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Sequel to Group Theory
As always, we start with reminding the reader about the "Important Remark" near the end of Part III of Group Theory (To see it, click HERE and then go to (end of) Part III ), a Remark concerning the direction of reading products of group elements, like, say, apq. We read such products (from that Remark onwards) from back to front. Thus (with respect to apq) first q, then p, and then a.
Infinite two-dimensional periodic patterns, or Ornaments (sequel)
In Part XXVIII we investigated antisymmetry, subpatterns and subgroups of a pattern representing the plane group P4 .
We obtain this pattern when we consider a different shape of the fundamental region (but equally valid as the one we considered earlier (previous document)), i.e. a different shape of the areas representing group elements. The next Figures indicate this.
Figure 1. Areas (in an alternative, but equivalent way) representing group elements of the generating P4 pattern of Figure 12 of the previous document Purple lines, together with the black lines (representing the motif units) delineate these areas. See also next Figure.
Figure 2. Same as previous Figure. A few areas representing group elements highlighted (blue).
The next two Figures again show the areas representing group elements.
Figure 3. Areas (yellow, green) representing group elements (as in the previous two Figures), i.e. group elements of the generating P4 pattern of Figure 12 of the previous document . The colors here do not signify symmetry features, but only serve to highlight the areas representing group elements.
Figure 4. Same as previous Figure. Alternative colors used to highlight areas representing group elements.
The next Figure highlights some 4-fold and 2-fold rotation axes of our P4 generating pattern (with its alternative but equivalent fundamental regions representing group elements, as just established).
Figure 5. Some 4-fold and 2-fold rotation axes of the generating P4 pattern with its new fundamental regions. (Recall : "generating", because from it an antisymmetry pattern will be generated).
In deriving the antisymmetry pattern we will use the color permutation (Light Blue Dark Blue), with respect to the background color. We will set the initial background color to be light blue :
Figure 6. Generating P4 pattern, with the shape of the fundamental regions (representing group elements) as established above. One such region (highlighted by yellow coloring) is chosen to represent the identity element. The initial background color is set to be light blue.
Figure 7. Generating P4 pattern, with the shape of the fundamental regions (representing group elements) as established above. Illustration of the generating translation.
Figure 8. Generating P4 pattern, with the shape of the fundamental regions (representing group elements) as established above. Illustration of the generating quarter-turn ( 900 anticlockwise rotation about the axis indicated by a small solid blue square). See also next Figure.
Figure 9. Generating P4 pattern, with the shape of the fundamental regions (representing group elements) as established above. Identity element indicated as 1 . Illustration of the generating quarter-turn p ( 900 anticlockwise rotation about the axis indicated by a small solid blue square). Repeated application of p yields p2 ( 1800 rotation about the same axis) and p3 ( 2700 anticlockwise rotation about the same axis).
The P4 symmetry pattern of the previous Figures can be generated as follows:
So indeed we have two generators of our initial P4 pattern, viz. the translation as indicated in Figure 7 , and the quarter-turn as indicated in Figure 9 .
Now we are ready to derive the (second) antisymmetry pattern from the generating P4 symmetry pattern : Like in the previous case we do this by replacing the generating translation by its corresponding antisymmetry translation, where the antiidentity transformation e1 is interpreted as the color permutation (Light Blue Dark Blue) (cycle notation) with respect to the background color, which is set as being initially light blue. Newly generated light blue elements will be colored yellow, and later be restored to light blue. We will derive the antisymmetry pattern in several steps.
Figure 9. First phase of the derivation of the antisymmetry pattern as defined above, from the generating P4 pattern with the fundamental regions shaped as established above. Successive application of the generating quarter-turn (where the identity element is supposed to be given) results in a four-fold rosette consisting of light blue elements (because the generating quarter-turn is not an antisymmetry transformation). These elements are, for the time being, colored yellow. Compare with Figure 9 .
Figure 10. Second phase of the derivation of the antisymmetry pattern as defined above. Application of the antisymmetry translation to the elements already obtained earlier, results in a band of elements involving color alternation.
Figure 11. Third phase of the derivation of the antisymmetry pattern as defined above. Application of the generating quarter-turn to the elements already obtained earlier, results in a second band of elements perpendicular to the first.
Figure 12. Final phase of the derivation of the antisymmetry pattern as defined above. Application of the antisymmetry translation to the elements already obtained earlier, completes the whole antisymmetry pattern.
