Symmetry Pattern Creator

Types of Symmetry

Symmetry is a fundamental concept in mathematics that describes the invariance of a shape or pattern under certain transformations.

Rotational Symmetry

A shape has rotational symmetry if it looks the same after being rotated by a certain angle. The order of rotational symmetry is the number of times the shape matches itself during a full rotation.

Reflectional Symmetry

Also called mirror symmetry, this occurs when a shape can be reflected across a line and still look the same. The line is called the line of symmetry or axis of symmetry.

Translational Symmetry

A pattern has translational symmetry if it can be translated (moved) by a certain distance in a particular direction and still look the same.

Applications in Nature and Art

Symmetry appears everywhere in nature and has been used in art and architecture throughout history:

• Crystals: Exhibit various types of symmetry based on their atomic structure
• Flowers: Many flowers have rotational symmetry (roses, daisies)
• Snowflakes: Display hexagonal symmetry due to water molecule structure
• Islamic Art: Complex geometric patterns based on symmetry principles
• Architecture: Buildings often use symmetry for aesthetic appeal
• Wallpaper Patterns: Based on the 17 wallpaper groups

Mathematical Notation

Symmetries are often described using group theory notation:

• Cn: Cyclic group with n-fold rotational symmetry
• Dn: Dihedral group with n-fold rotational and reflectional symmetry
• 17 Wallpaper Groups: All possible 2D patterns with translational symmetry