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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 the previous document we investigated antisymmetry, subpatterns and subgroups of a pattern represennting the plane group P4 . Here we will continue this investigation.
The third antisymmetry pattern to be derived from the generating P4 pattern (Figure 3 of previous document ) can be obtained by replacing the generating vertical translation tv by its corresponding antisymmetry translation e1tv , where the antiidentity transformation e1 is again interpreted as the color permutation (Blue Red) (cycle notation) with respect to the background color, which initially is set to blue. The other generators are not replaced.
Figure 1. First stage of the derivation of the antisymmetry pattern as specified above. The identity element is indicated by 1 , and the 4-fold generating rotation axis is indicated by a small solid yellow square. A four-fold rosette is generated, not involving color change, because the generating rotation is not an antisymmetry transformation. The vertical antisymmetry translation e1tv is indicated.
Figure 2. Second stage of the derivation of the antisymmetry pattern as specified above. The vertical antisymmetry translation e1tv is repeatedly applied to the elements of the 4-fold rosette, resulting in a vertical band of elements with color alternation, because (e1tv)2n+1 = e1tv2n+1 and (e1tv)2n = tv2n .
Figure 3. Third stage of the derivation of the antisymmetry pattern as specified above. Applying the generating rotation ( 900 anticlockwise rotation about the point indicated [yellow] ) to the elements already generated, creates new elements. Color alternation in the horizontal direction is automatically implied, meaning that the horizontal translation is in fact an antisymmetry translation.
Figure 4. Fourth stage of the derivation of the antisymmetry pattern as specified above. Applying the vertical antisymmetry translation to the elements already obtained, completes the whole antisymmetry pattern.
Restoring green to blue, and purple to red yields, finally, our antisymmetry pattern :
Figure 5. Antisymmetry pattern derived from the generating P4 pattern of Figure 3 of the previous document by replacing the generating vertical translation by its corresponding antisymmetry translation. Compare with the antisymmetry pattern derived earlier, as depicted in Figure 39 of the previous document . They are shifted with respect to each other.
The next Figure isolates the subpattern of blue elements of the just derived antisymmetry pattern.
Figure 6. Subpattern of blue elements isolated from the just derived antisymmetry pattern. Compare with that of the antisymmetry pattern derived earlier and depicted in Figure 41 of the previous document.
The next Figures (6a, 6b, 6c, 6d) analyse this subpattern. We start with giving this pattern without all auxiliary lines. But after that, we insert them again for reasons of clarity (because they help to see the structure of the pattern).
Figure 6a. Subpattern of blue elements. All auxiliary lines omitted.
Figure 6b. Point lattice (indicated by red connection lines) of the subpattern of blue elements. All auxiliary lines inserted again.
Figure 6c. Same as previous Figure. The areas representing group elements are isosceles right-angled triangles. One of them, representing the identity element, is indicated by alternative colors.
Figure 6d. Distribution of symmetry elements ( 2-fold and 4-fold rotation axes) in the subpattern of blue elements. It is the fingerprint of the plane group P4 .
As can be inferred from the above Figures, the subpattern's symmetry is that of the plane group P4 , and is a subgroup of the (group represented by the) generating P4 symmetry pattern [as well as of the (group represented by the) just derived antisymmetry pattern]. So the symbol for the antisymmetry pattern must read P4 / P4 .
The next Figures indicate the point lattice of the just derived antisymmetry pattern.
Figure 7. Square point lattice (indicated by strong dark blue connection lines) of the just derived antisymmetry pattern (Figure 5 ).
Figure 8. Square point lattice (indicated by strong dark blue connection lines) of the just derived antisymmetry pattern (Figure 5 ). A unit mesh is indicated by alternative colors.
The next Figures and accompanying text analyse two interesting examples (one from JABLAN, 2002) of P4 / P4 antisymmetry patterns. The first of these will be analysed in the present document, and the second in the next document.
With respect to the first antisymmetry pattern to be derived, we construct the generating P4 pattern (i.e. the pattern from which this antisymmetry pattern will be derived). We do this in several steps :
Figure 9. First phase in the construction of a generating P4 symmetry pattern. The color (blue) here only serves to visualize the pattern.
Figure 10. Second phase in the construction of a generating P4 symmetry pattern. The color (blue) here only serves to visualize the pattern.
Figure 11. Third phase in the construction of a generating P4 symmetry pattern. The color (blue) here only serves to visualize the pattern.
Figure 12. Final phase in the construction of a generating P4 symmetry pattern. As always, the pattern must be imagined to extend indefinitely over the plane.
