Symmetry Definition in Mathematics

Symmetry Definition

• Symmetry is the ability to divide an object into at least two identical portions (mirror images).
• Every existing or imaginary object has a shape that can be described by its edges and angles.
• Regular shapes, such as triangles, hexagons, and rectangles, tend to be symmetrical.

Line of Symmetry

• An imaginary line that cuts a shape into two exact or identical halves is named the line of symmetry.
• The line of symmetry represents the symmetry in shape along one axis only.
• Many shapes can be divided into halves by more than one line, so a shape can have multiple lines of symmetry.
• The properties of the line of symmetry are:
  • It always passes through the center of the shape.
  • The halves obtained by drawing the line of symmetry reflect each other.
  • The halves obtained by drawing the line of symmetry math are mirror images of each other.

Types of Line of Symmetry

• Vertical Line of Symmetry:
  • A vertical line runs from top to bottom.
  • If a shape can be divided into two halves by drawing a vertical line through it, this represents a vertical line of symmetry.
  • Examples: letter "M", heart shape.
• Horizontal Line of Symmetry:
  • A horizontal line goes from left to right or from right to left.
  • If a shape can be divided into two equal halves by such a line, the shape has a horizontal line of symmetry.
  • Examples: letter "B", irregular pentagon.
• Diagonal Line of Symmetry:
  • In a rectangle, the line that connects two opposite ends represents the diagonal.
  • The diagonal line of symmetry divides the rectangle into two halves.

One Line of Symmetry

• A shape has only one line of symmetry when it only has symmetry about one axis.
• The line of symmetry can be present in two ways:
  • Vertical line of symmetry, as in the letter "M".
  • Horizontal line of symmetry, as in the letter "B".

Two Lines of Symmetry

• Some shapes are symmetrical about two lines.
• Example: the letter "H" has two lines of symmetry; both a vertical and horizontal line will divide the letter into two equal parts.

Shapes That Have More than One Line of Symmetry

• Many regular shapes can be divided into halves by drawing lines in different directions at different angles.
• Examples: square, five-pointed star.
• A circle technically has infinite lines of symmetry; any line drawn through the center or a circle will result in two perfect semicircles.

Reflection Symmetry

• Reflection symmetry also represents line symmetry.
• If a shape remains unchanged after being reflected across a line, then it also has reflection symmetry.
• Examples: letter "T", crown shape.

Rotational Symmetry

• If the rotation of a shape around the center of its axis generates an indistinguishable image, then the figure has rotational symmetry.
• Features of rotational symmetry:
  • Rotational Angle: Every shape has a fixed angle of rotation.
  • Rotation with the same angle produces an identical image.
  • Order of Rotation: The number of times an image has to be rotated at an angle to reach the starting point gives the order of rotation.
  • Rotational Center: The point around which an image is rotated is called the rotational center.

Point Symmetry

• Point symmetry occurs when there is a point at the middle of a shape from which other points are equidistant.
• Example: letter "X".

Translation Symmetry

• Translation symmetry represents two or more images that are exact but have different positions.
• Features of translation symmetry:
  • The position of the object changes as it slides along an axis.
  • The angles of the object do not change.
  • The size of the object does not change.

Glide Symmetry

• The reflection of an image when combined with its movement (translation) represents glide symmetry.
• Example: the footsteps of a person are an excellent example of glide symmetry as one footstep is a reflection of another, but it is positioned at a different spot from the last one.