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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 two documents we derived the first antisymmetry pattern from the generating P4mm symmetry pattern as depicted in Figure 7 of Part XXXI , and analysed it as to its identity.
Here we will derive a second antisymmetry pattern from that same generating P4mm pattern.
This second antisymmetry pattern will be derived from the generating P4mm symmetry pattern by 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. The other generators are not replaced.
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. 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 .
The antisymmetry reflection e1mse is applied to the identity element, effecting a color change.
Figure 2. Second phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern as depicted in Figure 7 of Part XXXI .
Repeated application of the generating rotation p results in a four-fold rosette.
Figure 3. Third phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern as depicted in Figure 7 of Part XXXI .
The generating horizontal translation th is applied to elements already obtained earlier.
Figure 4. Beginning of fourth phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern as depicted in Figure 7 of Part XXXI .
The antisymmetry reflection e1mse is applied to elements already obtained earlier. Here demonstrated for two elements.
Figure 5. Completion of fourth phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern as depicted in Figure 7 of Part XXXI .
The antisymmetry reflection e1mse is applied to the rest of the elements obtained earlier (Figure 3 ).
Figure 6. Beginning of fifth phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern as depicted in Figure 7 of Part XXXI .
The generating horizontal translation th is applied to some elements already obtained earlier.
Figure 7. Completion of fifth and final phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern as depicted in Figure 7 of Part XXXI .
The generating horizontal translation th is applied to the rest of the elements already obtained earlier (Figure 5 ).
Restoration of purple to red and yellow to blue gives our antisymmetry pattern:
Figure 8. Antisymmetry pattern derived from the generating P4mm pattern as depicted in Figure 7 of Part XXXI , according to the above specifications (i.e. by replacing the generating reflection mse by its corresponding antisymmetry reflection e1mse ).
Figure 9. Same as previous Figure. Indications of symmetry elements (rotation axis, mirror line) omitted. The color of the group element boundaries changed from red to blue.
The next Figure gives the point lattice of the just derived antisymmetry pattern.
Figure 10. Point lattice (indicated by yellow connection lines) of the just derived antisymmetry pattern (Figure 9). Compare with the point lattice of the generating P4mm pattern as it is depicted in Figure 3 of Part XXXI . A unit mesh is indicated by alternative colors.
In order to identify our antisymmetry pattern, we investigate its subpattern of blue elements (which contains the identity element) :
Figure 11. Subpattern of blue elements of the antisymmetry pattern derived above (Figure 9 ).
The next Figure gives the point lattice of the subpattern of blue elements.
Figure 12. Point lattice (indicated by red connection lines) of the subpattern of blue elements of the antisymmetry pattern derived above (Figure 9 ). A unit mesh is indicated by alternative colors. Compare with the point lattice of the generating P4mm pattern as it is depicted in Figure 3 of Part XXXI .
From the above Figure we can see that the subpattern of blue elements consists of a C4 motif that is repeated according to a square lattice, and that there are no mirror lines. This means that its symmetry is according to the plane group P4 . Its rotations of 900, 1800 and 2700 are also symmetry transformations of the generating P4mm pattern, while the reflections of the latter are not symmetry transformations of the pattern of blue elements. So our subpattern of blue elements, having P4 structure, is a subgroup of the generating P4mm pattern. Therefore, the symbol of the antisymmetry pattern derived above (Figure 9 ) must read P4mm / P4 .
In the next document we will derive yet another antisymmetry pattern from our generating P4mm patttern.
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To continue click HERE for further group theoretic preparation to the study of the structure of three-dimensional crystals (crystallography) and the basic symmetry of organisms (promorphology).
Back to 3-dimensional crystals (conclusion), Organic Tectology and Promorphology
Back to the Beginning of the present Series on Subpatterns and Subgroups. There : LINK to Part XXVII
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Back to Part XXX of the present Series
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