Geometric Designs Chapter 3 Lesson 1

Geometric Designs Everything that surrounds us has shape, line, volume, surface area, curve, and some other aspects of geometry. Geometry has influenced the way we live. As infants we are interested in toys with shapes, patterns, and designs and now, whenever we do everyday tasks, we conform...

Geometric Designs Everything that surrounds us has shape, line, volume, surface area, curve, and some other aspects of geometry. Geometry has influenced the way we live. As infants we are interested in toys with shapes, patterns, and designs and now, whenever we do everyday tasks, we conform to geometric principles. Some professions use geometry to do their jobs properly, such as construction, weaving and sewing, computer imaging, art and aesthetics, and architectural designing. Geometry affects us even in the most basic details of our lives. Whatever form we use it helps us to understand specific phenomena and to uplift the quality of life. Learning Objectives At the end of this lesson, you are expected to: 1. Apply geometric concepts in describing and creating designs; and 2. Contribute to the enrichment of the Filipino culture and the arts using the concepts RECOGNIZING AND ANALYZING GEOMETRIC SHAPES Geometric shapes have fascinated many people throughout history in the field of art, science, engineering, interior designing, and many other professions. Mathematicians have constructed ideal representations of these shapes and developed methods in obtaining the measurements of lengths (one dimension), areas (two dimensions), and volume (three –dimensions). Polygons A polygon is a two-dimensional shape with straight sides. It can be classified according to the number of its sides, such as a three-sided shape called triangle and a four sided shape called quadrilateral. Others are pentagon, heptagon, hexagon, and so on Polygon are either simple or complex. A simple polygon has only one boundary and never crosses over itself while a complex polygon intersects Polygons are either concave or convex. A convex polygon has no angles pointing inward. More precisely, no internal angles can be more than 1800. If any internal angles are greater than 1800; otherwise , it is concave. Polygons are either regular or irregular. If all angles are equal and all sides are equal, it is regular; otherwise , it is irregular. The interior angles of a polygon are the angles inside the shape. In general, for a polygon with n sides, the sum of the internal angles is equal to ( − 2) × 1800and if the polygon is regular, the measurement of each angle is equal to ( −2)×1800 0. The sum of the exterior angles of a polygon is 360. The 0 interior and exterior angles of each vertex on a polygon add up to 180. SHAPE NUMBER OF SIDES SUM OF MEASUREMENT INTERNAL OF EACH ANGLE ANGLES FOR REGULAR POLYGON Triangle 3 1800 600 Quadrilateral 4 3600 900 Pentagon 5 5400 1080 Hexagon 6 7200 1200 : : : :  : : : : Any polygon n (n- 2) x 1800 (n-2) x 1800/n Solids: width, depth and height. A solid or form is the geometry of a three –dimensional space, the kind of space we live in. It is called three-dimensional or 3D because there are three dimensions; width, depth, and height. Solids have properties, such as volume (think of how much water it could hold) and surface area (think of the area you would have to paint). There are two main types of solids, namely: polyhedral and non-polyhedral. A polyhedron is a solid made of flat surfaces; each surface is a polygon, like the platonic solids, prisms, and pyramids. Non polyhedral are solids with curved surfaces, or a mix of curved and flat surfaces, such as spheres, cylinders, cones, and torus. Platonic solid is a convex polyhedron whose aces are all congruent convex regular polygons. None of it faces intersect except at their edges, and it has the same number of faces that meet at each of its vertices. There are five platonic solids, namely: Tetrahedron Cube Octahedron 4 faces 6 faces 8 faces 4 vertices 8 vertices 6 vertices 6 edges 12 edges 12 edges Dedecahedron Isosahedron 12 faces 20 faces 20 vertices 12 vertices  30 edges 30 edges Euler’s Formula deals with shapes called polyhedral. It states that F + V –E = 2 where F is the number of faces, V is the number of vertices or corners, and E is the number of edges. This formula works only on solids that do not have any holes and do not intersect itself. It cannot also be made up of two pieces stuck together, such as two cubes stuck together by one vertex. The Euler’s formula works on the platonic solids. POLYHEDRON FACES VERTICES EDGES F+V–E=2 Tetrahedron 4 4 6 4 + 4 -6 = 2 Cube 6 8 12 6 + 8 -12= 2 Octahedron 8 6 12 8 + 6- 12= 2 Dodecagon 12 20 30 12 + 20 -30=2 Icosahedron 20 12 30 20 +12 -30 =2 