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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)
Sequel to the Plane Group P4mm
In the previous documents we derived four antisymmetry patterns from the generating P4mm symmetry pattern as depicted in Figure 7 of Part XXXI , and analysed them as to their identity.
Here we will derive a fifth antisymmetry pattern from that same generating P4mm pattern.
This fifth antisymmetry pattern will be derived from the generating P4mm symmetry pattern by replacing the generating horizontal translation th by its corresponding antisymmetry translation e1th , and (at the same time) replacing the generating reflection mse by its corresponding antisymmetry reflection e1mse , 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 (Figure 1).
The antisymmetry pattern will be derived in several steps. Newly generated blue elements will initially be colored yellow, and later be restored to blue. Newly generated red elements will initially be colored purple, and later be restored to red.
Figure 1 depicts the generating P4mm pattern and its generators.
Figure 1. Generating P4mm pattern, and its generators, p ( 900 anticlockwise rotation about the axis indicated by a small solid red square), reflection in the line mse , and the horizontal translation th . The identity element is indicated (yellow).
Figure 2. First phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern as depicted in Figure 7 of Part XXXI (and again in Figure 1).
The antisymmetry reflection e1mse is applied to the identity element. Color change is involved.
Figure 3. Second phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
The two elements already present are subjected to the generating rotation p and its repetitions p2 and p3 , resulting in six more elements. The rotations do not involve color changes.
Figure 4. Beginning of third phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
Application of the antisymmetry translation e1th to one already existing element results in more elements that show color alternation.
Figure 5. Completion of third phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
Application of the antisymmetry translation e1th to the rest of the already existing elements (Figure 3 ) results in a horizontal band of elements that show color alternation (in the horizontal direction).
Figure 6. Beginning of fourth phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
Two elements already present, are subjected to the antisymmetry reflection e1mse . Color change involved.
Figure 7. Completion of fourth phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
The rest of the elements already present (Figure 5 ), are subjected to the antisymmetry reflection e1mse . Color change involved.
Figure 8. Same as previous Figure. Purple restored to red.
The next two Figures illustrate that the global (i.e. collective) effect of the antisymmetry reflection on the horizontal band of elements is identical to that of the generating rotation (900 anticlockwise rotation about the point indicated by a small red rectangle).
Figure 9.
Figure 10.
Figure 11. Beginning of fifth phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
Application of the antisymmetry translation e1th to one already existing element results in more elements that show color alternation.
Figure 12. Completion of fifth and final phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
Application of the antisymmetry translation e1th to the rest of the already existing elements (Figure 8 ) completes the pattern.
Restoring yellow to blue and purple to red, finally, gives our antisymmetry pattern :
Figure 13. Antisymmetry pattern derived from the generating P4mm pattern (Figure 1 ) by replacing the generating horizontal translation and the generating reflection by their corresponding antisymmetry transformations.
Figure 14. Same as previous Figure. Indications of symmetry elements (rotation axis, mirror line) removed.
The next Figures give the point lattice of the just derived antisymmetry pattern.
Recall that if, in a periodic pattern, we take a point (i.e. we just select a point of the pattern at our convenience), and then consider all points of that pattern that are equivalent to the chosen point, we have (chosen) a possible point lattice of that pattern, which describes its periodic repetition of motifs. This point lattice can be highlighted -- as we always do -- by indicating the lines that connect lattice points.
In the next Figure we have chosen a point in the just derived antisymmetry pattern. All other points that have identical surroundings as those of the chosen point are equivalent to the latter and together form a possible point lattice of the antisymmetry pattern.
Figure 15. A set of equivalent points (some indicated by dark blue dots), and thus having identical surroundings (highlighted by yellow coloring), of the above derived antisymmetry pattern (previous Figure). This set of points forms the point lattice of the pattern, as illustrated in the next Figures by connection lines.
Figure 16. Point lattice (indicated by yellow connection lines) of the above derived antisymmetry pattern (Figure 14 ).
If we interpret the pattern indeed as an a n t i s y m m e t r y pattern, the lattice, depicted in the present Figure, is a s q u a r e lattice, because the group representing the antisymmetry pattern is isomorphic to the plane group P4mm, which repeats its motifs according to a square lattice ( The reason for the isomorphy is that some symmetry transformations of the P4mm group are just replaced by other transformations -- antisymmetry transformations -- having the same periods).
If, on the other hand, the pattern is not interpreted as an antisymmetry pattern, but as a pattern in which the color change signifies an asymmetry, then the lattice (depicted in the present Figure) is a r h o m b i c lattice ( This will be explained further below). Compare with the point lattice of the generating P4mm pattern as it is depicted in Figure 3 of Part XXXI .
Figure 17. Same as previous Figure. A unit mesh is indicated by alternative colors.
The just derived antisymmetry pattern admits of a second, but equivalent, point lattice, namely a centered rectangular point lattice. See next two Figures.
