Transformations in Geometry

Transformations

• Transforming a geometric figure involves moving its points according to rules, creating a new figure called the image.
• A transformation establishes a correspondence between the points of the original figure and its image.
• Rigid transformations preserve the shape and size of the original figure.
• Non-rigid transformations change the size of the original figure, but not the shape.

Rigid Transformations (Isometries)

• Translations move a figure a fixed distance in a specific direction, keeping all points equidistant from their images.
• A translation vector indicates the distance and direction of the translation.
• Rotations turn a figure around a fixed center point by a specific angle, either clockwise or counterclockwise.
• Reflections flip a figure across a line of reflection, producing a mirror image. The line of reflection acts as a perpendicular bisector for segments joining points and their images.
• Glide reflections combine reflection and translation.

Non-Rigid Transformations

• Dilations expand or contract an object without changing its shape or orientation. This includes resizing, contraction, compression, enlargement, or expansion.
• Shearing "pushes" a side of the original shape while keeping a base fixed. Points move parallel to the fixed side in proportion to their distance from it.

Patterns and Diagrams

• Symmetry exists when an object can be divided into identical parts arranged in a specific pattern.
• Reflectional symmetry (mirror symmetry) occurs when half of an object is a reflection of the other half.
• Rotational symmetry involves rotating an object around an axis without altering its shape.
• Translational symmetry shifts a pattern or design from one location to another, maintaining the same orientation.

Rosette Patterns

• Rosette patterns are formed by rotating and/or reflecting a motif or element.
• Cyclic rosette patterns have rotational symmetry around a center point but no mirror lines.
• Dihedral rosette patterns possess both reflection and rotation symmetry around a center point, with mirror lines passing through the center.

Frieze Patterns

• Frieze patterns are repetitive designs on a two-dimensional surface, exhibiting translational symmetry in one direction.
• John Conway categorized frieze patterns based on their symmetries, using names like "hop," "step," "jump," "slide," "spinning hop," "spinning jump," and "spinning sidle."

Wallpaper Patterns

• Wallpaper patterns are repetitive designs covering a plane, exhibiting symmetries in multiple directions.
• Wallpaper patterns are categorized based on their symmetries.
• Symmetry group 1 (p1): Only translations, no reflections, glide reflections, or rotations.
• Symmetry group 2 (p2): Translations and 180° rotations, no reflections or glide reflections.
• Symmetry group 3 (pm): Reflections parallel to one translation axis and perpendicular to another.
• Symmetry group 4 (pg): Glide reflections parallel to one translation axis and perpendicular to another.
• Symmetry group 5 (cm): Reflections and glide reflections with parallel axes.
• Symmetry group 6 (pmm): Perpendicular axes of reflection, no glide reflections or rotations.
• Symmetry group 7 (pmg): Reflection and 180° rotations, centers of rotation not on reflection axes.
• Symmetry group 8 (pgg): Glide reflections with perpendicular axes and 180° rotations, centers of rotation not on reflection axes.