Generating patterns by introducing obliquity I Global outline

Introducing obliquity in a symmetrical universe evokes complexity. A man walking in a symmetrical universe, in a direction oblique to the lattice structure which represents the repetion pattern, alternately crosses a more vertical and a more horizonal side of the succesive repeatunits. The numer of times he crosses a more vertical side is L(arger) and the number of times he crosses a more horizontal line is S(maller). The obliquity of the direction in which he walks, relative to the arrangement of repeatunits can be expressed in the fraction S/L

In this concrete example the obliquity is indicated by the fraction 2/5 and creates the following sequence 
of crossings: LLLSLLS. After seven crossings, the sequence repeats itself. 

The obliquity related complexity (structure of the SL-series)can be transformed in funderfull shapes and quite intriguiing color mixurus, especially when we introduce this obliquity in the plane-symmetries P4 and P6. The relation between the obliquity (SL-series) and the resulting shapes can be immediately understood if one decomposes the concerning fraction in a series of quotients by applying a proces of continued fraction on it. That means: dividing the denominator of that fraction by its numerator, its numerator by the restof that first division, etc., until the rest becomes zero.In P4 this transformation can be characterized as 'the crossing-up of crosses'. In P6 it can be characterized as 'the cutting up the honeycomb'.

the crossing up of the cross

When we intruduce obliquity in the plane-symmetry P4, (two types of 4-fold rotation axes and one type of 2-fold axes) a complex intersection-patterns is generated in which lines in one orientation are crossed by lines in the other orientation according to a hierararchical layered 'jumping-pattern'. This complexity can directly be transformed to 'fractal-like' figures by successively drawing and not drawing in the succesievely subscribe and not subscipe the linesegments which are the result of the intersectionpattern.

How such a fractal-like figure looks, depending on the introduced obliquity, directly can be predicted by the q-series which is the result of applying contined fration to the ratio in which the obliquity is expressed. In principle, every extra q-value in the series of q-values, provided that it is even, yields an extra level in the 'crossing-up' of the outline of the cross .The value 2 is the basic value. By a higher value at a certtain level, the crossing up at that levels shows a repetition pattern, consistent with the amount of the value at that level. If the q-value is 8 at a cetrtain level, the crossing-up at thatlevel repeats itself 4 times.

When for the first time an unequal value occurs in the serie of q-values, the crossing-up-proces breaks down for a moment and 'ringing up' enters within the proces. After the rings are take up in the structure of the cross, by the following q-valuesd added, the proccess of crossing-up continues, but now at the level of the rings.

the cutting up of the honeycomb
Quite different patterns, but just as intriguing, can be generated when we introduce obliquity in P6. When we do so, an intersectionpattern arises like the one besides. In this case too, each line is divided in pieces by the intersecting lines. The rose and pink set of intersecting lines each show a specific jumping pattern. The jumping pattern can be transformed into figures by successively drawing and not drawing peaces of line. The resulting pattern can be explained by the ratio which represents the obliquity, by decomposing it in a series of q-values by means of contined fraction.

The relation between the series of q-values and the resulting pattern can be called miraculous. Startingpoint in every case is the honeycomb-like structure which is represented bij the q-series


in which cells occur which consist of one hexagon or a combination of three hexagons.
Patterns generated by other series of q-values can be conceived as being made up of excerpts from that honeycomb structure.

Below we explain how that fragmentation pattern is related to the series of q-values representing the obliquity. The explanation is given for the case in which the series of q-values consists of three values. How much cells each fragment includes, depends on the middel (for last) q-value, provided that the last q-value is 2. The greater in that case the middel q-value is, the more cells are included in each fragment. If the last q-value is greater then 2, no cells are included in the fragments (except the single hexagon). The first q-value adds fragments which can be conceived as 'narrows' of the fragments which are represented by the middle q-value

   q1=1 q2=3 q3=2 

    q1=1 q2=3 q3=4 

   q1=3 q2=3  q3=2 

II Further elucidation

In the following we shed more light on the methodological aspects of this approach. Global outline

1 Introducing obliquity

2 Lattice structures based on obliquity

3 Crossing-up of the cross

4 Cutting-up of the honeycomb

1. Introducing obliquity

Miller indices
Abbe Huay who was the founder of crystallography had been the first to presume that crystal faces
often held a slanting position in relation to the pattern in which the building stones of the crystals 
involved were arranged and that this obliquity could be expressed in simple integers. Though his 
presumptions were correct, he was not able to clarify them in a way which could withstand the test of time.