Figure 17a. Centered rectangular point lattice, indicated by green dots -- representing the lattice points -- and yellow connection lines. In the present case the meshes happen to be squares (which are a special type of rectangle).
Figure 17b. Same as previous Figure. For one unit mesh the lattice points -- as mutually equivalent points -- are highlighted by indicating identical surroundings.
The fact, that in addition to a primitive lattice, where the meshes have four equal edges, as in Figure 17 , a centered rectangular lattice is equally possible, indicates that the former must be a rhombic lattice, and not a square lattice. In the present case we have simply to do with the fact that the rhombuses have right angles.
We can say that if our antisymmetry pattern is indeed interpreted as a representation of an antisymmetry group, then it is isomorphic to the group represented by the generating P4mm pattern (from which the antisymmetry pattern was derived), because the color change (red <==> blue) is then a symmetry (namely an antisymmetry), which originates by replacing one or more generators by antisymmetry generators with the same period ( = number of repetitions necessary to become the identity element, 1 -- for example the period of a quarter-turn p is 4, because p4 = 1 ).
If, on the other hand, our (antisymmetry) pattern is not interpreted as an antisymmetry pattern, which means that the mentioned color change is then signifying an asymmetry : there then is no transformation defined that effects this color change, then the group representing the (above derived) pattern is not isomorphic to P4mm, but has the structure of the group C2mm. In this latter case the point lattice describing the periodic repetition of motifs, is a centered rectangular lattice or, equivalently, a rhombic lattice, while in the former case (i.e. the above derived pattern as antisymmetry pattern) it is a genuine square lattice.
Generally, however, this (loss of symmetry) need not be so : Some patterns, derived as antisymmetry patterns, but (then) not interpreted as an antisymmetry pattern -- this is, in effect, what we do when we isolate the subpattern of blue elements, i.e. the subpattern of elements not affected by color change -- are nevertheless isomorphic to their corresponding generating pattern, as we saw in the previous document (See there, Figure 14 ). But in the present case we see -- Figure 14 of present document -- that, when interpreting the color change as asymmetry, although certain horizontal and vertical reflections of the generating P4mm pattern are still preserved, the diagonal reflections (present in the generating P4mm pattern) have totally disappeared, resulting in the fact that the pattern cannot be isomorphic to the generating P4mm symmetry pattern. See next Figure.
Figure 17c. The antisymmetry pattern derived above.
If not interpreted as a n t i symmetry pattern, the diagonal reflections have totally disappeared, effecting that the pattern, so interpreted, is not isomorphic to the generating P4mm pattern. "Totally disappeared" here means : The original distribution of diagonal mirror lines across the pattern has not just become less dense (like we see it with respect to the horizontal and vertical mirror lines), it has disappeared completely. And the corresponding reflections are not (seen as being) replaced by other transformations.
In order to determine the identity of the above derived antisymmetry pattern, we will study its subpattern of blue elements (containing the identity element) :
Figure 18. Subpattern of blue elements of the above derived antisymmetry pattern (Figure 14 ).
The subpattern of blue elements still has reflection lines, as the next Figure shows.
Figure 19. Mirror lines (dark blue) of the subpattern of blue elements of the above derived antisymmetry pattern (Figure 14 ). These mirror lines do not define a point lattice of the pattern, because the motifs in the 'meshes' do not have the same orientation. They are enantiomorphic forms of each other.
Figure 20. Possible way to geometrically represent group elements of the group represented by the subpattern of blue elements. Because these areas tile the plane completely they can represent f u n d a m e n t a l r e g i o n s of the pattern. The colors red and green do not signify symmetry features. They only serve to highlight the areas representing group elements.
Figure 21. Alternative way to geometrically represent group elements of the group represented by the subpattern of blue elements. Because these areas do not tile the plane completely -- they leave gaps -- (if their content is taken into account, not just their shape), they cannot represent fundamental regions of the pattern, but can r e p r e s e n t group elements. The colors yellow and red do not signify symmetry features.
Figure 22. Subpattern of blue elements of the above derived antisymmetry pattern (Figure 14 ). All auxiliary lines removed, except those delineating areas representing group elements (as indicated in the previous Figure).
The next Figures give the point lattice of the pattern of blue elements.
Figure 23. Point lattice (indicated by red connection lines) of the subpattern of blue elements of the above derived antisymmetry pattern (Figure 14 ). It is (See below) a r h o m b i c lattice (of which the meshes happen to be squares -- which are a special form of rhombus, i.e. a rhombus with right angles).
Figure 24. Point lattice (indicated by red connection lines) of the subpattern of blue elements of the above derived antisymmetry pattern (Figure 14 ). A unit mesh is indicated by alternative colors.
In the above Figures we can see that the subpattern of blue elements consists of a D2 motif that is repeated along two independent translations. This means that the symmetry of the subpattern is either according to the plane group P2mm or to the plane group C2mm .
