SYMMETRY MATHEMATICAL CONCEPTS AND APPLICATIONS IN TECHNOLOGY AND ENGINEERING


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1 Daniel DOBRE, Ionel SIMION SYMMETRY MATHEMATICAL CONCEPTS AND APPLICATIONS IN TECHNOLOGY AND ENGINEERING Abstract: This article discusses the relation between the concept of symmetry and its applications in engineering. "Symmetry" is interpreted in a broad sense as repeated, coplanar shape fragments. An analysis of symmetry, which justifies its applications in engineering is given and discussed. After a brief explication of group theory and symmetry types, we show that there are industrial workpieces where symmetry is omnipresent. The analysis of symmetry can also be utilized for future research concerns the combining symmetry information with other functional characteristic of digital 3D design. Key words: symmetry, space group, automorphism, screw symmetry, regular polyhedron. 1. INTRODUCTION Symmetry is a vast subject, important in art and nature, based on mathematics. There are a wide variety of applications of the principle of symmetry in art, in the inorganic and organic nature. The notion of symmetry is equivalent to the harmony of proportions, defining the idea that man has tried, over time, cover and create order, beauty and perfection. It is seen in various forms: bilateral symmetry, translation symmetry, rotational symmetry, ornamental symmetry, crystallography symmetry etc. An object or structure is symmetrical if it looks the same after a specific of change is applied to it. The object or structure can be material, such as living organisms including humans and other animals, crystal, regular polyhedron, pavement tiles, or it can be an abstract structure such as a mathematical equation. The nature of the change can be similarly diverse, ranging from such simple operations as moving across a regularly patterned tile floor, to complex transformations of equations. This paper describes symmetry from two perspectives. The first is that of mathematics, in which symmetries are defined and categorized precisely. The second perspective looks at the application of symmetry concepts to engineering [2] [3]. displacement as composition, forms an infinite order commutative group, the continuous spatial displacement group. The group of similarities leaves the shape of a figure unchanged. The size of a figure is invariant with respect to the group of congruencies [1]. In mathematics, there is a major difference between discrete and continuous groups. Examples of discrete groups are the finite rotation groups of polygons and crystals. In one dimension, ornaments of stripes are classified by seven groups, which are systematically produced by periodic translations in one direction and reflections transverse to the longitudinal axis of translations (fig. 1). 2.2 Geometric symmetry In geometry, symmetric properties of figures and bodies indicate invariance with respect to automorphisms like rotations, translations and reflections. 2. MATHEMATICAL CONCEPT OF SYMMETRY An object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation does not change the object. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa) [1]. 2.1 Group theory In modern times, symmetries are defined mathematically by group theory. The symmetry of a set (e.g. points, numbers and functions) is defined by the group of automorphism that leaves unchanged the structure of the set (e.g. proportional relations in Euclidean space, arithmetical rules of numbers). The set of all displacement transformations in space, including the identity transformation, with consecutive Fig. 1 Symmetries in one dimension. (1translational symmetry; 2reflection symmetry; 3 7composite symmetries) JUNE 2009 NUMBER 5 JIDEG 21
2 In two dimensions, there is an axis of symmetry. The axis of symmetry of a twodimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that the shape was to be folded in half over the axis, the two halves would be identical: the two halves are each other s mirror image. Thus a square has four axes of symmetry because there are four different ways to fold it (fig. 2). A reflection in [P] is the application of space itself, S: A A 1, which leads arbitrary point A in reflection of A 1 vs. [P]. A reflection point reflects a point V to the opposite side of the point, as illustrated in fig. 5. All three reflection symmetries associate exactly one reflected point A 1 with each point A on the surface [4]. The axial symmetry associates each point A with an entire circle of points A 1, which includes A. The center of the circle intersects with the symmetry axis. The plane of the circle is orthogonal to this axis (fig. 6). Fig. 2 Symmetry lines of polygons. In R 3 there exist three basic symmetry features: a point, an axis and a plane. Each of these features establishes equivalence classes of multiple surface points. We also distinguish reflection, axial and spherical symmetries. In detail, reflection symmetry associated each point A on the object surface to another surface point A 1 on the opposite side of the object. For example, a reflection line reflects points across a line. Fig. 3 illustrates such a reflection line, which reflects a point D to a reflection point D 1 on the opposite side of the line. Fig. 4 Symmetry of pyramid given the symmetry plane. Fig. 5 Symmetry of pyramid given the symmetry point. Fig. 6 Axial symmetry. 