The next Figures analyse the just obtained generating pattern (that is, the pattern from which we will later derive an antisymmetry pattern).
Figure 13. A possible set of equivalent points (indicated by small solid blue squares) of the generating P4 symmetry pattern. These equivalent points are also locations of 4-fold rotation axes of the pattern. These are, however, not the only 4-fold rotation axes of the pattern : between every four such points lies another 4-fold rotation axis (but the points representing these latter axes are not equivalent to those of the set of points highlighted in the present Figure).
The set of equivalent points, as indicated in the above Figure, forms a point lattice of the pattern. The next Figure highlights this point lattice by means of connection lines.
Figure 14. Point lattice (indicated by small solid blue squares, and highlighted by red connection lines) of the generating P4 symmetry pattern. It is a square lattice.
Figure 15. Point lattice (indicated by small solid blue squares, and highlighted by red connection lines) of the generating P4 symmetry pattern. It is a square lattice. A unit mesh is indicated by alternative colors (blue and gray).
As we can see in the above Figures, the pattern consists of a repetition of a C4 motif :
It is a repetition according to a square lattice, which means that our generating pattern (Figure 12 ) is indeed a P4 pattern.
The next Figure shows a unit mesh, and the symmetry elements associated with it.
Figure 16. Symmetry elements associated with a unit mesh of the generating pattern of the above Figures.
At each corner of the mesh there is a 4-fold rotation axis (blue solid square).
In the center of the mesh there is also a 4-fold rotation axis (blue solid square).
At the center of each edge of the mesh there is a 2-fold rotation axis (green solid circle).
This is indeed the fingerprint of the planegroup P4 .
The next Figures show a possible fundamental region of the pattern. It is an asymmetric unit and is repeated according to the symmetry transformations of the group (four rotations and two translations). Therefore such regions can represent group elements.
Figure 17. Fundamental region, repeated according to the symmetry transformations of the group, resulting in our generating P4 pattern. The boundaries of the regions -- which (regions) tile the plane completely -- are indicated by thin purple and thick red lines. These regions can represent group elements. The point lattice of the pattern is indicated by the set of small solid blue rectangles and strong red connection lines.
Figure 18. Same as previous Figure. Some areas representing group elements are highlighted (light blue). These areas are isosceles right-angled triangles. One such area is magnified. The two black line segments (perpendicular to each other) inside such area form a motif unit.
Figure 19. Same as previous Figure. Indication of point lattice removed. The areas representing group elements are isosceles right angled triangles indicated by their boundaries (purple thin lines). Each area contains an asymmetric motif.
Figure 20. Same as previous Figure. Areas representing group elements indicated by colors (yellow, green). These colors do not signify symmetry features, i.e. they do not indicate an asymmetry of the pattern. The latter is evident from its motif units, each consisting of two unequal line segments making a right angle.
In order to derive an antisymmetry pattern from the above P4 pattern, we set the initial background color to be blue. We let the antisymmetry transformations effect a color change of this background according to the permutation (Blue Red) . So our initial pattern (provided with group element boundaries) will look as follows :
Figure 21. Generating P4 pattern with its intial background color set to be blue. Boundaries of areas representing group elements indicated by purple lines ( The black lines belong to the pattern's motifs). One of these areas is chosen to represent the identity element, and is indicated by yellow coloring. From this pattern the antisymmetry pattern will be derived.
Figure 22. Generating P4 pattern of the previous Figure.
The two generators of the pattern are :
900 anticlockwise rotation about the axis indicated (small solid blue square).
One of the two lattice translations, indicated by a red arrow.
The identity element is indicated (yellow).
Figure 23. Illustration of the above given generating translation.
We will now, finally, derive our antisymmetry pattern from the pattern of Figure 22 by replacing the generating translation (as given in the previous two Figures) by its corresponding antisymmetry translation. The antiidentity transformation e1 will -- as has been said -- be interpreted as the color permutation (Blue Red) with respect to the background color of the pattern, which (background color) is set to be initially blue.
Figure 24. First phase of the derivation of the antisymmetry pattern as defined above, from the generating pattern as given in Figure 22 . The generator p ( 900 anticlockwise rotation about the axis indicated) is applied to the identity element (as it is given in Figure 22 ). Then this generator is again applied to the result, and finally, this generator is again applied to the last obtained result. Because this generator is not an antisymmetry transformation, the result is a 4-fold rosette consisting of blue elements, here, for the time being, colored yellow.
Figure 25. Second phase of the derivation of the antisymmetry pattern as defined above, from the generating pattern as given in Figure 22 . The antisymmetry translation (indicated by a red arrow) is applied to the elements already obtained. It results in a band of elements involving color alternation.
