Geometric-Design

Geometric Designs - Venus L. Jucutan

Page 1

  • Course: GECC 103. Mathematics in the Modern World

  • Focus: Geometric Designs

  • Institution: Don Mariano Marcos Memorial State University - MLUC

Page 2: Geometric Design (GD)

  • Definition: Branch of computational geometry involving the construction and representation of free-form curves, surfaces, or volumes.

  • Application areas: Shipbuilding, aircraft, automotive industries, and architectural design.

  • Types of models: 2D and 3D geometric models

    • 2D models: Important in computer typography and technical drawing.

    • 3D models: Central to computer-aided design, manufacturing, geology, and medical image processing.

Page 3: Fascination with Geometric Shapes

  • Historical interest in geometric shapes across various fields.

  • Mathematicians developed ideal representations to measure:

    • Lengths (1D)

    • Areas (2D)

    • Volumes (3D)

Page 4: Types of Geometric Design - Circle

  • Definition: Shape traced as a curve equidistant from a point (center).

  • Characteristic:

    • Circumference: Distance around the circle.

Page 5: Types of Geometric Design - Triangle

  • Definition: Shape formed by 3 straight lines (sides).

  • Classification of triangles:

    • By Angles:

      • Right Triangle: One right angle.

      • Acute Triangle: All angles less than 90°.

      • Obtuse Triangle: One angle greater than 90°.

    • By Sides:

      • Equilateral: All sides equal.

      • Isosceles: At least two sides equal.

      • Scalene: No equal sides.

Page 6: Types of Geometric Design - Other Shapes

  • Rectangle: 4 sides, 90° angles.

  • Rhombus: 4 sides equal, non-right angles.

  • Square: A rectangle and rhombus, 90° angles, equal sides.

  • Trapezoid: 4 sides, 2 parallel sides.

Page 7: Polygons

  • Definition: Plane figure bounded by a finite chain of straight segments (sides).

  • Definition of edges (sides) and vertices (corners).

  • n-gon: A polygon with n sides.

Page 8: Characteristics of Polygons

  • Classification based on the number of sides: e.g., triangle (3), quadrilateral (4).

Page 9: Simple vs Complex Polygons

  • Simple Polygon: One boundary, non-intersecting sides.

  • Complex Polygon: Sides cross one another.

Page 10: Regular vs Irregular Polygons

  • Regular Polygon: Equal sides and angles.

  • Irregular Polygon: Unequal sides and angles.

Page 11: Concave vs Convex Polygons

  • Convex Polygon: No inward angles (all < 180°).

  • Concave Polygon: At least one angle > 180°.

Page 12: Angle Properties of Polygons

  • Definition: Interior angles - angles inside the shape.

Page 13: Interior Angles Sum Property

  • Formula: Sum of interior angles = (n-2) × 180°

  • For regular polygons: Exterior angle = 360°/n.

Pages 14-15: Sum of Internal Angles Table

Name

Sides

Sum of Angles

Each Angle

Triangle

3

180°

60°

Quadrilateral

4

360°

90°

Pentagon

5

540°

108°

Hexagon

6

720°

120°

Heptagon

7

900°

128.571°

Octagon

8

1080°

135°

Nonagon

9

1260°

140°

Decagon

10

1440°

144°

Page 16: Characteristics of Regular Polygons

  • Prism: Flat sides, two identical bases.

  • Regular Prism: Regular cross-section of equal edge lengths.

  • Irregular Prism: Irregular cross-section, not equal edge lengths.

Pages 17-21: Solids and Polyhedra

  • Solid Definition: 3-dimensional geometry (width, depth, height).

  • Polyhedra: Made of flat surfaces (polygons); includes:

    • Platonic Solids: All faces congruent and regular.

  • Five Platonic Solids: Cube, Tetrahedron, Dodecahedron, Octahedron, Icosahedron.

  • Vertices, Edges, Faces: Defined for polyhedra types.

Page 25: Non-polyhedra

  • Definition: Solids with curved surfaces.

    • Cylinder: Curved surface, two parallel circles.

    • Sphere: Round shape with no edges/vertices.

    • Torus: Formed by revolving a circle around an axis.

    • Cone: Formed by rotating a triangle about one edge.

Pages 27-36: Transformations in Geometry

  • Transformations: Movement of geometric objects (preimage to image).

    • Types:

      • Rigid Transformation: Shape/size unchanged.

      • Non-rigid Transformation: Changes size.

  • Types of Transformations:

    • Translation: Sliding without rotation.

    • Rotation: Turning around a fixed point.

    • Reflection: Mirroring over a line.

    • Glide Reflection: Translation followed by reflection.

    • Dilation: Resizing about a point (enlargement/reduction).

Pages 42-45: Isometry

  • Definition: Rigid transformation preserving lengths and angles.

  • Examples of identifying rigid transformations.

Pages 48-62: Rotational Symmetry

  • Definition: Shape remains unchanged under rotation through specific angles.

  • Examples include clock faces and flowers.

Pages 64-70: Patterns and Diagrams

  • Geometric Patterns: Repeating designs (e.g., tessellations).

  • Types of Symmetry:

    • Reflectional Symmetry: Mirrored halves.

    • Rotational Symmetry: Looks the same after rotation.

    • Translational Symmetry: Same pattern shifted.

    • Rosette Symmetry: Patterns with sequence rotations.

Pages 74-83: Tessellations

  • Definition: Covering surface with no overlaps/gaps.

  • Types: Regular tessellations (using regular polygons) vs semi-regular tessellations (mix of polygons).

Pages 85-92: Geometric Design in Art and Culture

  • Applications: Shipbuilding, architectural design, creating everyday objects.

  • Cultural significance: Traditional designs from tribes, especially from Mindanao, showcase geometric patterns woven into fabric.

  • Various indigenous groups from Mindanao with unique weaving traditions and designs.