Geometry Class: Polygons and Solids

Recognizing and Analyzing Geometric Shapes

• Polygon: A two-dimensional shape with straight sides. Classified based on the number of sides.
  • Simple: Single boundary, never crosses over itself.
  • Complex: Intersects itself.
  • Concave: Angles point inward.
  • Convex: No angles pointing inward.
  • Regular: All angles and sides equal.
  • Irregular: Not all sides or angles are equal.
• Interior angles of a polygon: Angles located inside the shape.
  • Sum of internal angles: (n - 2) x 180° (where n is the number of sides).
  • Measurement of each angle (regular polygon): (n - 2) x 180°/n
• Exterior angles of a polygon: Sum of exterior angles is always 360°.
• Platonic solids: Three-dimensional shapes with regular polygon faces, all angles equal, and all edges the same length.
  • Tetrahedron: 4 faces, 4 vertices, 6 edges.
  • Cube: 6 faces, 8 vertices, 12 edges.
  • Octahedron: 8 faces, 6 vertices, 12 edges.
  • Dodecahedron: 12 faces, 20 vertices, 30 edges.
  • Icosahedron: 20 faces, 12 vertices, 30 edges.
• Euler's Formula: F + V - E = 2 (where F is the number of faces, V is the number of vertices, and E is the number of edges).
  • Applies to: Polyhedrons with no holes, self-intersections or made up of two pieces stuck together.
• Prism: A polyhedron with flat sides. Has the same cross-section all along its length.
  • Regular: Cross-section is a regular polygon.
  • Irregular: Cross-section is an irregular polygon.
• Pyramid: A polyhedron made by connecting a base to an apex.
  • Types: Determined by the shape of its base, e.g., triangular pyramid, square pyramid.
• Non-polyhedral shapes: 3D shapes with curved surfaces.
  • Sphere: Perfectly round object, no edges or vertices.
  • Cylinder: Bounded by a curved surface and two parallel circles.
  • Torus: Solid formed by revolving a small circle along the line of another circle.
  • Cone: Made by rotating a right-angled triangle around one of its short sides.

Geometric Transformations

• Geometric transformation: Changes a shape's size, orientation, or position using mathematical techniques.
  • Object: Original shape.
  • Image: New shape.
• Types of geometric transformations:
  • Translation: Every point moves a fixed distance and direction.
  • Rotation: Object rotates about a point at a given angle.
  • Reflection: Every point is the same distance from a line as its image.
  • Glide reflection: Combination of translation and reflection.
  • Dilation: Resizing an object about a point by a certain factor.
    • Enlargement: Factor is greater than 1.
    • Reduction: Factor is between 0 and 1.

Patterns And Diagrams

• Patterns: Repeating elements in geometry found in everyday objects.
• Symmetry: The idea of order, beauty, and perfection.
  • Found in nature and arts.
• Symmetry groups: Collection of symmetries of a plane figure.
  • Cyclic symmetry group: Rotation symmetry only around a center point.
    • Cn: nth order rotation.
  • Dihedral symmetry group: Rotation symmetry with reflection lines.
    • Dn: nth order rotation with n reflection lines.
  • Rosette symmetry: Cyclic and dihedral symmetry groups.
• Frieze pattern: Infinite strip with a repeating pattern.
  • Frieze group: Symmetries of a frieze pattern.
    • International Union (IUC) notation: Classifies frieze groups by using symbols for vertical reflections, horizontal reflections, glide reflections, and rotations.
• Tessellation: Pattern of shapes that covers a plane without gaps or overlaps.
  • Examples: Pavements, patios, wallpapers, tiling.