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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 concluded our investigation of antisymmetry, subpatterns and subgroups of C2mm patterns. Here we will continue such investigation with respect to the next plane group, namely P4 .
Figure 1. Pattern representing plane group P4 . The motifs s.str. are (chosen to be) a set of four vanes, and as such placed inside the (square) lattice meshes. The vanes in a mesh are related to each other by a 4-fold rotation axis. A quarter of a square lattice mesh can represent a fundamental region (yellow or green), containing an asymmetric unit (vane) of the motif s.str. . Such a fundamental region thus represents a group element. A lattice mesh is indicated (blue). The pattern must be imagined to extend indefinitely over the plane.
The above Figure is here chosen to represent the plane group P4. Antisymmetry patterns will be derived from it by interpreting the antiidentity transformation e1 as the color permutation (Blue Red) (cycle notation) with respect to the background color, which initially is set to be blue :
Figure 2. Generating P4 symmetry pattern (i.e. a pattern from which antisymmetry patterns will be derived), with the background color set to blue.
Figure 3. Generating P4 pattern.
We recognize three generators :
Anticlockwise rotation by 900, quarter-turn, indicated by p , about the axis indicated by a small solid blue square.
A vertical and horizontal translation, indicated by their respective vectors (blue arrows).
We can, however, already generate the whole pattern with only two generators : With the rotation p (and its repetitions p2 and p3 ) we can generate the four-fold rosette. Then we repeatedly translate this rosette in the vertical direction. Then we rotate the resulting vertical stripe 900 about the generating 4-fold rotation axis (small solid blue square), resulting in a new stripe perpendicular to the initial stripe, and thus resulting in a cross. When we now subject all the elements of the horizontal stripe to the vertical translation repeatedly, we obtain the whole pattern. The set of generators accordingly was : One rotation (900) and a vertical translation.
The next two Figures indicate the point lattice of the generating P4 pattern.
Figure 4. Square point lattice (indicated by strong blue connection lines) of the P4 generating symmetry pattern. Thinner lines indicate boundaries of areas representing group elements. The horizontal and vertical translations are indicated (yellow), as well as the implied NE translation ( tvertthor = tne ). For one unit mesh the 2-fold and 4-fold rotation axes are indicated. The pattern of these axes is the finger print of a P4 pattern.
Figure 5. Square point lattice (indicated by strong blue connection lines) of the P4 generating symmetry pattern. The lines indicating group element boundaries removed.
The first antisymmetry pattern to be derived from the generating P4 pattern (Figure 3 ) can be obtained by replacing the generating rotation p by its corresponding antisymmetry rotation e1p , where the antiidentity transformation e1 is -- as has been said -- 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 6. 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, showing color alternation, in virtue of the fact that e1pe1p = (e1p)2 = p2 , and (e1p)3 = e1p3 . See for p, p2 and p3 Figure 3. Further, the relation p4 = 1 holds.
Figure 7. Second stage in the derivation of the antisymmetry pattern as specified above. The rosette is repeatedly subjected to the generating vertical translation, resulting in a vertical strip. The resulting copies of the rosette are identical in coloration because the vertical translation is not an antisymmetry transformation.
Figure 8. Third stage in the derivation of the antisymmetry pattern as specified above. All the elements already present, are rotated 900 anticlockwise about the rotation axis indicated by the small solid yellow square (i.e. they are subjected to the generating rotation), resulting in new elements.
The next Figures (9 -- 17) illustrate that in the just resulted pattern of generated elements (Figure 8), a (anticlockwise) rotation by 2700 (p3) results in color change, so also (anticlockwise) rotations by 900 (p), while rotation by 1800 (p2) does not result in a color change.
Figure 9. 2700 rotation anticlockwise : color change.
Figure 10. 2700 rotation anticlockwise : color change.
Figure 11. 2700 rotation anticlockwise : color change.
Figure 12. 2700 rotation anticlockwise : color change.
Figure 13. 2700 rotation anticlockwise : color change.
Figure 14. 2700 rotation anticlockwise : color change.
Figure 15. 900 rotation anticlockwise : color change.
Figure 16. 1800 rotation : no color change.
Figure 17. 1800 rotation : no color change.
When we now apply the generating vertical translation (which does not involve color change) to the elements already present, we get the whole antisymmetry pattern :
Figure 18. Final stage in the derivation of the above specified antisymmetry pattern : Subjecting the elements already present to the generating vertical translation, results in the complete antisymmetry pattern.
After restoring green to blue, and purple to red, we finally have our antisymmetry pattern :
Figure 19. Antisymmetry pattern, derived from the generating P4 pattern of Figure 3 by replacing the generating rotation (quarter-turn) by its corresponding antisymmetry transformation. The result is a checkerboard-like pattern.
In order to identify the just derived antisymmetry pattern, we isolate its subpattern of blue elements and assess its status :
Figure 20. Subpattern of blue elements isolated from the above antisymmetry pattern.
