The Symmetry of Two-dimensional Patterns

As prelude to the symmetry of three-dimensional crystals and organisms

Subpatterns and Subgroups Part XXXVII (Subgroups and Antisymmetry in Ornaments)
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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 in all five antisymmetry patterns from the generating P4mm symmetry pattern as depicted in Figure 7 of Part XXXI ,  and analysed them as to their identity.
They were the following :

P4mm / C2mm  ( Part XXXI )    click HERE to see it.
P4mm / P4        ( Part XXXIII )  click HERE to see it.
P4mm / P4mm  ( Part XXXIV )  click HERE to see it.
P4mm / P4mm  ( Part XXXV )   click HERE to see it.
P4mm / C2mm  ( Part XXXVI )  click HERE to see it.

Here we will derive yet another antisymmetry pattern, namely P4mm / P4gm . It originates from a different P4mm symmmetry pattern than the one that has been used in the foregoing.

In part XXXIII we considered a P4mm / P4 antisymmetry pattern derived from the generating P4mm symmetry pattern of Figure 7 of Part XXXI ,  as was used throughout in Parts XXXI--XXXVI,  by replacing the generating reflection by its corresponding antisymmetry reflection. The result was the following :

Figure 1.  Antisymmetry pattern derived earlier ( Part XXXIII ).
Let us look for glide lines.
The next Figures show that the antisymmetry pattern of Figure 1 has no glide lines at all, especially not at the positions and directions indicated in that Figure. If, however, the motifs s.str. (i.e. the commas) are removed, the resulting pattern has glide lines in the mentioned positions and directions, and this (antisymmetry) pattern, so obtained, then has a different identity, namely  P4mm / P4gm .

Figure 2.  Illustration of the  a b s e n c e  (upper image) of horizontal glide lines in the positions indicated in Figure 1 in the  P4mm / P4  antisymmetry pattern, and their  p r e s e n c e  (lower image) in the  P4mm / P4gm  antisymmetry pattern.  The same applies with respect to the corresponding vertical direction.

Figure 2a.  Illustration of the  a b s e n c e  (upper image) of diagonal glide lines in the positions indicated in Figure 1 in the  P4mm / P4  antisymmetry pattern, and their   p r e s e n c e  (lower image) in the  P4mm / P4gm  antisymmetry pattern. The same applies for the other diagonal direction.
From the above it seems that for the two antisymmetry patterns possessing glide lines at certain positions or not, has something to do with the presence of certain  motifs  in one of the patterns. But, as we will see further below, the other antisymmetry pattern has motifs too. And they too will forbid those glide lines.
Further, any antisymmetry pattern, can, as pattern, be interpreted in two ways, viz. (1) as genuine antisymmetry pattern, which means that the color change does not signify an asymmetry, but represents a symmetry transformation, namely an antisymmetry transformation, and (2) as just a symmetry pattern, where color change is supposed to indicate asymmetry, in the same way that going over the two parts of a comma (divided by a line, say) in the sense of going from one part to the other, is an asymmetry.  And this makes a difference as to whether certain glide lines are present or not. If the antisymmetry pattern represented in the upper image of the above Figure is indeed interpreted as an antisymmetry pattern, then it does possess diagonal glide lines, albeit at other locations (See next Figure). All this will be further worked out below.

Figure 3.  If the pattern of the upper image of Figure 2a is interpreted as an antisymmetry pattern, as in both images of the present Figure, then it has glide lines, but on different locations (One of them indicated). The color change is a true symmetry (See also lower image).
If, on the other hand, the pattern is not interpreted as an antisymmetry pattern, but just as a symmetry pattern, the color change is not a symmetry (but an asymmetry), implying that there are no glide lines present anywhere in the pattern. In fact if so interpreted, the pattern has a symmetry according to the plane group  P4  (which indeed has no glide lines at all) instead of  P4mm / P4 ,  and is not isomorphic to the plane group  p4mm ,  i.e. to the generating symmetry pattern from which the present pattern was derived.
So far so good with regard to the P4mm / P4 antisymmetry pattern.
Let us now prepare for a derivation of the  P4mm / P4gm  antisymmetry pattern, as alluded to above.
The appropriate generating P4mm symmetry pattern must then be the following :

Figure 3a.  Generating  P4mm  symmetry pattern from which an antisymmetry pattern will be derived. Each square represents a motif of this pattern, and because of the symmetry of this motif, it consists of eight triangular motif units (not shown), that represent group elements (Shown in the next Figure). The color blue can be considered as background color. The pattern is in fact (also) a  p l a n e   t i l i n g  where each tile is a square. These squares fill the plane completely, i.e. without leaving gaps, which is why it is a tiling.
The next Figure gives this same generating P4mm pattern, but now with the boundaries of the areas representing group elements indicated.

