repeatunit in P3 
repeatunit in P4 
repeatunit in P6  
There are 3 values in the outlines of the repeatunits: S, M and L. The value M always equals LS. In the repestunit, the values L, M and S are connected to the three types of rotationaxes we discriminate. In a stripe course, an Mevent always occurs between two Levents. The S and M values are exchangable. This means that in some cases the Svalues are connecting the t2fold rotationaxis in the outline of the repeatunit and in other cases the M/value. When the S/Lration is greater than 1/2, the Svalues are connecting the rotatationaxis; when the S/Lration is smaller than 1/2 the Mvalues provide for tyhe connection. 
Each level in the continued fraction has its own L, M and S values, as shown in the example beneath: 
In each of the repeatunits the latticestructure si divided in, we tolerate but one orientation of the linemotif. The results of this division of orientation of direction is thwe occurance of stripcourses or stripcircuits.  
In each of the three symmetry types, the unit generates a stripecourse with a closed character
, as shown beneath. In these examples, for P4 and P6 the lengthunits with a value M are not drawn in. They
play no role in the ' steering' of the stripcourse, because the course traverses these lenghtunits
under right angles, thus not influencing the course of the strip. But to get fully grasp on patterngeneration, one should take acount of the Mvalues
 

Centre of the stripecircuit
One of the three rotational axes lies in the centre of the stripcircuit, namely the axis that is connected to the value which is even. For every stripecircuit per definition only one of the three values  S, L or M  is even. In the example beneath this regularity is illustrated in P6
L= sharp; S= obtuse
S=1
M=2
L=3
Mvalues connected to the 2fold rotationaxis
S=1
M=1
L=2
Lvalues connected to the 6fold rotationaxis
S=0
L=1
M=1
Svalues connected to the 3fold rotationaxis
Wonderfull stripe courses can be generated when the length values in the repeatunit become larger, as shown beneath for the p6lattice: q1=2
q2=1
q3=1
q4=1
q5=1
q6=5 q1=2
q2=1
q3=1
44=1
q5=7
q1=4
q2=3
q3=5 q1=2
q2=3
q3=5
Representing stripecourse by LSseries
The course of a strip is dependent on the indices of obliquity. The relation between the S/Lfraction and the strip course becomes clear if we if we apply the proces of continued fraction on this fraction, which results in a series of qvalues. In series of L/Ssymbols there always are n or (n +1) Lsymbols between two Ssymbols. At the following levels the values n and 1 become symbols themselves; here too, the 'n symbol' shows up alternately 'n or ('n+ 1) times between two '1symbols'. At the last level, the number of occurrences of a "nsymbol" between two "1symbols" is constant ( does not alternate between 'n and 'n+1). The relation between the continued fraction and this rewritten L/Sseries, is now a matter of course: the number of chains in the continued fraction is equal to the number of strata in the L/Sseries and the quotient of each link is equal to the value n of the regarding stratum. Application of this general model to the stratified spiralling structure which is generated by the fraction 5/17 in P6. There are three strata in this spiralling structure. The sixfold rotation axes are centra of spiralling at level I end level III; the twofold axies at level II.
Hierarchically spiralling
The structure of the spiralling is determining the the esthetic quality of the resulting pattern, so we must shed some more light on it.At every layer in the spiralling, the shape of the spiralling follows the shape of the stripe circuit at the level beforde. And concequently, the axes which is centre of spiralling, is the axis which lies in the centre of this l of that stripcircuit on the level before. The first layer hower is exceptional and we must spend some words extra on it. The first level in the stripcircuit follows the shape of the stripcircuit indicated by the value L=1 and M=1 and S= 0 (Value S diminished to zero). In the case of p4 the shape of the spiralling at level 0 is in any case that of the square and in P3 that of the hexagon. In p6 it depends on the type of line unit the values L, M and S at higher levers lie on: Three options are possible:  if the Lvalues are connecting the 3fold rotationaxis and the Mvalues the 2fold axis, the shape at level 0 is that of the hexagon.  if the Lvalues are connecting the 6fold rotationaxes and the Mvalues the 2fold rotationaxes, the shape at level 0 is that of the trangle  if the Lvalue is connecting the 6fold rotationaxes and the Mvalues the 3fold axes, or the reverse, the shape at level 0 is that of the rhomb.