As insight in crystallography matured, it was a Professor of Mineralogy at the University of Cambridge, 
W.H.Miller, who - in the course of the 19th century, and profiting from the new insights in this matter - 
developed a useful system for the indexing of this obliquity of crystal faces by means of simple ratios.
The integers in these ratios are called 'Miller Indices'.
Miller started from the concept of repeat-units, as was traditional in crystallography. The various types 
of plane-symmetry are considered to be built up of repeat units which always have the shape of parallelograms.
Starting from this viewpoint, the indices of obliquity are equal to the number of times in which the vertical 
and the horizontal sides of the repeat unit are traversed by crystalfaces.


Walking in de direction 2/5 generates the sequence SLLSLLL.

The obliquiy in the example (besides) is reprsented by the fration 2/5. That means that seven events occur in cetrain succession and againand aguin in that succession while walking over the stroke-path: 5 L-events (=crossing a purple line) and 2 S-events (=crossing a green line).After seven events, the pattern is repeated The way in which the L- and S-events alternate is in principle hierarchically layered. The relation between this hierarchically layeredness and the fraction indicative of the obliquity can be made transparant by representing the fraction as a continued fraction.

Because we need to obey to the rules of symmetry, the in obliquedirection running stroke schould be repeated in a regular way. It intersects the puple sides of the repeatunts L +=5) times and the green sides S+=2) times.So the obliquity is also represented by this intersectio-ratio (indices of Miller) of these sides.

The effect of obliquity can be transformed in some sense by introducing repeateunits of differents size.Now the events which are generated bij walking over a stroke-path are crossing a large (L) and crossing a small(S) repeatunit. The pattern of alternation of L- en S events remains the same.

We can represent the obliquity by a line instead of by a stroke. The 7 events start and end at a corner of a repaetunit. When we decide that the event at the startingpoint is a S-event, that at the endingpoint musty be a L-event. Such a sequence of seven events , generated by the line while running through the plane, can be conceived as the 'motif' of the plane-symmetry. the endingpoint is a Levent.

Here too, the efefct of obliquty cn be transformed by introducing repeatunitsof different sizer.


Every direction (obliquity relative to arrangement of repeatunits) though a plane-symmetry, represented by an unique fraction, generates an unique sl-series:

1 Let y = L
2 For n= 1 to L
3 Let y = y + S
4 If y > L then let y = y - L : Print S
5 Print L
6 Next n

Continued fraction

The way the structure of the SL-serties is related to the fraction which represents the direction in the plane-symmetry  seems 
ondoorgrondelijk but becomes volstrekt clear if we rewrite the fraction as a continued fraction. Let us explian how that is done. 
However before we need to explain the continued fraction

Every fraction can be represented by a number of quotients (q1, q2, q3,.. ),resulting from a process of successive divisions. To begin with, the denominator of the fraction is divided by its numerator (divisor); then the divisor of that first division by its rest, etcetera, untill the rest is zero. This 'decomposition' of the fraction can represented as a continued fraction or 'Kettenbrüche'. In the example the fraction 103/356, this fraction can be 'decomposed' in the following series of q(uotient)-values: q1=3; q2=2; q3=5; q4=4; q5=2. Written in the notation of a 'Kettenbrüche':

103/356 = 1

3 + 1

2 + 1

5 + 1

4 + 1


On internet there can be fond a site onwhich for every fration the continued fraction is represented in this notation-systematiek.

Relation between the sl-series and the continued fraction

In series of L/S-symbols there always are n or n+1 'L'-symbols between two 'S'-symbols. At the following levels the values n and 1 become symbols themselves; here too, a 'n'-symbol shows up alternately n or n+1 times between two '1'-symbols. Ecetera. At the last level, the number of occurrences of a 'n'-symbol between two '1'-symbols is constant (does not alternate between n and n+1) and equals n at this level.The relation between the continued fraction and this rewritten L/S-series, is now a matter of course: the number of chains in the continued fraction is equal to the number of strata in the L/S-series and the quotient of each link is equal to the value n of the regarding stratum.
In the following some examples. First some examples of series with one strata, then onee with two stata, then some with three and finally one with four strata.

examples of Sl-series with one stratum







examples of Sl-series with two strata



q1 = 2
q2 = 2



q1 = 2
q2 = 3


q1 = 3
q2 = 2
example of Sl-series with three strata



q1 = 2
q2 = 4
q3 = 3
example of Sl-series with four strata



q1 = 2
q2 = 1
q3 = 1
q4 = 2

"q" or of "q+1" dominant

As may become clear from the preceding tabel, every stratum in a SL-series can be conceived as builded up out of "q" and "q+1" elements. One of the two is dominant at a certain stratum. As the next q value is 1, at the stratun belonging to the precweding q-value, q+1 is dominant.If that value is greater than 2, de "q"-element at the preceding stratum is dominant. If that value is 2 , nor the element "q" nor the elemant "q+1" is dominat. In this case theren is but one "q"-elemnt and one "q+1" elemant at tha preceding statum. So the value 2 has specific meaning, in the sense that the belong stratum represents exeptions. At the last strutom there is no "next q-value. this stratum consists of only one "q-element.