Prism is a polyhedron whose sides are all flat. It has the same cross section all along its length, and its shape is a polygon. Some examples of a prisms are as follows All the prisms above are classified as regular prisms because the cross section of each is a regular polygon. An irregular prism is one whose cross section is an irregular polygon. Pyramid is a polyhedron made by connecting a base to an apex. There are many types of pyramids, and they are named after the shape of their base, some of these are as follows: Non-polyhedral Sphere is a perfectly round object in a three-dimensional space. It is non-polyhedron because the surface is completely round. It is perfectly symmetrical with no edges or vertices. All points on the surface are the same distance from the center. Cylinder is a three-dimensional solid object bounded by a curved surface and two parallel circles of equal size at the ends. The curved surface is formed by all the line segments joining corresponding points of the two parallel circles. Because of its curved surface, it is not a polyhedron. Torus is a solid formed by revolving a small circle along a line made by another circle. It has no edges or vertices and, therefore, it is not a polyhedron. Cone is made by rotating a triangle. The triangle has to be a right-angled triangle, and it gets rotated around one of its two short sides. The side it rotates around is the axis of the cone. It has a flat base and has one curved side. Because of its curved surface, it is not a polyhedron. Transformations Geometric transformation of shapes is a change of its size, orientation, or position following certain techniques in mathematics. The original shape is called the object, and the new shape is called its image. Many objects around us are said to be symmetrical, and this symmetry resulted from geometric transformation. Some of the basic geometric transformations are as follows. Translation is a transformation of an object Rotation is a transformation of an Where every point of it moves a fixed distance object rotating about a given And a given direction through a given angle. Reflection is a transformation of an object Glide reflection is a composition of translation Where every point of it and its image are of and reflection in a line parallel to the direction Of the same distance from the line of of translation Symmetry. Dilation is a transformation of an object by resizing to either reduce it or enlarge it about a point with a given factor. The value of factor determines whether the dilation is enlargement. PATTERNS AND DIAGRAMS Patterns are one aspect in geometry, which are usually found and utilized. These are patterns around us at home, we see patterns on wallpapers, floor mats, bed sheets, window panes and pieces of furniture. Symmetry Symmetry is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection. -Herman Weyl Symmetries are an integral part of nature and the arts of cultures worldwide. They can be found in architecture, crafts, poetry, music, dance, chemistry, painting, physics, sculpture, biology, and mathematics. Rosette Patterns A symmetry group is the collection of all symmetries of a plane figure. The symmetry groups have all been one of two types: Cyclic symmetry group has rotation symmetry only around a center point. If the rotation has n order, the group is called Cn. Dihedral symmetry group has rotation symmetry around a center point with reflection lines through the center point. If the rotation has n order, there will be n reflection lines and the group is called Dn. The cyclic and dihedral symmetry groups are known as rosette symmetry groups, and a pattern with rosette symmetry is known as a rosette pattern. Rosette patterns have been used as architectural and sculptural decorations of the new century. Frieze Pattern An infinite strip with a repeating pattern is called a frieze pattern, or sometimes a Crystallography (IUC) border pattern or an infinite strip pattern. The term “frieze” is from architecture, where a frieze refers to a decorative carving or pattern that runs horizontally just below a roofline or ceiling. Here are some examples of frieze patterns. The patterns repeat and extend infinitely in both directions. A frieze group is the set of symmetries of a frieze pattern; that is geometric transformation built from rigid motions and reflections that preserve the pattern. This group contains translations and may contain glide reflections, reflections along the long axis of the strip, reflections along the narrow axis of the strip, and 1800rotations. Many authors present the frieze groups in a different order. Using the international Union (IUC) notation, the names of symmetry groups are listed In the table below. These names