22 Fig. 3 Symmetry of prism given the symmetry axis. Plane reflection symmetry reflects points to the opposite side of the symmetry plane. This is illustrated in fig. 4, which shows an object with an associated symmetry plane [4]. JUNE 2009 NUMBER 5 JIDEG The rotation transformation acts on a 3dimensional system by rotating the whole system through a given angle about a given axis, called the rotation axis. A 3dimensional system might be symmetric under rotations by any angle about one or more axes or only by
3 a minimum angle of 360 /n, where n is an integer greater than 1, and multiples of it. Such a rotation axis is called an axis of full or nfold rotational symmetry, respectively. A sphere, for example, has an infinite number of axes of full rotational symmetry all axes passing through its center. Systems having this type of symmetry are said to posses spherical symmetry. Spherical symmetries reflect each point A on the object surface to an entire sphere, whose center is the symmetry point. Figure 7 illustrates a spherical symmetry. A number of axes of full rotational symmetry are indicated. A regular tetrahedron has three axes of 2fold rotational symmetry (through the midpoints of pairs of opposite edges) and four 3fold rotational symmetry axes (through each vertex and the center of the opposite face). To calculate the order of the group, observe that a given vertex can be moved to one of four positions. Hence, the order of the group of direct symmetries (all rotations) is S(T) = 24. A cube has six 2fold axes (through the midpoints of pairs of opposite edges), four 3fold axes (through pairs of opposite vertices) and three 4fold axes (through the centers of pairs of opposite faces. Fig. 7 Spherical symmetry. A cylinder has full rotational symmetry only about a single axis, the longitudinal axis of the cylinder. Also a cone possesses this symmetry called axial symmetry (fig. 8) The symmetry group of the cube or octahedron S(C) They both have the same number of edges, being 12. The number of faces and vertices are interchanged. Because these two solids are dual to each other they have the same symmetry group. The order of the group of direct symmetries (all rotations) is S(C) = 24. The elements are:  3 rotations (by ± π/2 or π) about centers of 3 pairs of opposite faces;  1 rotation (by π) about centers of 6 pairs of opposite edges;  2 rotations (by ± 2π/3) about 4 pairs of opposite vertices (diagonals). Together with the identity this accounts for all 24 elements. Fig. 8 3dimensional systems with axial symmetry. An infinitely long cylinder with no ends has, in addition to its axial symmetry, displacement symmetry by any interval in the direction of its axis. This combination of symmetries, displacement symmetry by any interval along an axis of axial symmetry, is called cylindrical symmetry. 2.3 Symmetry of regular polyhedrons In three dimensions, rotations about a common axis give us the cyclic groups C n. For n 3, C n is the rotational symmetry group of the pyramid built on a regular n gonal base (with axis of rotational symmetry passing through the apex and the center of the base). All the Platonic solids are symmetric about their centers (fig. 9). Fig. 9 Regular polyhedrons The symmetry group of the dodecahedron or icosahedron S(D) They both have the same number of edges, being 30. Because these two solids are dual to each other they have the same symmetry group. The order of the group of direct symmetries (all rotations) is S(D) = 60. The elements are:  4 rotations (by multiples of 2π/5) about centers of 6 pairs of opposite faces;  1 rotation (by π) about centers of 15 pairs of opposite edges;  2 rotations (by ±2π/3) about 10 pairs of opposite vertices. Together with the identity this accounts for all 60 elements. We define the dual of a regular polyhedron to be another regular polyhedron, which is formed by JUNE 2009 NUMBER 5 JIDEG 23