The next Figures explain this final phase.
Figure 13. Explanation of the final phase of the derivation of the antisymmetry pattern as defined above. The antisymmwetry translation was applied to the elements newly obtained in Figure 11 (NE band of elements). This results in bands of elements (indicated by purple lines) with color alternation, bands parallel to the antisymmetry translation, as the next Figures show.
Successive generation of the bands of elements by and parallel to the antisymmetry translation ( The previous bands erased again for clarity) :
Figure 14.
Figure 15.
Figure 16.
Figure 17.
Figure 18.
Figure 19.
The next Figure gives the final result as obtained earlier.
Figure 20. Same as Figure 12 , but with the indication of the generating quarter-turn removed.
When we now restore yellow to light blue, we get the following :
Figure 21. Same as previous Figure, but with yellow restored to light blue.
If we now interpret the color permutation as being (White Black) (cycle notation) instead of (Light Blue Dark Blue), we get the antisymmetry pattern as it was depicted in JABLAN, 2002 :
Figure 22. Antisymmetry pattern derived from the generating P4 symmetry pattern as it was depicted in Figure 12 of the previous document . It can be obtained by replacing the generating SE translation by its corresponding antisymmetry translation, where the antiidentity transformation e1 is interpreted as the color permutation (White Black) (cycle notation) with respect to the background color which is set as initially being white.
( Adapted from JABLAN, 2002)
The next Figures give possible point lattices of our antisymmetry pattern.
Figure 23. Point lattice (indicated by strong red connection lines) of the above derived antisymmetry pattern. It is a square lattice. Compare with the lattice of the generating P4 pattern depicted in Figure 14 of the previous document .
There is, however, a possible point lattice, underlying our antisymmetry pattern, that has smaller meshes, and is therefore preferable :
Figure 24. Alternative point lattice (indicated by strong red connection lines) of the above derived antisymmetry pattern. It is also a square lattice, and its meshes are smaller than in the previous Figure. Compare with the lattice of the generating P4 pattern depicted in Figure 14 of the previous document . Translations of the pattern are given by the edges of a lattice mesh.
In order to identify the antisymmetry pattern, we isolate its subpattern of light blue elements (Figure 21 ) containing the identity element :
Figure 25. Subpattern of the antisymmetry pattern (as derived above) consisting of its light blue elements, one of which is the identity element (red).
We will determine the symmetry of this subpattern and whether or not it represents a subgroup of the generating P4 pattern. Recall that a subgroup S of a group G is a group such that all its elements (symmetry transformations) are also elements of G, and it is a subgroup in the strict sense if there are elements in G that do not occur in S.
Figure 26. Point lattice of the subpattern of the antisymmetry pattern (as derived above) consisting of its light blue elements, one of which is the identity element. The lattice is a square net. It is identical to that of the antisymmetry pattern (Figure 24 ), which means that the translations of the subpattern are also present as symmetry transformations in the antisymmetry pattern. What we see here is a pattern consisting of a D4 motif that is repeated according to a square point lattice. As such the pattern would have a symmetry according to the plane group P4mm . But, in the Figure we can see that the mirror lines of the motifs are not mirror lines of the whole pattern (See next Figure), which means that these mirror lines are absent in the pattern. Therefore the motif has -- a it is a constituent of the whole pattern -- no D4 but C4 symmetry (cyclic group of order 4). So what we have is a pattern consisting of a C4 motif that is repeated according to a square lattice. And thus the symmetry of our subpattern of light blue elements is according to the plane group P4 .
Figure 27. The mirror lines of the motifs (one set shown for one motif) of the subpattern of light blue elements are not mirror lines of the whole pattern. So they cannot count as symmetry elements of the pattern, making the symmetry of that motif in effect (i.e. in the symmetry context of the whole pattern) four-fold without reflection lines, i.e. a symmetry according to the point group C4 . This motif is repeated according to a square lattice, determining the symmetry of the whole pattern to be that of the plane group P4 .
The next two Figures show that the translations of the subpattern of light blue elements are also symmetry transformations of the generating P4 pattern.
Figure 28. The sum of two independent translations of the generating P4 pattern is also a symmetry transformation of this same pattern (because consecutive application of two elements of a group results in an(other) element of that same group). It is -- as turns out -- also a symmetry transformation of the subpattern of light blue elements. The small solid blue squares indicate equivalent points of the pattern.
