We take as the basis of every geometry the set of undefined elements (point, line, plane)which constitute space, the set of undefined relations (incidence, intermediacy, congruence) and the set of basic apriori assertions: axioms (postulates). All other elements and relations are defined by means of these primitive concepts, while all other assertions (theorems) are derived as deductive consequences of primitive propositions (axioms). So that, the character of space (this is, its geometry) is determined by the choice of the initial elements and their mutual relations expressed by axioms. The axioms of the usual approaches to geometry can be divided into a number of groups: axioms of incidence, axioms of order, axioms of continuity,axioms of congruence and axioms of parallelism.Geometry based on the first three groups of axioms is called"ordered geometry", while geometry based on the first four groups of axioms is called "absolute geometry "; to the latter corresponds the n-dimensional absolute space denoted by S^{n}.
With respect to congruence, we distinguish the analytic procedure with the introduction of space metric and the synthetic procedure, also called non-metric. The justification for the name “absolute geometry” is derived from the fact that the system of axioms introduced makes possible a branching out into the geometry of Euclid and that of Lobachevsky (hyperbolic geometry). This is achieved by adding the axiom of parallelism. By accepting the 5th postulate of Euclid (or its equivalent, Playfair's axiom of parallelism: "For each point A and line a there exists in the plane (a, A) at most one line p which is incident with A and disjoint from a", where line p is said to be parallel to a) we come to Euclidean geometry. By accepting Lobachevsky's axiom of parallelism, which demands presence of at least two such lines, we come to non-Euclidean hyperbolic geometry, i.e. the geometry of Lobachevsky and that of space L^{n}. In particular, for n = 0 all these spaces are reduced to a point, and for n = 1 to a line; their specific characteristics come to full expression for n = 2, and we distinguish the absolute (S^{2}), the Euclidean (E^{2}), and the hyperbolic plane (L^{2}). If there is no special remark, then the terms "plane" and "space" refer to the Euclidean spaces E^{2} and E^{3} respectively. By a similar extension of the set of axioms, ordered geometry supplemented with two axioms of parallelism becomes affine geometry.
Chapter 1.2 Transformations and Symmetry GroupsA function m is a mapping of a set A to a set B if for every element a ∈ A there exists exactly one element b ∈ B such that m(a) = b. The mapping m is one-to-one if m(a) = m(a') implies a = a', and it is onto if m(A) = B, where m(A) = {m(a) | a ∈ A}. A transformation is a mapping of a space to itself that is both one-to-one and onto, i.e. it is a one-to-one correspondence from the set of points in the space onto itself. If we denote a transformation of space by t, then for each point P which we call the original there exists exactly one point Q, the image of point P derived by transformation t and we write t(P) = Q. Each point Q of the space is the image of some point P derived by transformation t, where to equal images correspond equal originals. Points P, Q in the given order are called homologous points of transformation t.
A figure f is any non-empty subset of points of space. A figure f is called invariant with respect to a transformation S if S(f) = f; in this case the transformation S is called a symmetry of the figure f. The set of points invariant with regard to all the powers of a given symmetry S is called the element of symmetry of the figure f. The identity transformation of space is the transformation E under which every point of space is invariant, i.e. E(P) = P holds for each point P of the space. The identity transformation is a symmetry of any given figure. Any figure whoose set of symmetries consists only of the identity transformation E is called asymmetric; any other figure is called symmetric. For example, the capital letters A,B,C,D,E,K,M,T,U,V,W,Y are mirror-symmetric, H,I,O,X doubly mirror-symmetric and point-symmetric, N,S,Z point-symmetric, and F,G,J,P,Q,R asymmetric. The letters b d or p q form the mirror symetric pairs, and b q or p d the point-symmetric pairs.
For every two transformations S_{1}, S_{2} of the same space we define the product S_{1}S_{2}, as the composition of the transformations: S_{1}S_{2}(P) = S_{2}(S_{1}(P)). In other words, by product we mean the successive action of transformations S_{1}, S_{2}. As a symbol for the composition S∈S, where S occurs n times, we use S^{n}, i.e. the n-th power of the transformation S. The order of the transformation S is the minimal n (n ∈ ℕ) for which S^{n} = E holds. If there is no finite number n which satisfies the given relation, then the transformation S is called a transformation of infinite order. If n = 2, then the transformation S is called an involution. If transformations S_{1} and S_{2} are such that S_{1}S_{2} = E, then S_{1} is called the inverse of S_{2}, and vice versa. We denote this relationship as S_{1} = S_{2}^{-1} and S_{2} = S_{1}^{-1}. For an involution S we have S = S^{-1}, and for the product of two transformations (S_{1}S_{2})^{-1} = S_{2}^{-1}S_{1}^{-1} holds.
