Rotations in the Plane

This page delves into the concept of rotations in geometry. A rotation is described as a turn of a figure around a fixed point.

The key elements defining a rotation are:

1. The center point of rotation
2. The angle of rotation
3. The direction (clockwise or counterclockwise)

Highlight: Like translations, rotations also preserve lengths, alignments, angles, and areas of the original figure.

An example is provided showing a figure and its rotated image around a center point O.

Example: The diagram illustrates a shape (Figure 1) and its rotated version (Figure 4) around a center point, demonstrating a 180° clockwise rotation.

Note: If the angle and direction are not specified, a rotation is assumed to be 180°.

The page also introduces the concept of frises et pavages (friezes and tessellations), which are patterns created using these transformations.

Vocabulary: Rotation - A geometric transformation that turns a figure around a fixed point by a certain angle.

Friezes and Tessellations

This page explores the applications of geometric transformations in creating patterns, specifically frises et pavages (friezes and tessellations).

Tessellation (Pavage): A tessellation is a pattern that covers a plane without gaps or overlaps using one or more geometric shapes.

Definition: A tessellation is created by repeating a basic motif (elementary pattern) using translations and/or rotations.

Frieze (Frise): A frieze is a strip pattern that extends infinitely in one direction.

Example: The diagram shows a frieze pattern created by applying two translations to a basic motif.

Rosette (Rosace): A rosette is a circular pattern created by rotating a motif around a central point.

Example: The illustration demonstrates a rosette pattern formed by applying multiple rotations to a basic shape.

Vocabulary:

• Tessellation (Pavage) - A pattern that fills a plane without gaps or overlaps.
• Frieze (Frise) - A strip pattern extending infinitely in one direction.
• Rosette (Rosace) - A circular pattern created by rotations around a central point.

These patterns demonstrate practical applications of transformations géométriques (geometric transformations) in creating aesthetically pleasing and mathematically interesting designs.

Translations in the Plane

This page introduces the concept of translations in geometry. A translation is defined as a parallel shift along a straight line.

The key characteristics of a translation are:

1. Its direction
2. Its sense (orientation)
3. Its length

Highlight: Translations preserve lengths, alignments, angles, and areas of the original figure.

An example is provided with two figures, where Figure 2 is the image of Figure 1 after a translation that moves point A to point B.

Example: The diagram shows a triangle (Figure 1) and its translated image (Figure 2), clearly illustrating how the entire shape moves in a parallel manner.

Vocabulary: Translation - A geometric transformation that moves every point of a figure the same distance in the same direction.