Figure 4.  Generating  P4mm  symmetry pattern (as already depicted in the previous Figure), from which we will derive an antisymmetry pattern. All lines, whether black or (dark) blue, signify boundaries of the areas representing group elements. These areas have the shape of an isosceles right-angled triangle. They are at the same time fundamental regions of the pattern because they tile the plane completely. In fact, the pattern of the present Figure is also a genuine  t i l i n g  of the 2-dimensional plane, covering that plane with the same (i.e. congruent) isosceles right-angled triangles without leaving gaps. But in the present representation of the pattern these triangular tiles do not represent the pattern's motifs (as did the (square) tiles in the previous Figure). The motifs of our generating  P4mm  symmetry pattern are supposed to be the squares of the previous Figure (In fact squares with their empty content).
The next Figure gives again the generating P4mm symmetry pattern, but now with the motifs s.str. emphasized (strong black lines).

Figure 4a.  Generating  P4mm  symmetry pattern. Motifs s.str. emphasized (strong black lines). A motif in the broader sense is a black square together with its empty content.  Eight motif units are indicated (yellow, purple).
Let's analyse this generating P4mm pattern (Figures 5 -- 12), before we derive from it an antisymmetry pattern.

The next Figure indicates the motif and the motif unit of the generating pattern of Figure 4 and, equivalently, of Figure 3a, and especially Figure 4a.  The motif unit constitutes at the same time the fundamental region of the pattern, and it is an area representing a group element. A unit mesh of the pattern -- i.e. the smallest unit that is repeated periodically, and so builds up the whole patttern -- consists of eight such areas together forming a four-fold rosette, the (whole) motif. There is no motif s.str., unless we interpret the lines that border the motif, i.e. (that border) the square, as such. The motif unit is still symmetric, according to a reflection in the line bisecting the right angle of that motif unit, but this reflection is not a symmetry transformation of the whole pattern (See Figures 5a, 5b, and 5c, below). So this motif unit represents indeed the ultimate group element.

Figure 5.  Motif and motif unit of the generating  P4mm  symmetry pattern as given in Figure 4 and Figure 3a .
The next three Figures show that the reflection line of the motif unit (and that of any motif unit present in the pattern) is not a symmetry transformation of the whole pattern (with its square motifs).

Figure 5a.  Generating  P4mm  symmetry pattern. Group element boundaries indicated. Motifs s.str. emphasized (strong black lines). The reflection line  ( m' )  of a motif unit is not a reflection line of the whole pattern (with its square motifs). See also next Figure.

Figure 5b.  Generating  P4mm  symmetry pattern. Group element boundaries not (completely) indicated. The reflection line  ( m' )  of a motif unit is not a reflection line of the whole pattern (with its square motifs). This holds for all motif units (See next Figure).

Figure 5c.  Generating  P4mm  symmetry pattern. Group element boundaries indicated. The reflection line of  a n y  motif unit is not a reflection line of the whole pattern (with its square motifs).
The next Figure gives the point lattice of the generating pattern as depicted in Figure 4 .

Figure 6.  Point lattice (indicated by strong dark blue connection lines) of the generating  P4mm  symmetry pattern of Figure 4 .  A unit mesh is indicated by alternative coloring (green). Each mesh also represents the motif of the pattern.
This initial pattern can, like the pattern originally used (Figure 7 of Part XXXI ), be realized by the following set of generators :  {mse,  p,  th},  i.e. the reflection in the line  mse  (indicated in the next Figure), 900 anticlockwise rotation about the axis indicated in the next figure, and the horizontal translation.