Different form the concept of Miller indices When the first q-value in the continued fraction is one, the enumerator in the fraction is more than half the value of the numerator. In those cases we can conceive the builing up if the series as the number of times lm-elements are alternated by a ls-element. In these cases the lm -element occurs n or n-1 times between two LS- elements. The value s in these cases is represented by the diagonal line in the repatunits, which itself als can be cinceived as onde of the outlines of a larger repeatunit. In these cases, is one, n+1 is dominant in de SL-series at the different layers

2. Lattice structures based on obliquity

In p3 there occur 3 types of 3-fold rotational axes.
In p4 there occur 2 types of 4-fold rotational axes and 1 type of 2-fold rotational axes (the last in 2 orientations in space).
In P6 there occur 1 type of 6-fold axes, 2 types of 3-fold axes (in 2 orientions in space) and 1 type of 2-fold axes (in 3 orientations in space)

Escher realised beautiful expressions within these arrangements

the numer of oriemtations in pace of a motif

The number of orientations in space of the motif differs in each of these three symmetry types and equals the highest 'foldness' of rotationaxes which occur there. In p6, motifs have six different orientations in space; in p4 four different orientations and in pr 3 different oreintations and in p2 two different orientations.

Lattice-structures in P3, P4 and P6

Direction-related SL-series can get more 'richness in structure' in symmetry types in which rotations occur besides shifts. In such symmetries, directions ' run' in different orientatations which results in quite intriguing interactios between SL-series. Before we shed some light on this, we should spend some attention the concept of direction within these types of symmetry. In these types of symmetry the concept of direction is complicated and we shoud first shed some light on it.

what 's meant by the concept of "direction" in P3, P4 and P6?

The word "direction" in first instance indicates the degree of obliquity relative to the repeatunits which represent the regular shiftpattern which in every case underlies a planesymmetry (wallpapergroup). But dependent on the occurence of other element of symmetry (rotations, reflections, slide-reflections), a "direction" besides a degree of obliquity can possess different orientations in space. In those cases a direction is defined by its degree of obliquity and its orientation in space

6 orientations
in space

obliquity = 2/5

orientation = II

obliquity = 2/5

orientation = IV
The values 2 and 5 on the coordinates, for each orentation of course can be reversed. As long as there occur no reflections besides rotation in, the way the valyes are positioned needs to be consistent in the sense that it must be in accordance with the rotationsymmetry in question.

It is convenient to conceive the line representing the direction, marked out by two(highst foldness) rotationaxes as the "motif" (line-motif). On a mitif-line a number of events occurs, walking from the roationaxis at the 'departure' to the rotationaxis a the 'arrivel'

lattice structures generated by direction
Chosing a direction (line-motif) in P3, P4 or P6, this direction automatically occurs in four different orientations.The result is acomplex lattice-structure. To onderstand how patterns can be generated out of the laticestructures, i we need to shed som light on the structure of these lattiices. There are two important characteristics:, which both can be conceived at the level of the line-motif:
1) the pattern of shifts in the way a linemotif is traversed by line-motifs runningin other orientations
2) the pattern in the scatterin of sets of identical points on the line-motif. Beneth, these characteristics are elucidated for P4. Under the knob at the right, a closer look is offered at these characteristics in P3 and P6.

For elucidation of
in P3 and P6 click on the knop

pattern of shifts in the intersectionstructure

Every line-motif is intersected by other line-motifs at regular distances (one or two lin-emotifs intersect at ech point, depending on the symmetry type ). This results in a pattern of upward and downward jumps.
The buiding up of this shift-pattern is explained in more detail under the knob at the right

(click on the knop)
The smalles distance we conceive as the unit of distance.
The traversing line-motifs show shifts in their position relative to each other. This pattern of shifts in the traversing line-motifs can be represented by a series of symbols u (upward shift) and d (downward shift).

In p6 and P4 each line-motif inheres two orientations. On such lines at the midpoint there is reversion in the succession of SL-events,related to reversion of what's at left and at right.