A transformation t which maps every line l onto a line t(l) is a collineation. An affine transformation (or linear transformation) is a collineation of the plane that preserves parallels.
As a binary operation * we understand any rule which assigns to each ordered pair (A,B) a certain element C written as A *B = C, or in the short form, AB = C. A structure (G,*) formed by a set G and a binary operation * is a group if it satisfies the axioms:
a_{1}) (closure): for all A_{1},A_{2} ∈ G, A_{1}A_{2} ∈ G is satisfied;
a_{2}) (associativity): for all A_{1},A_{2},A_{3} ∈ G, (A_{1}A_{2})A_{3} = A_{1}(A_{2}A_{3}) is satisfied;
a_{3}) (existence of neutral element): there exists E ∈ G that for each A_{1} ∈ G the equality A_{1}E = A_{1} is satisfied;
a_{4}) (existence of inverse element): for each A_{1} ∈ G there exists A_{1}^{-1} ∈ G so that A_{1}^{-1}A_{1} = E is satisfied.
If besides a_{1}-a_{4}) also holds
a_{5}) (commutativity): for all A_{1},A_{2} ∈ G, A_{1}A_{2} = A_{2}A_{1} is satisfied, the group is commutative or abelian.
The order of a group G is the number of elements of the group; we distinguish finite and infinite groups. The power and the order of a group element are defined analogously to the definition of the power and the order of a transformation.
A figure f is said to be an invariant of the group of transformations G if it is invariant with respect to all its transformations, i.e. if A_{1}(f) = f for every A_{1} ∈ G. All symmetries of a figure f form a group, that we call the group of symmetries of f and denote by G_{f}. For example, all the symmetries of a square (Figure 1.1a) form the non-abelian group, consisting of identity transformation E, reflections R, R_{1}, R_{1}RR_{1}, RR_{1}R, and rotations RR_{1}, (RR_{1})^{2}, R_{1}R - the symmetry group of square D_{4}. The order of reflections is 2, the order of rotations RR_{1}, R_{1}R is 4, and the order of half-turn (RR_{1})^{2} is 2. This group consists of 8 elements, so it is of order 8. The elements of the same group, expressed as products of reflection R and rotation S of order 4 are: identity E, reflections R, RS, RS^{2}, SR, and rotations S, S^{2} and S^{3}. Instead of a square, we may consider the plane tiling having the same symmetry (Figure 1.1b).
A subset H of group G, which by itself constitutes a group with the same binary operation, is called a subgroup of group G if and only if (iff) for all A_{1}, A_{2} ∈ H, A_{1}A_{2}^{-1} ∈ H. Subgroups H = G and H = {E} of each group G are called trivial, while the other subgroups are nontrivial subgroups of the group G. In the symmetry group of square, identity transformation E and rotations S, S^{2}, S^{3} form the subgroup of the order 4 - the rotational subgroup of square C_{4}.
(a) Symmetric figure (square) consisting of equaly arranged congruent parts (1-8) and its symmetry transformations: identity transformation E ( 1 → 1, 2 → 2, 3→ 3, 4 → 4, 5 → 5, 6→ 6, 7 → 7, 8 → 8), reflections R ( 1 → 2, 3 → 8, 4→ 7, 5 → 6), R_{1} ( 1→ 4, 2 → 3, 5 → 8, 6→ 7), R_{1}RR_{1} ( 1 → 6, 2→ 5, 3 → 4, 7 → 8), RR_{1}R ( 1 → 8, 2 → 7, 3→ 6, 4 → 5), rotations R_{1}R ( 1→ 7, 2 → 8, 3 → 1, 4 →2, 5 → 3, 6 → 4, 7 → 5, 8→ 6), RR_{1} ( 1 → 3, 2 → 4, 3→ 5, 4 → 6, 5 → 7, 6 →8, 7 → 1, 8 → 2) and half-turn (RR_{1})^{2} ( 1 → 5, 2 → 6, 3 → 7,4 → 8). The order of the symmetry group of square D_{4} is equal to the number of congruent parts (8); (b) plane tiling with the same symmetry.
