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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 three 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 fourth antisymmetry pattern from that same generating P4mm pattern.
This fourth 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 rotation p ( 900 anticlockwise rotation about the axis indicated in the next Figure) by its corresponding antisymmetry rotation e1p , 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 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 rotation e1p is repeatedly applied to the identity element (i.e. the transformations e1p, p2 and e1p3 are applied to the identity element). Color alternation is involved.
Figure 3. Second phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
The identity element is reflected in the line mse , i.e. it is subjected to the generating reflection. The resulting element mse is then repeatedly subjected to the antisymmetry rotation e1p implying color alternation.
Figure 4. Same as previous Figure. Purple restored to red.
Figure 5. Beginning of third phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
Two elements (e1p, e1pmse ) are subjected to the antisymmetry translation e1th , resulting in new elements displaying color alternation.
Figure 6. Completion of third phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
The rest of the elements obtained in Figure 4 are subjected to the antisymmetry translation e1th , resulting in new elements displaying color alternation.
Figure 7. Beginning of fourth phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
Eight elements, that were obtained earlier, are subjected to the generating reflection mse , resulting in eight new elements. No color change involved.
The next Figure illustrates that reflection in the line mse of the horizontal band of elements gives the same collective result as applying to that band the antisymmetry rotation e1p (900 anticlockwise rotation accompanied by color change).
Figure 8. The reflection mse and the antisymmetry rotation e1p have, in the end, the same effect, when applied to the horizontal band of elements obtained earlier.
Figure 9. Completion of fourth phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
The rest of the elements, obtained earlier (Figure 6 ), are subjected to the generating reflection mse , resulting in a vertical band of new elements.
Restoring purple to red gives the following :
Figure 10. Same as previous Figure. Purple restored to red.
Figure 11. Beginning of fifth phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
The antisymmetry translation e1th is repeatedly applied to two elements already obtained earlier. A color alternation is involved.
Figure 12. Completion of fifth and final phase of the derivation of the antisymmetry pattern, as specified above, from the generating P4mm pattern.
The antisymmetry translation e1th is repeatedly applied to the rest of the elements already obtained earlier. A color alternation is involved.
Restoring yellow to blue and purple to red gives us, finally, our antisymmetry pattern.
Figure 13. Antisymmetry pattern derived (as specified above) from the generating P4mm pattern, by replacing the generating horizontal translation and the generating rotation by their corresponding antisymmetry transformations. Compare with the antisymmetry pattern obtained earlier : Figure 8 of Part XXXIV .
Figure 14. Same as previous Figure. Indications of symmetry elements (rotation axis, mirror line) removed.
The next two Figures give the point lattice of the just derived antisymmetry pattern.
Figure 15. Square point lattice (indicated by yellow connection lines) of the just derived antisymmetry pattern. Compare with the point lattice of the generating P4mm pattern as depicted in Figure 3 of Part XXXI .
Figure 16. Square point lattice (indicated by yellow connection lines) of the just derived antisymmetry pattern. A unit mesh is indicated by alternative colors.
In order to identify the just derived antisymmetry pattern we investigate its subpattern of blue elements (containing the identity element) :
Figure 17. Subpattern of blue elements of the above derived antisymmetry pattern.
Figure 18. Subpattern of blue elements of the above derived antisymmetry pattern.
A possible geometric representation of group elements. The areas -- the colors of which do not represent asymmetries -- are also possible fundamental regions, because they tile the plane completely, as the next Figure shows.
Figure 19. The areas, as indicated in the above Figure, tile the plane completely.
Figure 20. Subpattern of blue elements of the above derived antisymmetry pattern.
A second possible geometric representation of group elements. The areas indicated -- the colors of which do not signify asymmetries -- are not possible fundamental regions, because they do not tile the plane (if we take their content -- a comma -- into account), but they can geometrically r e p r e s e n t group elements, in the same way that a comma can represent a group element.
Along precisely the same lines as we did in the previous document, we can determine that the subpattern of blue elements has the structure of the plane group P4mm and is a subgroup of the generating P4mm symmetry pattern. Therefore the symbol of the above derived antisymmetry pattern must read P4mm / P4mm .
In the next document we will derive another antisymmetry pattern from the same generating P4mm pattern.
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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).
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