Geometric Designs PDF

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####### Mathematics in the Modern WorldModule 5GEOMETRICDESIGNS####### Percival B. CabangModule OverviewAlmost all things we find in our surroundings has shape, line, volume, surface area, curve, and some other aspects of geometry. Geometry has influenced the way we live. As a child, we were interes...

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MMW Module 5 – Geometric Designs

Mathematics in the Modern World

Module 5

GEOMETRIC DESIGNS Percival B. Cabang

Module Overview Almost all things we find in our surroundings has shape, line, volume, surface area, curve, and some other aspects of geometry. Geometry has influenced the way we live. As a child, we were interested in toys with shapes, patterns, and designs. Whenever we do our daily tasks, we conform to geometric principles. Some professions use geometry as a tool to do their jobs properly, such as construction, weaving and sewing, computer imaging, art and aesthetics, and architectural designing. The knowledge we learned through a complete understanding of geometric principles has provided not only safety but also increase in the creation of tools, skill level enhancement, and aesthetically pleasing arrangements. 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. Module Outcomes • •

Apply geometric concepts in describing and creating designs; and Contribute to the enrichment of the Filipino culture and the arts using the concepts in geometry.

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MMW Module 5 – Geometric Designs

Lesson 1

Recognizing & Analyzing Geometric Shapes

Learning Outcomes

The students will be able to analyze and recognize fundamental plane and solid geometrical shapes. Time Frame: Self-paced learning

Introduction Geometric shapes have fascinated, many people throughout history in the fields of art, science, engineering, interior designing, and many other professions. Mathematicians have constructed ideal representations of these shapes and developed methods in obtaining the measurement of lengths (onedimension), areas (two-dimension), and volumes (three-dimension).

Abstraction

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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 four-sided shape called quadrilateral. Others are pentagon, heptagon, hexagon, and so on. Polygons are either simple or complex. A simple polygon has only one boundary and never crosses over itself while a complex polygon intersects itself. 2|Page

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Illustration 1

Simple (Pentagon)

Complex (Pentagon)

Polygons are either concave or convex. A convex polygon has no angles pointing inward. More precisely, no internal angles can be more than 180. If any internal angles are greater than 180, it is a concave.

Illustration 2

Convex

Concave

Polygons are either regular or irregular. If all angles are equal and all sides are equal, it is regular; otherwise, it is irregular.

Illustration 3

Regular

Irregular

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MMW Module 5 – Geometric Designs

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 (𝑛 − 2) × 180° and if the polygon is regular, the measurement of each angle is equal to

(𝑛−2)×180° 𝑛

. The sum of the exterior angles of a polygon is 360. The interior

and exterior angles of each vertex on a polygon add up to 180.

Activity

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Activity 1. Complete the table below by supplying the appropriate information.

SHAPE

NUMBER OF SIDES

SUM OF INTERNAL ANGLES

MEASUREMENT OF EACH ANGLE FOR REGULAR POLYGON

Triangle

3

180

60

Quadrilateral

4

360

90

Pentagon

5

540

108

Hexagon

6

720

120

Pentagon

7 8 9 10

:

:

:

:

n-gon

N

(n – 2) x 180

(n – 2) x 180/n

A solid 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: polyhedra and non-polyhedra. A polyhedron is a solid made of flat surfaces; each surface is a polygon, like the platonic solids, prisms, and pyramids. Non-polyhedra are solids with curved 5|Page

MMW Module 5 – Geometric Designs

surfaces, or a mix of curved and flat surfaces, such as spheres, cylinders, cones, and torus.

Platonic solid is a convex polyhedron whose faces are all congruent convex regular polygons. None of its 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.

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MMW Module 5 – Geometric Designs

Euler’s Formula deals with three-dimensional shapes called polyhedra. It states that F + V - E = 2 where F is the number of faces, V is the number of vertices, 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.