Groups (G_{1}, *) and (G_{2}, ∘) are called isomorphic if there exists a one-to-one and onto mapping i of elements of the group G_{1} onto elements of the group G_{2}, so that for all A_{1},A_{2} ∈ G_{1}, i(A_{1} *A_{2}) = i(A_{1}) ∘i(A_{2}) holds; the mapping i is called an isomorphism. For example, by the mapping i(R) = R, i(R_{1}) = RS is defined the isomorphism of the symmetry group of square generated by reflections R,R_{1}, with the same group generated by reflection R and rotation S. Any isomorphism of a group G with itself is called an automorphism.
Instead of representing the group in the traditional way, by means of its Cayley table, which offers a listing of all the elements of the group and their compositions (products), complete information about the group is given more effectively and concisely by a group presentation (i.e. abstract, generating definition): a set of generators and defining relations. The group of transformations G is discrete if for each point P of the space in which the group G acts there is a positive distance d = d(P) such that no image of P (distinct from P) under an element of G is at distance less than d from P. The set { S_{1},S_{2},…,S_{m} } of elements of a discrete group G is called a set of generators of G if every element of the group can be expressed as a finite product of their powers (including negative powers). Relations g_{k}(S_{1},S_{2},…,S_{m}) = E, k = 1,2,…,s, are called defining relations if all other relations which S_{1}, S_{2},…, S_{m} satisfy are algebraic consequences of the defining relations. So that, in further discussions each discrete group will be given by a set of generators and defining relations, i.e. by a presentation.
The symmetry group of square is given by Cayley table:
E | R | R_{1} | R_{1}RR_{1} | RR_{1}R | RR_{1} | (RR_{1})^{2} | R_{1}R | |
E | E | R | R_{1} | R_{1}RR_{1} | RR_{1}R | RR_{1} | (RR_{1})^{2} | R_{1}R |
R | R | E | RR_{1} | (RR_{1})^{2} | R_{1}R | R_{1} | R_{1}RR_{1} | RR_{1}R |
R_{1} | R_{1} | R_{1}R | E | RR_{1} | (RR_{1})^{2} | R_{1}RR_{1} | RR_{1}R | R |
R_{1}RR_{1} | R_{1}RR_{1} | (RR_{1})^{2} | R_{1}R | E | RR_{1} | RR_{1}R | R | R_{1} |
RR_{1}R | RR_{1}R | RR_{1} | (RR_{1})^{2} | R_{1}R | E | R | R_{1} | R_{1}RR_{1} |
RR_{1} | RR_{1} | RR_{1}R | R | R_{1} | R_{1}RR_{1} | (RR_{1})^{2} | R_{1}R | E |
(RR_{1})^{2} | (RR_{1})^{2} | R_{1}RR_{1} | RR_{1}R | R | R_{1} | R_{1}R | E | RR_{1} |
R_{1}R | R_{1}R | R_{1} | R_{1}RR_{1} | RR_{1}R | R | E | RR_{1} | (RR_{1})^{2} |
and by the presentation:
or by Cayley table:
E | R | RS | RS^{2} | SR | S | S^{2} | S^{3} | |
E | E | R | RS | RS^{2} | SR | S | S^{2} | S^{3} |
R | R | E | S | S^{2} | S^{3} | RS | RS^{2} | SR |
RS | RS | S^{3} | E | S | S^{2} | RS^{2} | SR | R |
RS^{2} | RS^{2} | S^{2} | S^{3} | E | S | SR | R | RS |
SR | SR | S | S^{2} | S^{3} | E | R | RS | RS^{2} |
S | S | SR | R | RS | RS^{2} | S^{2} | S^{3} | E |
S^{2} | S^{2} | RS^{2} | SR | R | RS | S^{3} | E | S |
S^{3} | S^{3} | RS | RS^{2} | SR | R | E | S | S^{2} |
and by the presentation:
By "structure of the group" we understand its isomorphism with some of the basic, well known groups (e.g., cyclic group C_{n}, dihedral group D_{n},…) or with a direct product of such groups. The cyclic group C_{n} is given by the presentation: {S} S^{n} = E, and the dihedral group D_{n} can be given by two isomorphic presentations: {R,R_{1}} R^{2} = R_{1}^{2} = (RR_{1})^{n} = E or {S,R} S^{n} = R^{2} = (RS)^{2} = E. Hence, the structure of the symmetry group of square is D_{4}, and the structure of its rotational subgroup is C_{4}.