all begin with “p” followed by three characters. The first is “m” if there is a vertical reflection, and “I” if it has none. The second is “m” if there is horizontal reflection; “g” if there is a glide reflection, otherwise, use “I”. The third is “2” if there is an 1800rotation, and “I” if there is none. Mathematician John H. Conway also created nicknames for each frieze group that relate to footsteps. Tessellation A tessellation is defined as a pattern of shapes that covers a plane without any gaps or overlaps. Tessellations can be found on pavements, patios, and wallpapers. The tiled surface of flooring and walls is an example of tessellation where there are no tiles that overlap, and there are no gaps between shapes. In most cases, tessellations are formed by repeated pattern; however, some utilize pictures or designs, which in no way repeat.  Geometric transformation of polygon, such as translation, reflection, and rotation can be used to create patterns. Some patterns that cover a plane constitute tessellations. Examples Tessellation of triangles Tessellation of squares Tessellation of Hexagons Looking at these three samples of tessellations, you will notice that the squares can easily be lined up with each other while the triangles and hexagons involve translations. The vertex point is the point where the shapes come together. The sum of all the angles of each shape that come together at vertex point is 3600.The shapes will overlap if the sum is greater than 3600, otherwise, there will be gaps if the sum is less than 3600. Naming tessellation can be done by looking at one vertex point. Looking around vertex point, start with a shape with the least number of sides, and count the number of sides of each shape at each vertex point. The name of tessellation then becomes these numbers. For example in the tessellation of triangles, the number of sides is 3 and then are 6 shapes, therefore, it can be named as 3,3,3,3,3,3. For a square, it can be labeled as 4,4,4,4 and for hexagon, we can call it 6,6,6. Semi-regular tessellations can be formed using a variety of regular polygons and the arrangement of these polygons at every vertex is identical. Tessellations can be used to create art, puzzles, patterns, and crafts are as follows: designs. Some famous mathematicians and artist based their works on the concept of tessellation. One of them was Maurits The Mandaya people of Davao Oriental are known for their Cornelis Escher, a dutch graphic artist who made mathematically masterful weaving pattern ikat in abaca. One of their most inspired woodcuts, lithographs and mezzotints. popular textiles is called dagmay. It is distinguished from other any tribal weaving by the intricate figures and patterns depicting Mindanao Designs, arts, and culture the folklores and religion of the tribe. Mindanao is the home of eighteen tribal groups which have made weaving their identity, culture, and way of life. Some of their The Maranao are famous for their sophisticated weaving with designs and colors. Their textile weaving involves traditional Southeast Asian back strap loom weaving using native decorative ornamentation. A versatile garb malong is a hand-woven fabric with beautifully patterned designs. It can be worn by all genders and classes, with dominant hues of gold and purple. Malong made of high-quality silk or cotton are intended for special occasions, and a yellow making is considered to be royal or high class. Commented [WU1]: The Yakan are indigenous Muslim tribe native to the tropical island of Basilan in Sulu Archipelago. Yakan people are recognized for their remarkable Technicolor geometric weaves and distinctive face decorations used in their traditional ceremonies. The Yakan are kind and loving people that embody a non-materialistic culture and live in close-knit communities. Yakan weaving uses bright, bold, and often contrasting colors in big symmetrical patterns. The inspiration for designs comes from island living and Islamic sacred geometry. No other tribe in Davao is more recognized by their colorful clothing than the Bagobos. Whether they are of the giangan, obo or tagabawa tribe, they are usually spotted wearing head kerchiefs especially during celebrations or tribal festivities. These people also weave abaca cloths of earth tones and make baskets that are trimmed with beads, fibers, and horse’s hair. They have ornate traditions in weaponry and some other metal arts. Work for the following: (to be submitted through picture) 1. Explore the 5 platonic solids: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Look for pattern and relationship between faces, vertices, and edges. 2. Make a portfolio of the different patterns and designs made by the local indigenous tribes.