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MODULE 4 - GEOMETRIC DESIGNS
Geometric Design is a branch of computational geometry that deals with the construction and representation of free-form curves, surfaces, or volumes and is closely related to geometric modeling.
It focuses on the simplistic beauty of mixing certain shapes, lines, curves together for creative results. Geometric designs are one of the most visually appealing forms in graphic design. 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.
Geometric shapes have fascinated many people throughout history in the fields of art, science engineering, interior , designing, and many other profession. Mathematics have constructed ideal representations of these shapes and developed methods in obtaining the measurements of lengths (one-dimension), areas (two-dimension), and volumes (three-dimension).
It focuses on the simplistic beauty of mixing certain shapes, lines, curves together for creative results. Geometric designs are one of the most visually appealing forms in graphic design. 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.
Geometric shapes have fascinated many people throughout history in the fields of art, science engineering, interior , designing, and many other profession. Mathematics have constructed ideal representations of these shapes and developed methods in obtaining the measurements of lengths (one-dimension), areas (two-dimension), and volumes (three-dimension).
POLYGON
Polygon is is a two-dimensional geometric figure that has a finite number of sides. The sides of a polygon are made of straight line segments connected to each other end to end. Thus, the line segments of a polygon are called sides or edges. The point where two line segments meet is called vertex or corners, henceforth an angle is formed. An example of a polygon is a triangle with three sides. A circle is also a plane figure but it is not considered a polygon, because it is a curved shape and does not have sides or angles. Therefore, we can say, all the polygons are 2d shapes but not all the two-dimensional figures are polygons.
Polygon is is a two-dimensional geometric figure that has a finite number of sides. The sides of a polygon are made of straight line segments connected to each other end to end. Thus, the line segments of a polygon are called sides or edges. The point where two line segments meet is called vertex or corners, henceforth an angle is formed. An example of a polygon is a triangle with three sides. A circle is also a plane figure but it is not considered a polygon, because it is a curved shape and does not have sides or angles. Therefore, we can say, all the polygons are 2d shapes but not all the two-dimensional figures are polygons.
SIMPLE AND COMPLEX POLYGON
Simple polygon has only one boundary and never crosses over itself while complex polygon intersects itself or crosses its sides one or more times. Simple polygons have no self-intersecting sides. Complex polygons, also called self-intersecting polygons, have sides that cross over each other. The classic star is a complex polygon. Most people can doodle a star on paper quickly, but few people label it a pentagram, complex polygon, or self-intersecting polygon.
Simple polygon has only one boundary and never crosses over itself while complex polygon intersects itself or crosses its sides one or more times. Simple polygons have no self-intersecting sides. Complex polygons, also called self-intersecting polygons, have sides that cross over each other. The classic star is a complex polygon. Most people can doodle a star on paper quickly, but few people label it a pentagram, complex polygon, or self-intersecting polygon.
CONVEX AND CONCAVE POLYGON
A convex polygon closes in an interior area without looking "dented." None of its interior angles point inward. In geometry, you could have a four-sided polygon that points outward in all directions, like a kite, or you could have the same four sides so two of them point inward, forming a dart. The kite is convex; the dart is concave. Every interior angle of a convex polygon is less than 180°. A concave polygon has at least one angle greater than 180°. Think of a bowtie-shaped simple hexagon (6 sides). It will have two interior angles greater than 180°.
A convex polygon closes in an interior area without looking "dented." None of its interior angles point inward. In geometry, you could have a four-sided polygon that points outward in all directions, like a kite, or you could have the same four sides so two of them point inward, forming a dart. The kite is convex; the dart is concave. Every interior angle of a convex polygon is less than 180°. A concave polygon has at least one angle greater than 180°. Think of a bowtie-shaped simple hexagon (6 sides). It will have two interior angles greater than 180°.
REGULAR AND IRREGULAR POLYGON
Regular polygons have congruent sides and interior angles. Every side is equal in length to every other side, and every interior angle is equal to all other interior angles. The number of regular polygons is limitless. Irregular polygons do not have congruent sides and angles. Home plate on a softball or baseball field is an irregular pentagon, because it has five sides with two 90° angles.
