule
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
 1.
The relative frequency of the different type of tiles is variable.
 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.

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
The TwoLevel Tiles and the Interlacings
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
Fourth Inflation Rule: System V4
The Inflation Rule
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 XTiles 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 u_{n} be the number of fat tiles and v_{n} the number of slim tiles in a pattern made after n inflations.
Or, if r_{n} = u_{n}/v_{n}, r_{n + 1} = (20r_{n} + 12)/(12r_{n} + 8).
The serial {r_{n}} converges, its limit r satisfies the equation r^{2} – 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.
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.
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).
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 pentarhombuses. Including any generalized Penrose pattern, binary tiling, or periodic patterns.
 1.
There is no more orientation.
 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.
If the pattern is suitable for Persian artists, it is not still perfect for the Moroccans.
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
This pattern is made of a firstlevel 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 selfsimilar, even though only two levels are drawn.
The idea is to have all the different style of pentagonal patterns connected together inside a same selfsimilar system.
Because everything is in relation with everything, wish those relations be harmonious and peaceful.
Conclusion
The connection between the XTiles and the Tond patterns was very unexpected.
Not only it shows that a whole family of complexlooking 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 selfsimilar patterns.
CrossReferences
References
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 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
 With a special mention to the recent brilliant work of the French mathematician Armand Jaspar, available on line: http://patternsislamiques.fr/