The sum of the two independent translations (just referred to) of the generating P4 pattern can be seen to be present in it by considering an alternative lattice. The edges of the meshes of this lattice then represent this sum :
Figure 29. Alternative point lattice (indicated by red connection lines) of the generating P4 pattern. The edges of the meshes of this lattice represent translations. These latter are sums of translations also occurring in this pattern (Figure 28 ). And they also occur in the subpattern of light blue elements, as can be seen in Figure 26
The smaller translations, as they are present in the generating P4 pattern (Figure 28 ), do not occur in the subpattern of light blue elements. See next Figure.
Figure 30. Subpattern of light blue elements. The smaller translations, which are symmetry transformations of the generating P4 pattern, are not symmetry transformations (and therefore not elements) of the group representing the subpattern of light blue elements. As one can see in the Figure, these smaller translations (red arrows) (of the generating pattern) do not effect repetitions in the subpattern of light blue elements, and thus are not translations of this subpattern.
The next Figure gives the distribution of symmetry elements (4-fold and 2-fold rotation axes) of the subpattern of light blue elements.
Figure 31. Subpattern of light blue elements. Distribution of its symmetry elements ( 2-fold and 4-fold rotation axes). It is the fingerprint of the plane group P4 . Compare with the distribution of symmetry elements of the generating P4 pattern as it is depicted in Figure 39 of the previous document .
All the above demonstrates that the subpattern of light blue elements has the structure of the plane group P4 and is a subgroup of the generating P4 pattern (as well as of the antisymmetry pattern that implies this subgroup). Therefore the symbol for the antisymmetry pattern (Figure 21 ) must read P4 / P4 .
The subpattern of light blue elements -- where the blue color just indicates its clusters of group elements -- is, as has been said, a s u b g r o u p of the generating P4 pattern, which (generating pattern) is depicted in Figure 12 of the previous document , but which can also be given as in Figure 6 . As such the subpattern is a subgroup of itself, because it has the same algebraic structure as its corresponding full group, namely that of the plane group P4.
Subgroups
Here we will give some subgroups -- not necessarily connected with antisymmetry -- that can be represented by subpatterns of our P4 pattern as it was depicted in Figure 3 of the previous document.
Figure 32. If we apply the generating rotation p ( 900 anticlockwise rotation about the axis indicated (small solid dark blue square)) to the identity element (1) , and apply that rotation on the result, and again apply it to the last obtained result, we get three new elements, that -- together with the identity element -- make up a four-fold rosette. This repeated application of the rotation comes to an end after three times doing so, because p4 = 1 . The subgroup (yellow) has the structure of the cyclic group C4 , which is a point group ( = under the symmetry transformations of the group -- which here is the subgroup C4 -- only one point remains invariant).
Figure 33. If we apply to the identity element the symmetry transformation p2 , i.e. a half-turn about the point indicated (dark blue solid square), which is a symmetry transformation of the full group, we get a second element. And if we again apply this transformation to the result, we end up at the identity element again (p2p2 = p4 = 1 ). So we have the subgroup {1, p2} , the structure of which is that of the cyclic group C2 .
Figure 34. When we apply the horizontal translation th repeatedly to the identity element -- i.e. when we apply successively . . . th-2 , th-1 , th , th2 , th3 , . . . etc., to the identity element -- we get a frieze pattern (yellow). In the present case it is a translational repetition of an asymmetric motif, so its symmetry is that of the line group p11 .
Figure 35. When we apply the half-turn p2 (which we get by two consecutive applications of the generating quarter-turn) to the identity element, we get the subgroup {1, p2} , and when we apply the horizontal translation repeatedly to the elements of this subgroup, we obtain a frieze pattern. It consists of a linear repetition of a motif with C2 symmetry, so its structure is that of the line group p12 .
Figure 36. When we apply the half-turn (previous Figure) to the identity element, we get the subgroup {1, p2} , and when we apply the vertical translation repeatedly to the elements of this subgroup, we obtain a frieze pattern. It consists of a linear repetition of a motif with C2 symmetry, so its structure is that of the line group p12 .
This concludes our investigation concerning antisymmetry, subpatterns and subgroups of patterns representing the plane group P4.
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