Composition of Transformations Glide Relfection Art Glide Reflection Art
As an illustration of the methodological arroyo used in this work, we will requite the example of one symmetry group and its complete comparative assay from the signal of view of the theory of symmetry and ornamental art.
Let the discrete group of isometric transformations of the plane Due east ii, generated by a glide reflection P and a reflection R 1, be given by the presentation:
{P,R i}R one 2 = (R 1 P)2 = East.
(R 1 P)two = E � R 1 PR 1 = P -1 � R 1 -ane PR 1 = P -1 � P R = P -1.
The group discussed possesses an invariant space E 2 - the airplane, an invariant subspace Due east 1 - a line, and has no invariant points. Therefore it belongs to the category of symmetry groups of friezes Grand 21 - the line groups of the airplane E 2 (S ii) without invariant points. Distinguishing betwixt the spaces E 2, S 2, L 2 is non necessary because nosotros are dealing with the line groups. Because this group is generated past a glide reflection P perpendicular to the reflection R 1, its crystallographic symbol will exist pmg, or in short grade mg (One thousand. Senechal, 1975). Within the crystallographic symbol pmg, p denotes the presence of a translation X = P two, i.eastward. the translational subgroup 11={X}; the symbol m denotes a reflection R one perpendicular to this translation, and the symbol g denotes the glide reflection P. In the short symbol mg, the translation symbol p is omitted.
Since the set P,R one is a generator set of the grouping mg, after concluding that the reflection line of R one is perpendicular to the axis of the glide reflection P, we tin can construct an appropriate ornamental motif, the visual model of the frieze symmetry group mg. This is achieved by applying the transformations P and R ane to the called asymmetric figure, which belongs to a fundamental region of the symmetry group mg (Figure 1.19).
Using the substitution T = R 1 P we come to an algebraic equivalent of the previous presentation of the group mg - a new presentation of the same group:
{S 1,S 2}S ane two = S two 2 = (Due south 1 S ii) northward = E.
{S 1,Southward ii} S 1 2 = Due south 2 two = E
Instead of the disproportionate figure, which under the action of the group mg gives the frieze pattern, by considering the orbit of the closure of a central region of the group mg we obtain the corresponding frieze tiling. The key region of the group mg and all other frieze symmetry groups, is unbounded and allows the variation of all boundaries which do not vest to reflection lines. Figure one.21 shows two of these possibilities.
The Cayley diagram of the group mg is derived as the orbit of a point in general position with respect to the group mg. Instead of a straight mutual linking of all vertexes (i.e. orbit points) and obtaining the complete graph, we can, aiming for simplification, link only the homologous points of the group generators. By denoting with the cleaved oriented line the glide reflection P, and with the dotted non-oriented line the reflection R one, nosotros get the Cayley diagram which corresponds to the first presentation of the grouping mg (Figure i.22a).
By an analogous procedure nosotros come up to the graph which corresponds to its second presentation with the generator set {R 1,T}, where a one-half-plow is indicated with the dot-dash line (Figure one.22b).
Let us notation besides, that the defining relations tin be read off directly from the graph of the group. Each cycle, i.e. closed path in which the showtime point coincides with the endpoint, corresponds to a relation between the elements of the group and vice versa. Cayley diagrams (graphs of the groups) may also very efficiently serve to make up one's mind the subgroups of the given symmetry grouping. Namely, every connected subgraph of the given graph satisfying the following condition determines a certain subgroup of the grouping discussed, and vice versa. The condition in question is: an element (transformation) is included in the subgraph either wherever it occurs, or not at all (i.due east. information technology is deleted). Of course, to be able to determine all the subgroups of a given group, it is necessary to use its complete graph equally the footing for defining the subgraphs.
Since in the group mg there are indirect isometries, this group does not requite enantiomorphic modifications. For the groups consisting only of direct symmetries, the enantiomorphic modifications can be obtained by applying the "left" (e.g., b) and "right" (d) form of an simple asymmetric figure. For example, for the group 11, generated by a translation X, this results in the enantiomorphic friezes: bbbbbbbbbbbbbbbbbbbbbbbbbbbbb and dddddddddddddddddddddddddddddd. The translation axis l of the grouping mg is not-polar, because at that place exists an indirect transformation, the reflection R 1 for which the relation R one(l) = -l holds. Rotations of the club 2 in the group mg are polar because each circle c drawn effectually the center of rotation of the order ii is invariant only with respect to this rotation and to the identity transformation E, so that the group Cii ( 2) (generated by the half-turn T) of transformations preserving the circumvolve c invariant, a rosette subgroup C2 (2) of the group mg, consists of direct transformations. Besides the rosette subgroups C2 ( two), the group mg has as well the rosette subgroups Dane (yard), namely the one generated past the reflection R one, or by its conjugates.
