Symmetry in mathematics

E8Petrie

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Symmetry in mathematics refers to a property by which an object remains invariant under certain transformations, such as reflection, rotation, or translation. Symmetry is a fundamental concept in various branches of mathematics, including geometry, algebra, and analysis.

Types of Symmetry

Symmetry can be classified into several types based on the nature of the transformation:

Reflection Symmetry

Reflection symmetry, also known as mirror symmetry, occurs when an object is identical to its image in a mirror. The line of reflection is called the line of symmetry.

Rotational Symmetry

Rotational symmetry exists when an object can be rotated about a central point and still look the same. The angle of rotation that maps the object onto itself is called the angle of symmetry.

Translational Symmetry

Translational symmetry occurs when an object can be translated (slid) in a particular direction and remain unchanged. This type of symmetry is common in tessellations and crystallography.

Glide Reflection Symmetry

Glide reflection symmetry combines reflection and translation. An object with glide reflection symmetry can be reflected over a line and then translated along that line to coincide with itself.

Rotoreflection Symmetry

Rotoreflection symmetry involves a combination of rotation and reflection. An object with this symmetry can be rotated about an axis and then reflected across a plane perpendicular to that axis.

Symmetry in Geometry

In geometry, symmetry is often studied in the context of shapes and figures. Common geometric shapes with symmetry include circles, regular polygons, and platonic solids.

Symmetry in Algebra

In algebra, symmetry is related to the concept of group theory. A group is a set equipped with an operation that combines any two elements to form a third element, satisfying certain conditions. Symmetry groups, such as the dihedral group and the cyclic group, describe the symmetries of geometric objects.

Symmetry in Analysis

In analysis, symmetry can be found in the solutions to differential equations and in the properties of functions. For example, an even function is symmetric about the y-axis, while an odd function is symmetric about the origin.

Applications of Symmetry

Symmetry has numerous applications in various fields, including physics, chemistry, biology, and art. In physics, symmetry principles are used to derive conservation laws. In chemistry, molecular symmetry helps in understanding the properties of molecules. In biology, symmetry is observed in the body plans of organisms. In art, symmetry is used to create aesthetically pleasing designs.

Related Pages

• Group theory
• Geometry
• Algebra
• Analysis
• Even function
• Odd function
• Tessellation
• Crystallography
• Dihedral group
• Cyclic group