In the above Figure it is easily seen that the subpattern consists of a two-fold motif (i.e. a motif with C2 symmetry) that is repeated along two independent translations. These latter, as well as the half-turns of the subpattern, are also elements of the generating P4 pattern (Figure 3 ) and of the antisymmetry pattern (Figure 19 ). And the identity element (1) in the latter two patterns is present in the subpattern. So this subpattern has P2 symmetry and is a subgroup of the generating (P4) pattern as wel as of the antisymmetry pattern. Therefore the symbol for our antisymmetry pattern must read P4 / P2 .
The next two Figures indicate the point lattice of the just derived P4 / P2 antisymmetry pattern.
Figure 21. Point lattice (indicated by strong dark blue connection lines) of the above derived antisymmetry pattern P4 / P2 (Figure 19 ). It is a square lattice. Compare with the point lattice of the generating P4 symmetry pattern (Figure 4 ).
Figure 22. Same as previous Figure. A unit mesh is indicated by alternative colors.
The next antisymmetry pattern can be obtained from the generating P4 pattern of Figure 3 by replacing the generating rotation p and the generating vertical translation by their corresponding antisymmetry transformations, where again the antiidentity transformation e1 is interpreted as the color permutation (Blue Red) (cycle notation) with respect to the background color which initially is set to blue (while, for reasons of clarity, the motifs s.str. are colored black).
Figure 23. 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, showing color alternation. The generating vertical antisymmetry translation is indicated by its vector (red arrow).
Figure 24. Second stage of the derivation of the antisymmetry pattern as specified above. The identity element (1) is subjected to the vertical antisymmetry translation (red arrow), which effects color alternation.
Figure 25. Third stage of the derivation of the antisymmetry pattern as specified above. The remaining elements of the rosette are subjected to the vertical antisymmetry transformation (resulting in color alternation).
Figure 26. Fourth stage of the derivation of the antisymmetry pattern as specified above. Applying the h o r i z o n t a l translation, which is not an antisymmetry transformation, to the elements already generated, creates new elements.
The new elements, obtained in the above Figure, can (and should) also be interpreted as being obtained by the generating antisymmetry rotation ( 900 anticlockwise about the axis indicated by the small solid yellow square). And this implies that color changes must be involved. But, in fact, it turns out not to be so, as the next two Figures show.
Figure 27. 900 anticlockwise : no color change.
Figure 28. 900 anticlockwise : no color change.
We see that we end up with a contradiction, which means that we must try to redefine our antisymmetry pattern that is to be derived from our generating P4 pattern (Figure 3 ) :
Let us then define this antisymmetry pattern as obtainable from the generating P4 pattern by replacing the generating rotation p , the generating vertical translation and the horizontal translation by their corresponding antisymmetry transformations, where the antiidentity transformation e1 is again interpreted as the color permutation (Blue Red) (cycle notation) with respect to the background color, which is initially set to blue (and where, for clarity, again the motifs s.str. are colored black) :
We can pick the derivation up from Figure 25 where, by repeatedly applying (to the identity element) the generating antisymmetry rotation -- which means by applying the transformations e1p, p2 and e1p3 -- a 4-fold rosette was generated. And this was followed by applying to that rosette the generating vertical antisymmetry translation, resulting in a broad vertical band :
Figure 29. Third stage of the generation of the redefined antisymmetry pattern from the generating P4 symmetry pattern (Figure 3 ). The pattern, so far obtained, resulted from repeated application of the generating antisymmetry rotation and the generating vertical antisymmetry translation.
Figure 30. Fourth stage of the generation of the redefined antisymmetry pattern.
The horizontal antisymmetry translation is applied to the elements already present, resulting in a horizontal stripe, where color alternation is involved.
The following Figures (31 -- 37) show that the result, obtained sofar, complies with the effect of the antisymmetry rotation being automatically implied (and thus not constituting a contradiction) :
Figure 31. 900 anticlockwise rotation : color change.
Figure 32. 900 anticlockwise rotation : color change.
Figure 33. 900 anticlockwise rotation : color change.
Figure 34. 900 anticlockwise rotation : color change.
Figure 35. 1800 rotation : no color change ( because indeed it should be so that 1800 antisymmetry rotation = e1pe1p = e1e1pp (because e1 and p commute) = pp = p2 , so no e1 is involved ).
Figure 36. 1800 rotation : no color change.
Figure 37. 2700 anticlockwise rotation : color change ( because indeed it should be so that 2700 anticlockwise antisymmetry rotation = (e1p)3 = e1p3 ).
So now we can safely proceed further :
Figure 38. Final stage of the derivation of the antisymmetry pattern as redefined above. Applying the generating vertical antisymmetry translation to all the elements already present, completes the antisymmetry pattern.
When we now restore green to blue and purple to red, we finally obtain our antisymmetry pattern :
Figure 39. Antisymmetry pattern derived from the generating P4 pattern of Figure 3 by replacing the generating rotation, the generating vertical translation, and the horizontal translation by their corresponding antisymmetry transformations. The result is a checkerboard-like pattern. Compare with the antisymmetry pattern derived earlier (Figure 19 ).