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Activity

Activity 2. Verify the Euler Formula for each of the regular polyhedron by supplying the necessary information in the table below. POLYHEDRON

Tetrahedron Cube Octahedron Dodecahedron Icosahedron

FACES

VERTICES

EDGES

4

4

6

F+V–E=2

4+4–6=2 =2 =2 =2 =2

Prism is a polyhedron whose sides are all flat. It has the same cross section all along its length, and its shape is polygon. All the prisms are classified as either regular prism because the cross section of each is a regular polygon or irregular prism because its cross section is an irregular polygon.

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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.

Non-polyhedra

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 polyhedron. 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 cured surface, it is not a polyhedron. 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. 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 polyhedron. 8|Page

MMW Module 5 – Geometric Designs

Lesson 2

Geometric Transformations

Learning Outcomes The students will be able to analyze and recognize the four geometric transformations. Time Frame: Self-paced learning Introduction Geometric transformation of shapes is a change of its size, orientation, or position following certain techniques in mathematics. The original shape is called 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, Rotation, Reflection, and Dilation.

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MMW Module 5 – Geometric Designs

Abstraction

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Translation is a transformation of an object where every point of it moves a fixed distance and a given direction.

Rotation is a transformation of an object by rotating about a given through a given angle.

Reflection is a transformation of an object where every point of it and its image are of the same distance from the line of the symmetry. Glide Reflection is a composition of translation and reflection in a line parallel to the direction of translation.

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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 (r) determines whether the dilation is enlargement or reduction.

Activity

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Activity 3. Underline the correct word which describes how the shape at the left has been transformed to the shape at the right in just one transformation. Note: There is more than one correct answer for some of the transformations.

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Lesson 3

Patterns and Diagrams

Learning Outcomes The students will be able to analyze and recognize symmetries along with the Rosette and Frieze patterns. Time Frame: Self-paced learning Introduction Patterns are one aspect in geometry, which are usually found and utilized. There are patterns around us; at home, we see patterns on wallpapers, floor mats, bed sheets, window panes and pieces of furniture. Patterns are also profuse in nature: on flowers, in leaves, on animals, and all on other places. Patterns can be simple or complex, and they sometimes use the same object or color more than once. 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. Because symmetric designs are so natural pleasing, symmetric symbols are very popular.

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Abstraction

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Symmetry When a figure undergoes an isometry and the resulting image coincides with the original, the figure is symmetrical. Different isometries yield different types of symmetry.

If a figure can be reflected over a line in such a way that the resulting image coincides with the original, the figure has reflection symmetry. Reflection symmetry is also called bilateral symmetry. The reflection line is called the line of symmetry.

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You can test a figure for reflection symmetry by tracing and folding it. If you can fold it so that one half exactly coincides with the other half, the figure has reflection symmetry.

Activity

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Activity 4. State the order of rotational symmetry for each shape below.

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Activity 5. Reflect each of these shapes in the dotted lines. No. 1 has been done for you as an example.

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Rosette Pattern A symmetry group is the collection of all symmetries of a plane figure. The symmetry groups have all been one of two types: 1. Cyclic symmetry group has rotation symmetry only around a center point. If the rotation has n order, the group is called Cn. 2. Dihedral symmetry group has rotation symmetry around the 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 decoration of the new century.

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You can create your own rosette pattern using a pattern generator. Visit the site http://math.hws.edu/eck/jsdemo/rosette.html. Frieze Pattern An infinite strip with a repeating pattern is called a frieze pattern, or sometimes a 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:

Examples of Frieze Pattern

(The patterns repeat and extend infinitely in both directions) A frieze group is the set of symmetries of a frieze pattern; that is, geometric transformations 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 180 rotations. Many authors present the frieze groups in a different order. Using the International Union of Crystallography (IUC) notation, the names of symmetry groups are named that begins with “p” followed by three characters. The first is “m” if there is a vertical reflection, and “1” if it has none. The second is “m” if there is a horizontal reflection; or “g” if there is a glide reflection, otherwise, use “1”. The third is “2” if there is a 180 rotation, and “1” if there is none. 17 | P a g e