Regular polygons have congruent sides and interior angles. Every side is equal in length to every other side, and every interior angle is equal to all other interior angles. The number of regular polygons is limitless. Irregular polygons do not have congruent sides and angles. Home plate on a softball or baseball field is an irregular pentagon, because it has five sides with two 90° angles.
Here are the first ten polygons from triangle with three sides to dodecagon with twelve sides.
TYPES OF TRIANGLES ACCORDING TO LENGTH OF SIDES
There are three types of triangles according to length of sides namely, scalene, isosceles, and equilateral triangle. A scalene triangle has three sides of different length and three angles of different measures. An isosceles triangle has exactly two sides of equal length and the angles opposite the equal sides are of equal measures. Meanwhile, an equilateral triangle has three sides of equal length and three angles of equal measure.
There are three types of triangles according to length of sides namely, scalene, isosceles, and equilateral triangle. A scalene triangle has three sides of different length and three angles of different measures. An isosceles triangle has exactly two sides of equal length and the angles opposite the equal sides are of equal measures. Meanwhile, an equilateral triangle has three sides of equal length and three angles of equal measure.
TYPES OF TRIANGLES ACCORDING TO ANGLES
There are three types of triangles according to angles namely, acute, obtuse, and right triangle. An acute triangle is a triangle whose all the three interior angles are acute or less than 90 degrees. An obtuse triangle is a triangle in which one of the three interior angles has a measure greater than 90 degrees or is an obtuse angle. Further, a right triangle is a triangle in which one of the interior angles is equal to 90 degrees or is a right angle.
There are three types of triangles according to angles namely, acute, obtuse, and right triangle. An acute triangle is a triangle whose all the three interior angles are acute or less than 90 degrees. An obtuse triangle is a triangle in which one of the three interior angles has a measure greater than 90 degrees or is an obtuse angle. Further, a right triangle is a triangle in which one of the interior angles is equal to 90 degrees or is a right angle.
QUADRILATERALS
A quadrilateral is a general term for a polygon with four sides and four vertices. Both triangles and quadrilaterals include sides, vertex, interior angles, height and base except that a quadrilateral has a diagonal. Here are the properties of a quadrilateral:
A quadrilateral is a general term for a polygon with four sides and four vertices. Both triangles and quadrilaterals include sides, vertex, interior angles, height and base except that a quadrilateral has a diagonal. Here are the properties of a quadrilateral:
- They have four sides.
- They have four vertices.
- They have two diagonals.
- The sum of all interior angles is 360°.
INTERIOR ANGLES OF A POLYGON
It is angles inside the shape. In general, for a polygon with n sides, the sum of the internal angles is equal to (n − 2) x 180° and if the polygon is regular, the measurement of each angle is equal to (n − 2) x 180° /n . The sum of exterior angles of a polygon is 360°. The interior and exterior angles of each vertex on a polygon add up to 180°.
It is angles inside the shape. In general, for a polygon with n sides, the sum of the internal angles is equal to (n − 2) x 180° and if the polygon is regular, the measurement of each angle is equal to (n − 2) x 180° /n . The sum of exterior angles of a polygon is 360°. The interior and exterior angles of each vertex on a polygon add up to 180°.
SOLIDS
A solid or form is the geometry of a three-dimensional space, the kind of space we live in. It is three-dimensional with width, depth, and height. Solids have properties, such as volume and surface area. Two types of solids are Polyhedral and Non-polyhedral.
A solid or form is the geometry of a three-dimensional space, the kind of space we live in. It is three-dimensional with width, depth, and height. Solids have properties, such as volume and surface area. Two types of solids are Polyhedral and Non-polyhedral.
PLATONIC SOLID
Platonic solid is a convex polyhedron whose faces are all congruent convex regular polygons. There are five platonic solids namely, tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
Platonic solids were studied by the ancient Greek who also call these solids cosmic solids and are of 5 types. It is believed that the five platonic solids that exist in nature represent the five elements i.e. earth, air, fire, water, and the universe.
