Chasles' Theorem

Summary Facts about Isometry.

Multiplication Table for Isometries.

Types of Transformations.

There are different types of isometric transformations. Some of the basic movements or mapping of a figure include sliding, flipping, turning, and enlarging or reducing the size.

1. Translation

Translation is the movement of figures by sliding them upwards, downwards, sideways, or diagonally. Under translation, there are no fixed points; each point moves by an equal distance, which is determined by the vector. Additionally, the pre-image and the image must be congruent with each other. Hence, it can be concluded that in a translation, a figure does not change its orientation and size. It is merely the exact copy of the figure but in a different location of the plane.

2. Rotation

It is the process of turning or rotating the pre-image around the axis. This type of transformation has exactly one fixed point. Thus, there must be no changes in terms of size or shape, and must be congruent to its pre-image. The rotation can be determined by its center point according to its angle of rotation. 

3. Reflection

This transformation is the flipping of a figure over a line and produces a mirror image. The line in which points from the pre-image and the image intersect acts as the perpendicular bisector. The distance from the image to the line is the same as the pre-image but on the other side of the axis. When a figure is reflected, no changes in size or shape should be made. The image must be congruent to the pre-image, but orientation reversed.

4. Dilation

This transformation is the enlarging or reducing of the size of the figure. Unlike translation, rotation, and reflection which somehow preserves the pre-image, in dilation, the pre-image and the image are not similar in terms of fixed points and are not congruent. The transformations change the dimensions of the figure, which makes it a non-isometric transformation. 

5. Glide Reflection

This comprises two rigid transformations: translation and reflection. This happens when a pre-image is reflected over a line and then translated horizontally or vertically to form the image. The resulting image will have a “gliding effect.”

Proper and Improper Isometry.

• If an isometry cannot be applied when lifting the plane into space, is it considered to be an improper isometry. Otherwise, it is considered proper isometry.

• Proper Isometry includes: translation and rotation

• Improper Isometry includes: reflection and glide reflection

Patterns (Overview). 

Trivia! Motif vs Design vs Pattern

A motif is the basic unit of design. A design starts with a motif. If a motif is repeated several times, it produces a pattern. A repetition of this pattern, creates a design. In simple words, the repetition of the motif creates a pattern, which is used to create designs.

Types of Symmentric Patterns.

1. Rossete Patterns

Rosette patterns are patterns that consist of either Cyclic Symmetry or Dihedral Symmetry.

2. Frieze Patterns

Frieze patterns, also known as border patterns, are infinite long strip patterns printed with a repeating motif. It is a pattern that extends to a certain direction such that the patterns are mapped onto themselves through a translation (T). Additionally, frieze patterns can also be mapped through rotation (R), glide reflection (G), vertical reflection (V), and horizontal reflection (H).

The Seven Frieze Patterns

3. Wallpaper Patterns

Wallpaper patterns cover the plane mapped onto themselves in more than one direction of translation symmetry. For it to be considered wallpaper it must at least maintain the basic elements. One copy of the figure by translation and another copy of the figure through another translation in the next row. Thus, there must be at least two rows with two units long. The multiple directions of the translation force the pattern to fill in the entire plane. 

The lattice of translation in wallpaper is the point at which each pattern is collected. The lattices are connected forming a grid structure. The red dots shown below are the lattices of the wallpaper.

4. Tessellations

Tessellations are repeating patterns of a figure that completely covers a plane without gaps or overlaps. They can be created with translation, rotation, and reflection. 

A tessellation that is made from one repeating shape/polygon is called a regular tessellation. There are only three regular tessellations. These three shapes are triangles, squares, and hexagons. An example of a regular tessellation is a honeycomb. 

On the other hand, semi-regular tessellation arises made of two or more regular polygons, which have eight types.

Symmentrical Patterns in Real life.