Figure 26. Third phase of the derivation of the antisymmetry pattern as defined above, from the generating pattern as given in Figure 22 . The generating rotation is applied to the already obtained elements. A second band (perpendicular to the first one) is obtained.
Figure 27. Fourth phase of the derivation of the antisymmetry pattern as defined above, from the generating pattern as given in Figure 22 . The antisymmetry translation is applied to some of the already obtained elements. A second band in the direction of the translation is obtained.
Figure 28. Final phase of the derivation of the antisymmetry pattern as defined above, from the generating pattern as given in Figure 22 . The antisymmetry translation is now applied to the rest of the elements already obtained in Figure 26 , completing the whole pattern ( All these bands must, of course, be imagined to extend indefinitely along their directions).
When we now restore yellow to blue we, finally, get our antisymmetry pattern :
Figure 29. Antisymmetry pattern derived from the pattern of Figure 22 , by replacing the generating translation (yellow arrow) by its corresponding antisymmetry translation.
The next Figure depicts the same as the previous Figure does, but has changed red to yellow in order to let the boundaries of the areas representing group elements clearly stand out.
Figure 30. Antisymmetry pattern derived from the pattern of Figure 22 . Red changed to yellow, in order to let the boundaries (purple lines) of the areas representing group elements (fundamental regions) clearly stand out.
The next Figure indicates the point lattice of the just derived antisymmetry pattern.
Figure 31. Point lattice (indicated by strong red connection lines) of the antisymmetry pattern derived from the pattern of Figure 22 . It is a square lattice. Compare with the point lattice of the generating P4 pattern as depicted in Figure 17 .
In order to identify our antisymmetry pattern, we investigate the subpattern of blue elements that is involved in this antisymmetry, and which contains the identity element. First we give the antisymmetry pattern again, but now without the indications of its generators (half-turn and antisymmetry translation) and without emphasizing the point lattice.
Figure 32. Antisymmetry pattern as derived above. Without indication of generators and point lattice. The set of blue elements, containing the identity element, forms a subpattern of the antisymmetry pattern.
The next Figure isolates this subpattern of blue elements.
Figure 33. Isolated subpattern (of the antisymmetry pattern derived above) of blue elements. For a part of the pattern the boundaries of the areas representing group elements are indicated (purple lines).
Figure 34. Point lattice (indicated by red connection lines) of the isolated subpattern. Compare with the point lattice of the antisymmetry pattern as given in Figure 31 . Compare also with the point lattice of the generating P4 pattern as depicted in Figure 14 . The translations of our subpattern of blue elements are given by the sides of the lattice meshes. We can see that these translations are also symmetry transformations (and thus elements) of the generating as well as of the antisymmetry pattern. See next two Figures.
Figure 35. Generating P4 pattern. The Figure shows that each translation of the subpattern of blue elements (Figure 34 ) is equal to the sum of the two translations of the generating P4 pattern, which means that the translations of the subpattern are also symmetry transformations of the generating pattern. See also next Figure.
Figure 36. Generating P4 pattern. Indication and illustration of possible translations of this pattern. They are identical to the translations of the subpattern of blue elements as depicted in Figure 34 .
Figure 37. Indication of the pattern of 4-fold rotation axes and 2-fold rotation axes of the isolated subpattern of blue elements.
Figure 38. Isolated unit mesh of the subpattern of blue elements. The location of the 4-fold and 2-fold rotation axes is indicated. It is the fingerprint of the plane group P4 . These rotation axes also represent symmetry transformations of the generating P4 pattern as well as of the antisymmetry pattern. See next Figure.
Figure 39. Distribution of symmetry elements -- 2-fold and 4-fold rotation axes -- in the generating P4 pattern. Compare with that of the subpattern of blue elements as depicted in Figure 37 . Compare also with that of the antisymmetry pattern derived above, Figure 40 .
Figure 40. Distribution of symmetry elements -- 2-fold and 4-fold rotation axes -- in the antisymmetry pattern derived above.
From the above Figures it is clear that the symmetry of the subpattern of blue elements (as was isolated from the above derived antisymmetry pattern) is according to the plane group P4 , and that it is a subgroup of the generating P4 pattern (as well as of the antisymmetry pattern). Therefore the symbol of the antisymmetry pattern (depicted in Figures 30 and 29 ) must read P4 / P4 .
In the next document we will derive a second P4 / P4 antisymmetry pattern from the generating P4 symmetry pattern of Figure 12 .
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