The fact that our subpattern of blue elements contains glide reflections (shown below) rules out the plane group P2mm. Therefore it must have the structure of C2mm, which means that in addition to describing the periodic repetition of its motifs by a rhombic lattice, as we did in Figure 24 , we can also (and equivalently so) describe this repetition by a centered rectangular lattice, as the next three Figures show.
Figure 25. Centered rectangular point lattice (indicated by red connection lines) of the subpatttern of blue elements. In this case the meshes happen to be squares (which are special versions of rectangles). Each lattice mesh consists of five equivalent points : One at each corner and one in its center. See next two Figures.
Figure 26. Centered rectangular point lattice (indicated by red connection lines) of the subpatttern of blue elements. Each lattice mesh consists of five equivalent points : One at each corner and one in its center. For one mesh these points are highlighted (by coloring their identical surroundings). See also next Figure.
Figure 27. Centered rectangular point lattice (indicated by strong blue connection lines) of the subpatttern of blue elements. Lattice points indicated by red dots. Compare with the centered rectangular lattice of the corresponding antisymmetry pattern, as it is depicted in Figure 17a .
The next two Figures indicate the four glide lines in a unit mesh of the centered rectangular lattice of the subpattern of blue elements.
Figure 28. The two horizontal glide lines ( g ) of a unit mesh of the centered rectangular lattice of the subpattern of blue elements. Illustrated for three meshes.
Figure 29. The two vertical glide lines ( g ) of a unit mesh of the centered rectangular lattice of the subpattern of blue elements. Illustrated for two meshes.
The next two Figures show the symmetry elements (reflection lines, glide lines, rotation axes) of our subpattern of blue elements.
Figure 30. Reflection lines (blue, m) and glide lines (red, g) of the subpattern of blue elements. The thick blue lines are, in addittion to representing part of the pattern's reflection lines, also lattice connection lines, i.e. indicating the (centered) lattice meshes of the centered rectangular lattice. One such mesh is highlighted. The next Figure adds the rotation axes possessed by the subpattern (shown for one unit mesh only).
Figure 31. Same as previous Figure. For one unit mesh : all its rotation axes added. In Figure 32 such a unit mesh is isolated.
Figure 31a. Same as previous Figure. For one axis its two-fold nature is illustrated.
Figure 32. Isolated unit mesh of the centered rectangular lattice (Figure 27 ) of the subpattern of blue elements. The pattern of symmetry elements (rotation axes, glide lines and mirror lines) is the fingerprint of the plane group C2mm .
The next Figure shows those rotation axes of the subpattern of blue elements that are located at the intersections of the pattern's glide lines. They seem to be four-fold rotation axes (instead of two-fold rotation axes).
Figure 33. Four-fold rotation axes are seemingly present in the subpattern of blue elements (two such axes highlighted).
In the subpattern of blue elements, as we have it, there are indeed four-fold rotation axes, and this is incompatible with the symmetry content of the plane group C2mm. The only other plane group that could be a candidate for expressing the group theoretic identity of our subpattern of blue elements is the plane group P4gm. But this group is ruled out by the fact that it does not possess rotation axes at the intersection of its glide lines (whereas our subpattern does have rotation axes at those points), and by the fact that our subpattern of blue elements does not possess a glide line bisecting the 900 angle between the other glide lines, as is the case for P4gm.
Figure 33a. Two-fold and four-fold rotation axes.
So we have now firmly established that our subpattern of blue elements has the algebraic structure of the plane group C2mm. But, in order to determine the identity of the above derived antisymmetry pattern (Figure 14 ) we must assess whether our subpattern is a subgroup of the generating P4mm symmetry pattern (Figure 1 ) from which the antisymmetry pattern was derived.
The translations of our subpattern of blue elements -- indicated by the edges of the lattice mesh (See Figure 23 ) -- are also symmetry transformations of the generating P4mm pattern. This was illustrated in Figure 17 of Part XXXIV . On the other hand the unit translation of the generating P4mm symmetry pattern is absent in the subpattern of blue elements. Further, one can see that the reflections of the subpattern are also reflections of the generating P4mm pattern.
It can be seen that all the half-turns (represented by the 2-fold rotation axes) of the subpattern of blue elements (Figure 23 ) are also present in the generating P4mm symmetry pattern. See for this Figure 4 of Part XXXI : at the corresponding locations we see there 4-fold rotation axes, and these imply quarter-turns, three-quarter-turns and . . . half-turns.
All this shows that our subpattern of blue elements has the structure of the plane group C2mm and is a subgroup of the (group represented by the) generating P4mm pattern from which the antisymmetry pattern (Figure 14 ) was derived. So the symbol for the latter must read P4mm / C2mm .
In the next document we will consider yet another antisymmetry pattern. It originates from another P4mm pattern than the one we have used in the foregoing.
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