Example
In the example beneath , the utlimate stripecircuit is 13/69. There are three layers in the building up of the S/Lseries:
q1=5To make clear the buiding up of the spiralling structure we step by step build op the stripcourse in its three layers:
q2=3
q3=4
As already stated, at every layer the stripecircuit,in its spiralling, follows the shape of the stipcircuit at th level before.
Hierarchically colormixuring
Number of colors in symmetrical arrangement
(p.m.)
Hierarchically mixuring
The stratified property of a stripcourse and the resulting strippattern can also find expression in stratified colourclustering.
If more than two colours are introduced in a pattern, of which not all are involved in the spiralling action around a certain type of axis, colourclustering may occur. In the example two patterns in p3 are shown: the number of strata in pattern a is two ; in pattern b four. At level II simple colourclusering presents itself. At level IV an extra stratum in colourclusteringshows up. The structure of the stratified colourclustering at level IV is as follows: Both levels 'take place' around the same type of threefold rotation axis , which is involved at two levels in the spiralling structure. At the higher level of involvement in this structure, the lower level in colour clustering takes place :
 colour 1 and colour 2 wind around each other;
 colour 1 and colour 3 wind around each other
 colour 2 and colour 3 wind around each other
At a lower level of involvement in this structure, the higher lever in colour clustering takes place. Now the three clusters (each composed of two colours) wind as a whole around each other:
 cluster (1+2) and cluster (1+3) wind around each other;
 cluster (1+2) and cluster (2+3) wind around each other;
 cluster (1+3) and cluster (2+3) wind around each other.
Reversion of sharp and obuse
In P6, for each strip circuit a complementary stripcircuit can be generated, in which the angles of 60 degrees are replaced by angles of 120 degrees and vice versa, as is shown by this overview of some simple fractions.The fraction remains the same, then, but the number of obtuse angles is linked to the other value in the fraction, and vice versa. So at one extremity of the value spectrum (number of obtuse angles zero) the triangle appears. At the other extremity (number of acute angles zero) the hexagon appears. In de middle of the valuespectrum (equal number of obtuse and acute angles ), only one stripcircuit exists, the diamond, that has no complement
Escher, too, had experimented with the possibility of reversion of the acute/obtuse ratio in strippatterns, as appears from one of his notebooks.
In p4 and p6 there is  naturally  no reversion of sharp and obuse:
In decorational art through the ages there occur patterns inP3, P4 and P6which can be positioned within the summary tables
example in P3:
example in P4:
example in P6:
Swastica
The swastica in many versions appears as result of patterngeneration within direction based latice structures. It is regrettable that the Nazis in the past century has besmirched the symbol of the Swastica so much. It's a symbol deeply rooted in cultural expression of mankind . It was used by many cultures around the world, including in China, Japan, India, and southern Europe. The swastika symbolized life, sun, power, strength, and good luck, until the Nazis used this symbol to represent the darkest side of humanity. The picture at right shows a decoration with swasticas on a miedieval mosque, dating from far before the rise of the Nazism.
Intriguing colormixures
Branching
OSreversion quite complex
By the introduction of more variables in the area outline, it becomes possible to realize quite complex forms in the acute/obtusereversion, related to the symmetry of the stripcircuits which are generated by these variables, as this overall picture of some simple examples, based on the areaoutline presented in fig. 10a, shows:  For some fractions the reversion takes place in one and the same stripcircuit. In these cases, in certain parts of the stripcircuit the pattern of acute/obtuse alternation is the reverse of that in other parts. Nice examples may be found at the one extremity and in the middle of the valuespectrum: the diamond and the trapezium. The first is symmetrical and consists of four parts two of which (in opposite positions) are equal and the reverse of the other two.This causes the sequence acute, obtuse, acute , obtuse, that is typical for the diamond. The last is asymetric and consists of two parts which are the reverse of each other : the acute/obtuse sequence in the one part is followed by the obtuse/acute sequence in the other part, which causes the trapeziumlike outline of this stripcircuit.  For other fractions two different circuits occur in a pattern. In the one, the obtuse angles come in the place of the acute angles in the other, and vice versa. The combination of hexagon and triangle is a nice example, in the other extremity of the valuespectrum.