4 connecting the centers of the faces of the original polyhedron (fig.10). Fig. 10 Dual of a regular polyhedron. 3. SCREW (HELICAL) SYMMETRY Helical symmetry is the kind of symmetry seen in everyday objects such as springs, drill bits and spiral staircases. It can be thought of as rotational symmetry along with translation along the axis of rotation (fig.11). interval h. Since there are n steps per turn, each step is essentially a wedge of angle 360 /n (fig. 11). So a rotation of the staircase about its axis by 360 /n puts each step exactly in a position either above where the step below it was. A displacement by interval h/n along its axis can return the staircase to its original appearance. Three main classes of helical symmetry can be distinguished based on the interplay of the angle of coiling and translation symmetries along the axis:  infinite helical symmetry (an object has infinite helical symmetries if for any small rotation of the object around its central axis there exists a point nearby the translation distance on that axis at which the object will appear exactly as it did before);  nfold helical symmetry (objects of this type will exhibit a symmetry after a rotation by some fixed angle θ and a translation by some fixed distance, but will not in general be invariant for any rotation angle;  nonrepeating helical symmetry. Fig. 12 Helical staircase. 4. SYMMETRY IN ENGINEERING 24 Fig. 11 Screw transformation. Consider a helical staircase (fig. 12). Let h denote the change of height involved in each complete turn of the staircase and n the number of steps in one complete turn. The height of each step above the one below it is then h/n. If it were infinitely long, it would have displacement symmetry along its axis with minimum displacement JUNE 2009 NUMBER 5 JIDEG Lots of metallic parts such as flexible membranes from the structure of elastic couplings, sprocket wheels, cars rims etc, used as pieces for devices, mechanisms and machines have been designed according to the principle shape follows function and the beauty of these objects increased at the same time with their functional efficiency [3]. An aspect of beauty is symmetry, which represents relative simplicity within complexity. We illustrate this in fig The membrane is formed from thin spokes by making radial indents in the central portion, connected by inner and outer diameters of the shape (fig. 13) [2]. Many logos start with a basic shape, a rectangle, a diamond or an oval, and then the graphic artist uses symmetry to create the design.
5 The Mitsubishi company logo (fig. 18) began with a diamond that was rotated 120 degrees, than another 120 degrees from that. Toyota logo has horizontal reflection symmetry across a vertical line through its center. The logo is made up of three ellipses. The two inner ellipses are 90 0 rotations of each other. Fig. 13 Spoked membrane for elastic coupling. Translation Rotation Fig. 14 Three dimensional object with translational and rotational symmetries. Fig. 17 Rims with multiple spokes (rotational symmetry). Fig. 15 3D model with axial symmetry [3]. 5. CONCLUSION Fig. 18 Corporate logos. Fig. 16 Objects with helical and rotational symmetry [5]. Symmetry is a very important concept in mathematics and can be applied in many different areas including JUNE 2009 NUMBER 5 JIDEG 25
6 equations, shapes, workpieces and aero dynamical buildings. Symmetry also plays an important role in human visual perception and aesthetics. Knowing that a shape or object has symmetry can help us solve problems involving that shape (e.g. technique for segmenting objects into parts characterized by different symmetries). We believe that for many common objects, the construction of 3D surface shape using symmetries types is necessary for practical applications. The use of 3D modelling and simulation concepts and tools can highlight the design in the machine building process REFERENCES [1] Thrun S., Wegbreit B. (2005), Shape from symmetry, Proceedings of the 10 th IEEE International Conference on Computer Vision (ICCV), pp [2] Dobre, D., (2008), Development Basics of a Product (Bazele dezvoltarii de produs), Bucharest, Romania. [3] Dobre D., Simion I., (2009), Special applications of fair surfaces representation, The 3 rd International Conference on Engineering Graphics and Design, in Acta Technica Napocensis, Series: Applied Mathematics and Mechanics, no. 52, vol. Ia, pp , ClujNapoca, Romania. [4] Aldea, S., (1984), Descriptive geometry. Bodies and surfaces study (Geometrie descriptivă. Studiul corpurilor şi al suprafeţelor), U.P.B., Romania. [5] Simion, I., (1998), Engineering Graphic (Grafică inginerească), Bren Publishing House, Bucharest, Romania. Authors: Eng. Daniel Dobre, Ph.D, Lecturer, University POLITEHNICA of Bucharest, Department of Descriptive Geometry and Engineering Graphics, Eng. Ionel Simion, Ph.D, M Eng, Professor, University POLITEHNICA of Bucharest, Head of Descriptive Geometry and Engineering Graphics Department, E mail: 26 JUNE 2009 NUMBER 5 JIDEG
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