The next Figure shows that the antisymmetry character of the generating antisymmetry quarter-turn is still preserved.
Figure 40. Antisymmetry pattern of the above Figure.
900 anticlockwise rotation : color change.
In order to correctly identify our just derived antisymmetry pattern, we isolate its subpattern of blue elements (containing the identity element) :
Figure 41. Subpattern of blue elements of the above derived antisymmetry pattern isolated.
The next three Figures consider the point lattice of the just derived subpattern of blue elements.
Figure 42. Point lattice (indicated by strong red connection lines) of the subpattern of blue elements of the above derived antisymmetry pattern. It is a square lattice.
Figure 43. Alternative, but equivalent, point lattice (indicated by strong red connection lines) of the subpattern of blue elements of the above derived antisymmetry pattern. It is a square lattice.
Figure 44. Same as Figure 43. A unit mesh is indicated by alternative colors.
From the above Figures, it is clear that the subpattern of blue elements consists of a 4-fold motif (Here a motif with C4 symmetry) that is repeated according to a square lattice. So the subpattern represents a plane group, namely the plane group P4 .
Figure 45. Same as Figure 42. Subpattern of blue elements.
The original translations (vertical and horizontal) of the generating P4 symmetry pattern -- (Figure 4 , from which the antisymmetry pattern (Figure 39 ) was derived) -- are indicated (dark blue arrows). As can be seen, they are not symmetry transformations of the subpattern. Their vector sum is equal to the NE translation of the subpattern of blue elements, which means that this NE translation is also an element of the generating P4 pattern. The quarter-turns of the subpattern are also elements of the generating P4 pattern. In fact all the symmetry transformations of the subpattern are also symmetry transformations of the generating P4 pattern. In both cases the square unit mesh has 4-fold rotation axes at its corners and one at its center, and 2-fold rotation axes at the centers of its edges (See for the subpattern Figure 49 and for the generating symmetry pattern Figure 4 ), while, as has been just found out, the NE and SE translations of the subpattern also occur in the generating pattern (Figure 4 ). The original translations (horizontal and vertical) of the generating P4 pattern (Figure 4 ) are, however, not elements of the subpattern (See present Figure ). But in both patterns one vane (i.e. a fourth part of the motif s.str.) can represent the identity element. All this proves that the subpattern of blue elements represents a s u b g r o u p of the (group representing the) generating P4 pattern. Moreover, in virtue of the same reasons (See below) our subpattern also represents a s u b g r o u p of the (group representing the) antisymmetry pattern.
All the above implies that the symbol for our derived antisymmetry pattern (Figure 39 ) must read P4 / P4 .
The next Figures show possible point lattices of the P4 / P4 antisymmetry pattern, and explain why its subpattern of blue elements is a subgroup of it.
Figure 46. Square point lattice (indicated by strong dark blue connection lines) of the antisymmetry pattern of Figure 39 . Two translations (horizontal and vertical) of the pattern are indicated by the sides of a lattice mesh.
Figure 47. Alternative square point lattice (indicated by strong dark blue connection lines) of the antisymmetry pattern of Figure 39 . Two diagonal translations (NE and SE) of the pattern are indicated by the sides of a lattice mesh.
Figure 48. Same as Figure 47. A unit mesh is indicated by alternative colors. Notice that this unit mesh is smaller than the mesh of the other possible lattice depicted in Figure 46. A mesh in the present Figure contains eight units (vanes) of the motif s.str., while there are sixteen of them in a lattice mesh of Figure 46. So we prefer the lattice choice as is done in the Figures 47 and 48.
If we look to the subpattern of blue elements (Figure 45 ) we see that it has four-fold rotation axes, 2-fold rotation axes and two diagonal (SE and NE) translations (which are indicated by the sides of the lattice meshes) :
Figure 49. Pattern of the symmetry elements (4-fold and 2-fold rotation axes) -- as given for one unit mesh -- possessed by the subpattern of blue elements, identifying its symmetry as that of the plane group P4 . As can be seen, a unit mesh of this pattern has four 4-fold rotation axes at its corners and one at its center. It further has a 2-fold rotation axis at the center of each of its edges. This indeed is the finger-print of a P4 pattern. The two independent translations are indicated by the edges of a unit mesh (a NE and a SE edge).
Precisely the same pattern of symmetry elements (4-fold and 2-fold rotation axes), and precisely the same translations, are possessed by the antisymmetry pattern, derived earlier (Figure 39 ), proving that the two patterns both are P4 patterns.
Figure 50. Pattern of symmetry elements (4-fold and 2-fold rotation axes) of the above derived antisymmetry pattern. This pattern of symmetry elements, and its translations, are the same as in the subpattern of blue elements implied by this antisymmetry pattern [Recall that "symmetry elements" of a pattern are those geometric entities (points, lines, planes, etc) to which certain symmetry transformations (like reflections, rotations, etc) refer.]. Compare with Figure 49 (subpattern of blue elements).
In the next document we will continue our investigation of the plane group P4 as to its antisymmetries, subpatterns and subgroups.
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