Geometric Design PDF

MATHEMATICS IN THE MODERN WORLD GEOMETRIC DESIGN GEOMETRIC DESIGN After completing this chapter, the you will be able to: ✔Recognize and appreciate geometrical shapes. ✔Discuss the different forms of transformation. ✔Solve problem involving patterns. ✔Apply geometry concepts, especiall...

MATHEMATICS IN THE MODERN WORLD GEOMETRIC DESIGN GEOMETRIC DESIGN After completing this chapter, the you will be able to: ✔Recognize and appreciate geometrical shapes. ✔Discuss the different forms of transformation. ✔Solve problem involving patterns. ✔Apply geometry concepts, especially isometries in describing and creating designs. ✔Identify patterns in nature and regularities in the world. ✔Appreciate various forms of art. ✔Contribute to aesthetics. ✔Articulate the importance of mathematics in one's life. ✔Express appreciation for mathematics as a human endeavor. ✔Support the use of mathematics in various aspects and endeavors in life. GEOMETRIC DESIGN Chapter Outline 5.1: Recognizing and Analyzing Geometric Shapes 5.2: Transformation 5.3: Patterns and Diagrams 5.4: Designs, Arts, and Culture RECOGNIZING AND ANALYZING GEOMETRIC SHAPES Mathematicians instinctively search for geometrical and numerical patterns and for symmetry. The discovery of patterns and symmetries enable us to understand some practical problems and as geometry evolves people begin to clearly understand and control the world around us.. RECOGNIZING AND ANALYZING GEOMETRIC SHAPES Humans are fascinated in the nature of symmetry. From the time of the Greeks to the present generation we tend to side with symmetry in everything from planning our house layout to the way we dress. RECOGNIZING AND ANALYZING GEOMETRIC SHAPES Nature displays an infinite number of geometrical shapes from a small atom to the greatest of the spiral across the galaxies. RECOGNIZING AND ANALYZING GEOMETRIC SHAPES Romanesco broccoli RECOGNIZING AND ANALYZING GEOMETRIC SHAPES the honeycomb of the bee RECOGNIZING AND ANALYZING GEOMETRIC SHAPES the spiral growth structure of a sunflower RECOGNIZING AND ANALYZING GEOMETRIC SHAPES pinecones and pineapple RECOGNIZING AND ANALYZING GEOMETRIC SHAPES the spiral of the nautilus shell RECOGNIZING AND ANALYZING GEOMETRIC SHAPES the feather of a peacock's tail RECOGNIZING AND ANALYZING GEOMETRIC SHAPES the spider's web RECOGNIZING AND ANALYZING GEOMETRIC SHAPES Patterns abound a nature, and geometry paves the way for us to understand it more as we experience it. Perhaps there is no subject has intrigued the human race through centuries as much as geometry and see the wonder in nature, arts, design, and in other aspects of human experience. TRANSFORMATION In geometry, we don't only study figures but we also dwell on the movement of figures. Moving each point of a geometrical figure according to set of rules we can create a new geometric figure. TRANSFORMATION The movement establishes a correspondence between the set of points in the original figure and the set of points of the new figure which we called image. If we can pair each point of a figure with exactly one point of its image on a Euclidean plane and vice versa, then the correspondence is called transformation. TRANSFORMATION According to Jennifer Beddoe, (2003) in geometry, transformation refers to the movement of objects in the coordinate plane. Geometric transformations involve taking a preimage and transforming it in some way to produce an image. TRANSFORMATION There are two different categories of transformations: 1. The rigid transformation, which does not change the shape or size of the preimage. 2. The non-rigid transformations, which will change the size but not the shape of the preimage. TRANSFORMATION An isometry (or rigid transformation) is a transformation that preserves size and shape. The image of a figure is always congruent to the original figure in an isometry. There are three types of isometry: translation, rotation, and reflection. TRANSFORMATION A translation is the simplest type of isometry, when a figure moves a fixed distance in a fixed direction. Meaning, all the points in the original figure are equidistant from their images. We can use an arrow to show the distance and direction which is referred to as translation vector. TRANSFORMATION TRANSFORMATION A rotation is an isometry in a turning motion, when points in the original figure rotate or turn an identical number of degrees about a fixed center point. A rotation has a center of rotation and an angle of rotation (clockwise or counterclockwise). TRANSFORMATION TRANSFORMATION A reflection is an isometry when a reflection flips across an axis of reflection or when a reflection produces a figure's "mirror image." A line reflection defines reflection, it serves as perpendicular bisector reflection to every segment joining a point in in the figure with the image of the point. Each point a figure is reflected or flipped over the reflection line as if the reflection line were a mirror. TRANSFORMATION TRANSFORMATION A glide reflection is a special type of two- step isometry, uses a combination of a reflection and translation. TRANSFORMATION TRANSFORMATION A non-rigid transformation can change the size or shape, or both size and shape, of the preimage. Two non-rigid transformations are dilation and shear. The image resulting from the transformation will change its size, its shape, or both. TRANSFORMATION Dilation is expanding or contracting an object without changing its shape or orientation. This is called resizing, contraction, compression, enlargement or even expansion. The shape becomes bigger or smaller. TRANSFORMATION TRANSFORMATION Shearing makes it look like you are "pushing" one part of the original shape, while keeping the bottom cemented to the floor. All the points along one side of a preimage remain fixed while