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Mathematician John H. Conway also created nicknames for each frieze group that relate to footsteps. IUC Notation

EXAMPLE/NICKNAME

p111

DESCRIPTION

Translations only Hop Glide-reflections and Translations

p11g Step

Vertical reflection lines and Translations

p1m1 Sidle

Translations and 180 Rotations

P211 Spinning Hop

P2mg Spinning Sidle

Vertical reflection lines, Glide reflections, Translations, and 180 Rotations

Translations and Horizontal reflections

p11m Jump

P2mm Spinning Jump

Horizontal and Vertical reflection lines, Translations, and 180 Rotations

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Lesson 4

Tessellation

Learning Outcomes The students will be able to analyze and recognize tessellation patterns. Time Frame: Self-paced learning Introduction A tessellation is defined as a pattern of shapes that covers a plane without any gaps or overlaps. Tessellation 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.

Abstraction

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Geometric transformation of polygons, such as translation, reflection, and rotation can be used to create patterns. Such patterns that cover a plane constitute tessellation. Examples of tessellation of regular polygons are shown below.

Squares

Triangles

Hexagons

Looking at these three regular 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 360. The shapes will overlap if the sum is greater than 360; otherwise, there will be gaps if the sum is less than 360. Naming tessellation can be done by looking at one vertex point. Looking around a 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 there are 6 shapes; therefore, it can be named as 3,3,3,3,3,3. For the squares, it can be labeled as 4,4,4,4 and for hexagon, we can call it 6,6,6. 20 | P a g e

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Semi-regular tessellation can be formed using a variety of regular polygons and the arrangement of these polygons at every vertex is identical. Some examples of semi-regular tessellation are as follows:

Semi-Regular Tessellations

Tessellation can be used to create art, puzzles, patterns, and designs. Some famous mathematicians and artists based their work on the concept of tessellation. One of them was Maurits Cornelis Escher, a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints. Bruce Bilney is best known for tessellations with Australian themes and for realistic animal shapes in natural, comfortable poses. That is not as common as you would expect. It is a sad fact that most tessellation artists are satisfied with blocky, almost unfinished looking or overly stylized unnatural awkward shapes. You can view more animal tessellations at Bruce Bilney’s Tessellation Art Gallery here: http://www.tessellations.org/bruce-bilney-tgp.shtml.

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Activity

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Activity 5. Write the name the following semi-regular tessellations below.

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Lesson 5

Mindanao Designs, Arts and Culture

Learning Outcomes The students will be able to analyze and recognize geometric patterns among Mindanao designs, arts and culture. Time Frame: Self-paced learning Introduction With over 7,100 islands in the Philippines and three major island groups, it’s no wonder that different cultural practices, traditions, and groups are present in the country. Among the archipelago’s existing communities, there are indigenous tribes who have managed to keep their cultural identity, despite the nonrecognition and marginalization they’re facing. Although there are quite a number of indigenous tribes or ethnic groups in the country, they remain some of the poorest, least privileged, and impeded members of society. They mostly reside in the mountains, and hence were not affected by Spanish or American colonization, which is the primary reason they were able to retain their customs and traditions. There are two main ethnic groups comprising several upland and lowland indigenous tribes living within the Philippines – from the northern and southern parts of the Philippines. The indigenous people living in the northern part of the country are called the Igorots, whereas those non-Muslim indigenous tribes living in the south are referred to as Lumad.

Abstraction

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Mindanao is the home of 18 Lumad and 13 Muslim indigenous people’s groups, which have made weaving their identity, culture, and way of life. For these indigenous communities, woven textile conveys their creativity, beliefs, and 23 | P a g e

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ideologies, and there are some very interesting geometric...