The group mg contains as subgroups the following symmetry groups of friezes: p1 (xi) generated by the translation Ten = P ii, p1g (1g) generated past the glide reflection P, pm1 (m1) generated by the translation 10 and the reflection R one, and itself. Besides the list of all frieze groups, subgroups of the grouping mg, the table of the minimal indexes of subgroups of the given group points out the possible desymmetrizations which atomic number 82 to this subgroup. In particular, considering the use of antisymmetry and colour- symmetry desymmetrizations, from this table nosotros can come across that antisymmetry desymmetrizations of group mg result in the subgroups of the alphabetize 2: 1g, 12 and m1. This can be accomplished past a blackness-white coloring (or, e.g., 1-two indexing) according to the laws of antisymmetry, using the following systems of (anti)generators: {P,e ane R one} or {e 1 R 1,e ane T} for obtaining the antisymmetry desymmetrization mg/1g; {eastward i P,R ane} or {R 1,e i T} for obtaining the antisymmetry desymmetrization mg/m1; {e one P,due east 1 R ane} or {e i R 1,T} for obtaining the antisymmetry desymmetrization mg/12, where eastward1 = (12), i.eastward. the group of colour permutations P N = P 2 = C 2 (Figure 1.23).
The junior antisymmetry groups obtained can be understood also equally adequate visual interpretations of the symmetry groups of bands Thousand 321 - as the Weber diagrams of the symmetry groups of bands p2111, pm11 and p112 respectively. In this case the alternation of colors white-black is understood in the sense "higher up-under" the invariant aeroplane of the frieze, i.e. as the identification of the antiidentity transformation e ane with the airplane reflection in the invariant aeroplane of the group mg. The seven generating symmetry groups of friezes G 21, seven senior antisymmetry groups and seventeen junior antisymmetry groups correspond to the 31 groups of symmetry of bands, offer complete information on their presentations and structures.
Using N = four colors and the system of colored generators {c ane P,c two R 1} or {c 2 R 1,c 1 c two T}, we get the colour-symmetry desymmetrization mg/11, where c 1 = (12)(34) and c two = (thirteen)(24); hence, the group of color permutations is P Due north = P iv = C two×C ii = D 2 (Figure 1.24).
In all the antisymmetry and colour-symmetry desymmetrizations mentioned, for which the grouping P N is regular, the subgroup H derived past the desymmetrization is a normal subgroup of the grouping mg (1g, m1, 12, 12). Because of this, complete information on the antisymmetry or colored symmetry group, i.east. on the corresponding desymmetrization, is given past the number Due north and by the group/subgroup symbol G/H. The next case of coloring with N = iii colors, the irregular group P Due north and the subgroup H which is not a normal subgroup of the group G, demands the symbols G/H/H 1. In this case, as well the number North, the group of colored symmetry G * , i.e. the corresponding color-symmetry desymmetrization is uniquely defined by the generating group 1000, the stationary subgroup H of G * , which maintains every individual index (colour) unchanged and its symmetry subgroup H 1 which is the final result of the color-symmetry desymmetrization. The index of the subgroup H in the group G is equal to North and the product of the index of the subgroup H 1 in grouping H and the number N is equal to the order of the grouping of color permutations P Northward , i.e. [G:H] = N, [H:H 1] = N 1, and the order of the group P N is NN one.
Every bit an example of the irregular case we tin use the color-symmetry desymmetrization of the group mg obtained by N = three colors, i.due east. past the system of colored generators: {c one P,c 2 R 1} or {c two R 1,c 1 c 2 T}, which results in the color-symmetry desymmetrization mg/mg/1g, where c 1 = (123), c 2 = (23), P N = P iii = D three and [mg: mg]=3, [mg:1g]=ii. This color-symmetry desymmetrization mg/mg/1g, N = 3 is shown on Figure 1.25a, while the stationary subgroup H (mg) which maintains each individual index (color) unchanged is singled out on Figure 1.25b. All cases of subgroups which are not normal subgroups of the given group are denoted in the tables of (minimal) indexes of subgroups in groups by italic indexes (e.k., [mg:mg]=3).
In terms of construction, for frieze grouping mg we tin also distinguish the rosettal method of construction - the multiplication of a rosette with the symmetry group C2 (2) (generated by the half-turn T) or D1 ( 1000) (generated by the reflection R 1) by the glide reflection P (Effigy ane.26a, b). Like all other symmetry groups of friezes, the group mg is the subgroup of the maximal symmetry group of friezes mm generated past reflections. Since it is the normal subgroup of the index 2, the antisymmetry desymmetrization of the generating group mm with a set of generators {Ten,R,R i} or {R,R 1,R 2} where X is the translation, R the reflection in translation axis line, and R ane, R two reflections with reflection lines perpendicular to the translation axis, can exist used.
Past means of the system of (anti)generators: {e 1 X,east i R,R ane} = {e ane X, e 1 R,e i R one} or {due east one R,R ane,eastward 1 R 2} the antisymmetry desymmetrization mm/mg is obtained (Effigy ane.27), where east i = (12), P N = P 2 = C ii, [mm:mg] = two.
Many visual properties of the group mg, e.chiliad., a relative constructional and visual simplicity of corresponding friezes conditioned by a high degree of symmetry, specific residue of the stationariness conditioned by the presence of reflections, by the non-polarity of the glide reflection axis, by the absence of enantiomorphism, and the dynamism conditioned by the presence of glide reflection and by polar, oriented rotations, are the direct consequences of the algebraic-geometric characteristics mentioned. Also, the unlike possibilities that the grouping mg offers, e.g., the possibilities for antisymmetry and color-symmetry desymmetrizations, the ways of varying the form of the cardinal region, construction possibilities etc., get evident afterwards the analysis of this symmetry grouping of friezes from the point of view of the theory of symmetry.
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