For groups G and G_{1}, G ∩G_{1} = {E}, given by presentations (1), (2) we define the direct product G×G_{1} as the group with the set of generators {S_{1},S_{2},…,S_{m},S_{1}',S_{2}',…,S_{n}'}, the set of defining relations of which is, besides the relations (1), (2), made up of relations S_{i}S_{j}' = S_{j}'S_{i}, i = 1,2,…,m, j = 1,2,…,n. For each group G we can discuss the possibility of it being decomposed, i.e. represented as the direct product of its nontrivial subgroups. A group which allows such a decomposition we call reducible, otherwise it is called irreducible. For example, the direct product of two cyclic groups, C_{3} given by the presentation {S} S^{3} = E and C_{2} given by {T} T^{2} = E is the group {S,T} S^{3} = T^{2} = E ST = TS. By the substitution U = ST, this results in the presentation {U} U^{6} = E, so C_{3}×C_{2} @ C_{6}, showing that the group C_{6} is reducible.
The term "decomposition" can be used in another sense. Each group can be decomposed according to its subgroup H:
According to those basic geometric-algebraic assumptions, we can consider as the subject of this study the analysis of plane figures - ornamental motifs and their invariance with respect to symmetry groups.
The set of points G(P) = {g(P) | g ∈ G}, obtained from a point P by all transformations of the group G, is called the orbit of P with respect to G; it is the set of points equivalent to point P (or the transitivity class of P) with respect to the group G. Analogously we can also define the orbit (or transitivity class) of any figure f with respect to the group G and denote it by G(f). A point P which is invariant with respect to a transformation S, i.e. a point for which S(P) = , is also called singular. A figure f is invariant with respect to a transformation S if S(f) = f. A point P is a singular (invariant) point of a group G if it is a singular (invariant) point of all transformations of G. A point which is not an invariant point of a transformation S is also called a point in general position with respect to the transformation S. A point is said to be a point in general position with respect to a group of transformations G if it is in general position with respect to all the transformations of the group G, i.e. if it is not an invariant point of any transformation of the group G. For example, the singular (invariant) point of the symmetry group of square is the center of square. The points belonging to the mirror-reflection lines are the invariant points of the corresponding reflections. All other plane points, are the points in general position with respect to the symmetry group of square (Figure 1.1).
The orbit of some point P in general position with respect to the discrete group of transformations G makes possible a schematic interpretation of the group G: a Cayley diagram or a graph of the group G - a visual model of discrete group of transformations G. To each vertex of the graph corresponds exactly one element of the group, and to each edge corresponds one transformation. The edges which connect the homologous points of the same transformation are denoted by the same type of line (full, broken, dotted). The non-oriented edges correspond to the involutions. For any other, oriented edge, the motion in the direction of the arrow indicates the multiplication by the corresponding transformation from the right, and the motion in the opposite direction of the arrow corresponds to multiplication by the inverse of the corresponding transformation on the right. A Cayley diagram is a connected graph, i.e. there exists a path which connects every two vertexes of the graph. It represents the direct visual interpretation of the presentation of the group, since to every closed cycle there corresponds one defining relation (1). A complete graph is considered to be the graph in which every two vertexes are directly linked by the edge (Figure 1.2).
(a) Graph of the group C_{4} given by the presentation {S} S^{4} = E; (b) the complete graph of the same group. |
For a discrete group G it is possible to define a fundamental region of G. A fundamental region F is a figure which satisfies the following conditions:
a) for each point P of the space where the group of transformations G acts, there exists S � G that P � S(F);
b) for each S � G\{E} holds int(F)�int(S(F)) = �. If Cl(F) is the closure of F, the orbit G(Cl(F)) represents a tiling of the space on which the group G acts. A space tiling or tessellation is a countable family of closed sets T = {T_{1},T_{2},�} covering space without gaps or overlaps. More explicitly, the union of the sets T_{1}, T_{2},�, which are known as the tiles of T, is to be the whole space, and the interiors of the sets T_{i} are to be pairwise disjoint (B. Grünbaum, G.C. Shephard, 1987). Since a fundamental region F has no points which are equivalent under any transformation of the group G, unless they are on the boundary, each internal point of F is a point in general position with respect to the group G. Regarding the extent of the fundamental region we distinguish between groups with bounded and unbounded fundamental regions. A discrete group of transformations G usually does not determine uniquely the fundamental region, or the induced tiling G(Cl(F)). Therefore, it is of interest to inquire about the different possible shapes of the fundamental region. In the tiling G(Cl(F)) the intersection of tiles of any finite set of tiles (containing at least two distinct tiles) may be empty or may consist of a set of isolated points (vertices) and arcs (edges). When discussing variations of the form of the fundamental region F we distinguish between two aspects of change: the change in the number of vertices and edges of the fundamental region F, and the change of the form of the edges (arcs) themselves in which the number of vertices and edges remains unchanged. As the result of the action of the symmetry groups we have tile-transitive or isohedral tilings. Their tiles belong to the same class of transitivity G(Cl(F)), since for every two tiles of G(Cl(F)) there exists a transformation of group G which maps one tile onto the other (Figure 1.3).