The complexity of the u/d-series can be derived from the ratio between the shiftdistances of the u-shifts and the d-shifts. These distances can be calculated on base of the following two formula's:

(1) ax +by= + 1

(2) s=bx+ay

where x and y are the values in the ratio for obliquity. In the example besides s(d)/s(u)=12/17. When we represent this fraction as a continued fraction (series of q-values), the relation between the structureof the u/d-series and the indices for obliquity become clear.

3. Crossing-up of crosses

Mathematial rules

In this approach of generatiing patterns out of obliquity based lattice-structures, the motif is a line-segment between 2 rotation axes. Such a line is intersected many times by line-segments running in an other orientation. So the line is diveded in lots of equal pieces. Of these , we successively draw one and pass over the next . The result are cristal like figures, the shape of which depends on the obliquity of the line-motif (S/L-ratio.)

In the scheme beneath, some basic elements of these figures are elucidated in relation to the series of q-vaqlues which represent the S/L-series as indication of the obliquity.
Relation between S/L-series and U/D-series
The structure of the U/D-series resembles that of the S/L-series The series of q-values representing the U/D-series, is composed of two times the series of q-valoues representing the S/L-series, with a point of reversal half way. But the first q-value within the total series of q-values always has to be diminished with 1 (see examples). The picture beneath explains this reversal in the q-value-series.

The fraction for obliquity = 2/5. There are two levels in the corresponding S/L-series and within each, half of the pattern of the U/D is generateed in such way that de SL-series and UD series match.

Example 2/7,

S/L = 2/7

q1 = 3
q2 = 2

q1 = 2 (-1)
q2 = 3
q3 = 3
q4 = 2

Example 4/11

S/L = 4/11

q1 = 2
q2 = 1
q3 = 3

q1 = 3 (-1)
q2 = 1
q3 = 2
q4 = 2
q5 = 1
q6 = 3

At the left the basic elements in buiding up of cristalline patterns are presented: squares and 'notched' lines. In the categorie of noched elements we discriminate z-wise-notching and u-wise-notching. The occurence isrelated to the u/d ratio (explanation u/d-ratio under Euler-knob2. The relative frequency of occurense of thsee three elements in a cristalline pattern and the types of combinations between them, is directly related tot the u/d-ratio.
To elucidate this relation we first must focus for a while on the u/d-ratio itself. Togethe these values represent the total number of line-units on a line motif . We call that total numer T. So U + D = T). The value t always must be uneven, otherwise the line-motif has a fourfold rotation axis in the mid. And in that case we are taking of two linemotifs with each T/2 line units. So in every case the value u must be even and the value d uneven, or the reverse.

For understanding the structures which are genereated , it is conceniant two look how pairs of linemotifs running in one orientation intersect with pairs of lines running in the other orientation
In the position of pairs of line-motifs running in one one orientation are two options: the segments which are drawn lie on the same level or they lie or the differ one unit of distance. But the can only have that position to same extent, because there is a shift-relation between them. So the position shifts after a distanc of u. The same applies to pairs of lines running in the other orientation.
If the valu D (largest disctance is even and value U (smallest disctance is uneven, the main pattern is that of a zigzag-wise nothching, resulting fron intersection of pairs of lines which have unequal levels of drawing in line-segments. In the same sense, when the value U is ebven and D is uneven, the sguares dominate in the pattern
In both cases, some of the interactions between pairs results in a u-wise notching, resulting from an intersection in whihich a pair of lines in one orientation is equal levveled and in the othor direc tion is un equal levelled. But the elemnts are never dominant and are connected to te S-value in the S/L-fraction.

The different level combination within pairs of line-motifs at different levels in the U/D-series can occur and cause combinations in combinations which were created at the level beforefore When the u value at a certain leven is even, at the next level the genereated combinations are like shown in figure a).

When the u value at a certain leven is uneven, at the next level in which the u-value occurs, the generated combinations are like shown in figure b) This last type of combinations is far more scanty, becaus of the fact that even T-values are ' notin the game' .

different levels of cristalline figures occur besides and within each other. The U/D-series generates different ' circuits of right/left/bendig of successive linesegments. When the rotational axes are connected with the first line-segment drawn in, all lower level circuits are enclosed within the highest level circuit. In the other case there occur also circuits besides the highest level circuit. Thenumer and types of circuits can be graspecd by analysing the way the successeve intersectionpoint at the strat of a segment ar linking together ongroun of value enlargement becaus of shiftdistance. in the higest level circuit

II Examples

Hierarchy and redundancy


Every time a link is added to the continued fraction, another layer is added to the figure complexity that was being generated.




Succession of of even and oneven q-values in the series of q-values

Relation between grid in the outline and overall contour of the outline