A platonic solid is a 3D shape where each face is the same as a regular polygon and has the same number of faces meeting at each vertex. A regular, convex polyhedron with identical faces made up of congruent convex regular polygons is called a platonic solid.
Platonic solid is a convex polyhedron whose faces are all congruent convex regular polygons. There are five platonic solids namely, tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
Platonic solids were studied by the ancient Greek who also call these solids cosmic solids and are of 5 types. It is believed that the five platonic solids that exist in nature represent the five elements i.e. earth, air, fire, water, and the universe.
A platonic solid is a 3D shape where each face is the same as a regular polygon and has the same number of faces meeting at each vertex. A regular, convex polyhedron with identical faces made up of congruent convex regular polygons is called a platonic solid.
EULER'S POLYHEDRA FORMULA
Euler's polyhedra formula states that V - E + F = 2. The formula is named after Leonhard Euler, a well-known Swiss mathematician (1707 - 1783)
Euler's polyhedra formula states that V - E + F = 2. The formula is named after Leonhard Euler, a well-known Swiss mathematician (1707 - 1783)
PRISM
Prism is a three-dimensional solid object in which the two ends are identical. It is the combination of the flat faces, identical bases and equal cross-sections. The faces of the prism are parallelograms or rectangles without the bases. And the bases of the prism could be triangle, square, rectangle or any n-sided polygon. For example, a pentagonal prism has two pentagonal bases and 5 rectangular faces. Prism has two types, regular prism and irregular prism. It is classified as regular prism when the cross section of each is a regular polygon. An irregular prism is one whose cross section is an irregular polygon or not “regular” in shape.
Prism is a three-dimensional solid object in which the two ends are identical. It is the combination of the flat faces, identical bases and equal cross-sections. The faces of the prism are parallelograms or rectangles without the bases. And the bases of the prism could be triangle, square, rectangle or any n-sided polygon. For example, a pentagonal prism has two pentagonal bases and 5 rectangular faces. Prism has two types, regular prism and irregular prism. It is classified as regular prism when the cross section of each is a regular polygon. An irregular prism is one whose cross section is an irregular polygon or not “regular” in shape.
PYRAMID
A pyramid is a polyhedron that has a base, which can be any polygon, and three or more triangular faces that meet at a point called the apex. These triangular sides are sometimes called the lateral faces to distinguish them from the base. In a nutshell, a pyramid is a polyhedron made by connecting a base to an apex.
A pyramid is a polyhedron that has a base, which can be any polygon, and three or more triangular faces that meet at a point called the apex. These triangular sides are sometimes called the lateral faces to distinguish them from the base. In a nutshell, a pyramid is a polyhedron made by connecting a base to an apex.
NON-POLYHEDRON
Non-polyhedron is a geometric solid with side or sides that are not polygons. Some examples of non-polyhedron are cones, spheres, cylinders, and torus.
Non-polyhedron is a geometric solid with side or sides that are not polygons. Some examples of non-polyhedron are cones, spheres, cylinders, and torus.
REFLECTION
In this module, I gained an important realization about how important geometric design is especially in the field of engineering. At some point, it came to my mind that in the coming years, we will be dealing with drawing and planning which I think require knowledge in geometric design. Since geometric design focuses on the simplistic beauty of mixing certain shapes, lines, curves together for creative results, it would be better for me to familiarize the lesson as a form of framework towards creating my own creative results. It was nice that I was able to grasp lesson with full understanding given that the topics were just in textual form and I just need to be familiar with the terms discussed such as the polyhedron, platonic solid, quadrilateral, polygon, and others. I learned a lot in this module.
In this module, I gained an important realization about how important geometric design is especially in the field of engineering. At some point, it came to my mind that in the coming years, we will be dealing with drawing and planning which I think require knowledge in geometric design. Since geometric design focuses on the simplistic beauty of mixing certain shapes, lines, curves together for creative results, it would be better for me to familiarize the lesson as a form of framework towards creating my own creative results. It was nice that I was able to grasp lesson with full understanding given that the topics were just in textual form and I just need to be familiar with the terms discussed such as the polyhedron, platonic solid, quadrilateral, polygon, and others. I learned a lot in this module.
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