(a) Isohedral plane tiling corresponding to the symmetry group D_{4}; (b) two isohedral plane tilings with different shape of the fundamental region, corresponding to its rotational symmetry subgroup C_{4}. |
If the symmetry group G_{T} contains also transformations which map any vertex of tiling T onto any other vertex, i.e. if the vertices make up one class of transitivity, the tiling is said to be isogonal. By a flag in a tiling we mean a triple (V,E,T) consisting of a vertex V, an edge E and a tile T which are mutually incident. A tiling is called regular if its symmetry group is transitive on the flags of the tiling. In particular, for the symmetry groups of ornaments there exist exactly three regular tilings (regular tessellations) by means of regular polygons. Each of them can be denoted by a Schläfli symbol {p,q} denoting regular p-gons, where q of them are incident with each vertex of the regular tessellation: {4,4} , {3,6} , {6,3} . A dual of regular tiling {p,q} is the regular tiling {q,p} (Figure 1.4).
Regular tilings {4,4} , {3,6} and {6,3}. |
A uniform or Archimedean tiling is an isogonal plane tiling by regular polygons, which is edge-to-edge, i.e. in which every vertex and edge of a tile is a vertex and edge of the tiling. Each of the 11 types of uniform tilings can be denoted by the symbol (p_{1}^{q1} p_{2}^{q2} �p_{n}^{qn}) where p_{1},p_{2},�,p_{n} denote regular p-gons, and q_{1},q_{2},�,q_{n} the number of adjacent regular p-gons of the same type which are incident with one vertex. Besides regular tessellations (3^{6}) = {3,6} ,(6^{3}) = {6,3} and (4^{4}) = {4,4} the family of uniform tilings consists of (3^{4}.6), (3^{3}.4^{2}), (3^{2}.4.3.4), (3.4.6.4), (3.6.3.6), (3.12^{2}), (4.6.12) and (4.8^{2}) (J. Kepler, 1619) (Figure 1.5). The Archimedean tiling (3^{4}.6) occurs in two enantiomorphic forms - "left" and "right".
Archimedean tilings. |
An open circle (or open circular disk) is the set of points X such that OX < r, where O is a fixed point and r is a positive number. For OX � r, the circle (circular disk) is called a closed circle.
A transformation t is continuous if for any two points P, Q of the plane it is possible to make t(P) and t(Q) as close together as we wish, by taking P and Q sufficiently close, and bicontinuous if both t and t^{-1} are continuous. A homeomorphism or topological transformation is any bicontinuous transformation. The open (closed) topological disk is any plane set which is homeomorphic image of an open (closed) circle.
A tiling T is normal if:
a) every tile of T is a topological disk;
b) the intersection of every two tiles of T is a connected set, i.e. does not consist of two closed and disjoint subsets;
c) the tiles of T are uniformly bounded, i.e. there exist circles c and C, with fixed radiuses, such that every tile T_{i} of tiling T contains a translate of c and is contained in a translate of C.
A tiling T is called homeohedral if it is normal and is such that for any two tiles T_{1}, T_{2} of T there exists a homeomorphism of the plane that maps T onto T and T_{1} onto T_{2}. A normal tiling is called two-homeohedral if its tiles form two transitivity classes under a homeomorphism mapping T onto itself. For example, all non-regular Archimedean tilings (Figure 1.5) are two-homeohedral.
A continuous set of points is any set of points which satisfies the axiom(s) of continuity. Every continuous set of points is a homeomorphic image of a line. Alongside the discrete groups of transformations, continuous symmetry groups may also be discussed. A symmetry group G of the space E^{2} or E^{2}\{O} is called continuous if the orbit G(P) of a point in general position P with respect to the group G satisfies one of the following conditions:
(i) G(P) is the complete space on which G acts; or
(ii) G(P) can be divided into disjoint continuous sets of points, and for every point of each of these sets there is a positive distance d = d(P) such that the circle c(P,d) contains no points of any other of the sets mentioned. By the terms "continuous group of translations, rotations, central dilatations and dilative rotations" we mean that all translations along one line, all rotations around one center, all central dilatations with a common center, and all dilative rotations with a common center and with a fixed angle, are elements of such a group. In particular, the continuous symmetry groups of ornaments, depending on whether they satisfy condition (i) or (ii), are called the symmetry groups of continua or semicontinua.