Figure 7.  The above given generating  P4mm  pattern and its possible generators :  The reflection  mse ,  the rotation  p  ( 900 anticlockwise rotation about the axis indicated by a small solid red square) and the horizontal translation  th .  The identity element is indicated (yellow).
The next Figures show how the generating P4mm pattern, as given in the above Figures, can indeed be generated by the three mentioned generators.

Figure 8.  First phase of the construction of the above given generating  P4mm  pattern by the above proposed generators.
The identity element is reflected in the line  mse  yielding one more element.

Figure 9.  Second phase of the construction of the above given generating  P4mm  pattern by the above proposed generators.
The two elements already present are subjected to the repeated rotation  p  (i.e. to p,  p2  and  p3  [p4 = 1] ),  resulting in a  D4  rosette (yellow), consisting of eight elements in all.

Figure 10.  Third phase of the construction of the above given generating  P4mm  pattern by the above proposed generators.
The horizontal translation is repeatedly applied to the elements already present, resulting in a horizontal band of elements.

Figure 11.  Fourth phase of the construction of the above given generating  P4mm  pattern by the above proposed generators.
The rotation  p  is applied to the elements already present, resulting in a vertical band of new elements.

Figure 12.  Fifth and final phase of the construction of the above given generating  P4mm  pattern by the above proposed generators.
The horizontal translation is repeatedly applied to the elements already present, resulting in the whole pattern.
Now that we have analysed our generating P4mm pattern, we are ready to derive from it an antisymmetry pattern having a structure not yet encountered in the previous documents. The derivation goes along the same lines as was done in Part XXXIII for the derivation of the pattern redepicted in Figure 1 .

This new antisymmetry pattern will be derived from the generating P4mm symmetry pattern of Figure 3a and (with boundaries of group elements added) Figure 4 ,  by replacing the generating reflection  mse  by its corresponding antisymmetry reflection  e1mse  (See Figure 7 ),  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 13.  First phase of the derivation of the antisymmetry pattern, as specified above, from the generating  P4mm  pattern as depicted in Figure 4 .  See also Figure 7 ,  where the generators of the P4mm pattern are indicated.
The antisymmetry reflection  e1mse  is applied to the identity element, effecting a color change.

Figure 14.  Second phase of the derivation of the antisymmetry pattern, as specified above, from the generating  P4mm  pattern as depicted in Figure 7 .
Repeated application of the generating rotation  p  results in a four-fold rosette.

Figure 15.  Third phase of the derivation of the antisymmetry pattern, as specified above, from the generating  P4mm  pattern as depicted in Figure 7 .
The generating horizontal translation  th  is applied to elements already obtained earlier.

Figure 16.  Beginning of fourth phase of the derivation of the antisymmetry pattern, as specified above, from the generating  P4mm  pattern as depicted in Figure 7 .
The antisymmetry reflection  e1mse  is applied to elements already obtained earlier. Here demonstrated for two elements.

Figure 17.  Completion of fourth phase of the derivation of the antisymmetry pattern, as specified above, from the generating  P4mm  pattern as depicted in Figure 7 .
The antisymmetry reflection  e1mse  is applied to the rest of the elements obtained earlier (Figure 15 ).

Figure 18.  Beginning of fifth phase of the derivation of the antisymmetry pattern, as specified above, from the generating  P4mm  pattern as depicted in Figure 7 .
The generating horizontal translation  th  is applied to some elements already obtained earlier.

Figure 19.  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 .
The generating horizontal translation  th  is applied to the rest of the elements already obtained earlier (Figure 17 ).
Restoration of purple to red and yellow to blue gives our antisymmetry pattern:

Figure 20.  Antisymmetry pattern derived from the generating  P4mm  pattern as depicted in Figure 7 ,  according to the above specifications (i.e. by replacing the generating reflection  mse  by its corresponding antisymmetry reflectiom  e1mse ).

Figure 21.  Same as previous Figure.  Indications of symmetry elements (rotation axis, mirror line) omitted.
The next Figure gives the point lattice of the just derived antisymmetry pattern.

Figure 22.  Point lattice (indicated by yellow connection lines) of the just derived antisymmetry pattern (Figure 21 ).  A unit mesh is indicated by alternative colors. Compare with the point lattice of the generating  P4mm  pattern as it is depicted in Figure 6 .
In the next document we will analyse this derived antisymmetry pattern.

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