all other points of the preimage move parallel to that side in proportion to the distance from the given side TRANSFORMATION TRANSFORMATION PATTERNS AND DIAGRAMS Symmetry An object is said to be symmetric if it can be divided into two or more identical parts that can be arranged in an organized fashion. In other words, symmetry is an exact configuration of an image around an axis of symmetry. The word "symmetry" came from Greek word "symmetrein" which means "to measure together." There are three types of geometrical symmetry namely: reflection, rotational, and translational. PATTERNS AND DIAGRAMS Reflectional symmetry (or mirror symmetry) is a symmetry in which half of the image of an object is exactly same as the other half. It is either left portion of an image which is the reflection of the right image or the upper portion of an object is the reflection of lower portion of the object. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Rotational symmetry is a symmetry in which the image is rotated to a certain degree about at axis and does not affect the shape of the image. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Translational symmetry is a symmetry in which a particular pattern or design is shifted from one place to another, meaning the same exact image is found on another location, even the orientation of the image is the same. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Rosette Pattern Rosette pattern is a pattern consist of taking a motif or element and rotating and/or reflecting that element. The fact that they can only have these two forms of this rigid motion is called Leonardo's Theorem. The theorem was named after Leonardo Da Vinci who formulated it, he needed to make sure cathedrals remained symmetrical when additions were added to the chapels. There are two types of rosette patterns: cyclic and dihedral. PATTERNS AND DIAGRAMS A cyclic rosette patterns are rosette patterns which do not contain reflection symmetry, rotation symmetry around a center point, but no mirror lines. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS A dihedral rosette patterns are rosette patterns which have reflection symmetry/rotation symmetry around a center point with mirror lines through the center point. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS ▪ PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Frieze Pattern In mathematics, frieze pattern is a design or pattern on a two-dimensional surface that is repetitive (or translational symmetry) in one direction. It can be imagined that the frieze pattern goes infinitely in both directions or wrap around. PATTERNS AND DIAGRAMS Many of the rosette patterns occur frequently in architecture and decorative art. Mathematician John Conway created names to relate to footsteps for each of the frieze groups. The only possibilities are 180° rotations, reflections with vertical axes, a reflection with horizontal axis, and a glide reflection with horizontal axis. He named it according to symmetries such as hop, step, jump, slide, spinning hop, spinning jump, and spinning sidle. PATTERNS AND DIAGRAMS Hop it contains only translational symmetry, in the horizontal direction. PATTERNS AND DIAGRAMS Mosaic Border Alcazar de los Reyes Cristianos Cordoba, Spain PATTERNS AND DIAGRAMS Spinning hop it has translation and rotation (by a half- turn) symmetries. PATTERNS AND DIAGRAMS Meander Frieze San Giorgio Maggiore Venice, Italy PATTERNS AND DIAGRAMS Step it has a translation and glide reflection symmetry. PATTERNS AND DIAGRAMS Downtown Richmond PATTERNS AND DIAGRAMS Jump it is a reflection across a horizontal symmetries. PATTERNS AND DIAGRAMS Ceiling Mezquita Cordoba, Spain PATTERNS AND DIAGRAMS Spinning jump it has all symmetries (translation, horizontal and vertical reflection, and rotation). PATTERNS AND DIAGRAMS Back of a Bench Banos de la Maria de Padilla Reales Alcazares Seville, Spain PATTERNS AND DIAGRAMS Sidle it has translation and vertical reflection symmetries. PATTERNS AND DIAGRAMS Tile Frieze Palacio de Velazquez Parque de Retiro Madrid, Spain PATTERNS AND DIAGRAMS Spinning sidle it has translation, glide reflection and rotation (by a half-turn) symmetries. PATTERNS AND DIAGRAMS Mosaic Nuestra Senora de la Almundena Madrid, Spain PATTERNS AND DIAGRAMS Wallpaper Pattern Wallpaper pattern is a mathematical classification of a two-dimensional repetitive pattern which covers a plane and can be mapped based on the symmetries on the pattern in more than one direction. The multiple directions force the pattern to cover the entire infinite plane, while a finite portion of a wallpaper pattern is enough to establish the translation symmetry which is used to cover the entire plane. PATTERNS AND DIAGRAMS Wallpaper patterns can be categorized based from their symmetries. The difference may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group. PATTERNS AND DIAGRAMS Symmetry group 1 (p1) This is the simplest symmetry group. It consists only of translations. There are neither reflections, glide-reflections, nor rotations. The two translation axes may be inclined at any angle to each other. Its lattice is parallelogrammatic, so a fundamental region for the symmetry group is the same as that for the translation group, namely, a parallelogram. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 2 (p2) This group differs only from the first group is that it contains 180° rotations, that is, rotations of order 2. All symmetry groups there are translations, but there neither reflections nor glide reflections. The two translations axes may be inclined at any angle to each other. The lattice is a parallelogrammatic. A fundamental region for the symmetry group is half of a parallelogram that is a fundamental region for the translation group. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 3 (pm) This is the first group that contains reflections. The axes of reflection are parallel to one axis of translation and perpendicular to the other axis of translation. The lattice is rectangular. There are neither rotations nor glide reflections. A fundamental region for the translation group is a rectangle, and one can be chosen that is split by an axis of reflection so that one of the half rectangles forms a fundamental region for the symmetry group. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 4 (pg) This is the first group that contains glide reflections. The direction of the glide reflection is parallel to one axis of translation and perpendicular to the other axis of translation. There are neither rotations nor reflections. The lattice is rectangular, and a rectangular fundamental region for the translation group can be chosen that is split by an axis of a glide reflection so that one of the half rectangles forms a fundamental region for the symmetry group. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 5 (cm) This group contains reflections and glide reflections with parallel axes. There are no rotations in this group. The translations may be inclined at any angle to each other, but the axes of the reflections bisect the angle formed by the translations, so the fundamental region for the translation group is a rhombus. A fundamental region for the symmetry group is half the rhombus. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 6 (pmm) This symmetry group contains perpendicular axes of reflection. There are no glide-reflections or rotations. The lattice is rectangular, and a rectangle can be chosen for the fundamental region of the translation group so that a quarter-rectangle of it is a fundamental region for the symmetry group. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 7 (pmg) This group contains both a reflection and a rotation of order 2. The centers of rotations do not lie on the axes of reflection. The lattice is rectangular, and a quarter-rectangle of a fundamental region for the translation group is a fundamental region for the symmetry group. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 8 (pgg) This group contains no reflections, but it has glide-reflections and half-turns. There are perpendicular axes for the glide reflections, and the centers of the rotations do not lie on these axes. Again, the lattice is rectangular, and a quarter-rectangle of a fundamental region for the translation group is a fundamental region for the symmetry group. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 9 (cmm) This group has perpendicular reflection axes, as does group 6(pmm), but it also has rotations of order 2. The centers of the rotations do not lie on the reflection axes. The lattice is rhomic, and a quarter of a fundamental region for the translation group is a fundamental region for the symmetry group. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 10 (p4) This is the first group with a 90° rotation, that is, a rotation of order 4. It also has rotations of order 2. The centers of the order-2 rotations are midway between the centers of the order-4 rotations. There are no reflections. The lattice is square, and again, a quarter of a fundamental region for the translation group is a fundamental region for the symmetry group. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 11 (p4m) This group differs from 10 (p4) in that it also has reflections. The axes of reflection are inclined to each other by 45° so that four axes of reflection pass through the centers of the order-4 rotations. In fact, all the rotation centers lie on the reflection axes. The lattice is square, and an eighth, a triangle, of a fundamental region for the translation group is a fundamental region for the symmetry group. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 12 (p4g) This group also contains reflections and rotations of orders 2 and 4. But the axes of reflection are perpendicular, and none of the rotation centers lie on the reflection axes. Again, the lattice is square, and an eighth of a square fundamental region of the translation group is a fundamental region for the symmetry group. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 13 (p3) This is the simplest group that contains a 120°-rotation, that is, a rotation of order 3, and the first one whose lattice is hexagonal. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 14 (p31m) This group contains reflections (whose axes are inclined at 60° to one another) and rotations of order 3. Some of the centers of rotation lie on the reflection axes, and some do not. The lattice is hexagonal. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 15 (p3m1) This group is similar to the last in that it contains reflections and order-3 rotations. The axes of the reflections are again inclined at 60° to one another, but for this group all of the centers of rotation lie on the reflection axes. Again, the lattice is hexagonal. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 16 (p6) This group contains 60° rotations, that is, rotations of order 6. It also contains rotations of orders 2 and 3, but no reflections. Its lattice is hexagonal. PATTERNS AND DIAGRAMS PATTERNS AND DIAGRAMS Symmetry group 17 (p6m) This most complicated group has rotations of order 2, 3, and 6 as well as reflections. The axes of reflection meet at all the centers of rotation. At the centers of the order-6 rotations, six reflection axes meet and are inclined at 30° to one another. The lattice generator is hexagonal. PATTERNS AND DIAGRAMS REFENCES: Winston S. Shrug, PhD Mathematics in the Modern World CHED Curriculum Compliant, 2018. MINDSHAPERS CO., INC. The 17 plane symmetry groups https://www2.clarku.edu/faculty/djoyce/wallpaper/ seventeen.html