What is the importance of symmetry

Symmetry¶

Symmetry is the property of some geometric shapes to appear unchanged after a certain transformation.

Axial symmetry¶

Any straight line or segment running across a geometric figure is called a transversal. If the figure can be divided by one of these transversals into two equally large sub-figures, which are completely congruent with one another by folding over along the transversals, then the figure is called axially symmetrical. The two parts of the figure are referred to as corresponding to one another and called the transverse axis of symmetry.

A figure can also have several axes of symmetry. For example, a rectangle has two axes of symmetry, a circle even has any number.

You can create two figures of the same size by folding them around a straight line between the two figures bring them to coincide, both figures are said to be axially symmetrical to one another. Axisymmetric figures always have the following properties:

  • Corresponding points are equidistant from the axis of symmetry away.
  • Connecting lines between corresponding points are perpendicular to the axis of symmetry.
  • Corresponding straight lines (as well as segments and their extensions) intersect the symmetry axis at the same point and at the same angle.
  • The direction of rotation of both figures is reversed, corresponding corners follow one another in one figure clockwise and counterclockwise in the other.

Axisymmetrical figures can also be understood as pairs of axially symmetrical figures whose surfaces partially overlap.

Point symmetry¶

Is a figure with a -Rotation around a point in the interior completely congruent with itself, so the figure is called (simply) point-symmetrical. The point of rotation is called the center of symmetry and the angle of rotation is called the angle of symmetry.

Is a figure already after being rotated by an angle congruent with itself, whereby is any natural number, the figure is called multiple point symmetry. It is on further rotations around the angle always congruent again.

Can two figures of the same size be replaced by one - Rotation around a point between the two figures bring them to coincide, both figures are referred to as being point-symmetrical to one another. Point symmetrical figures always have the following properties:

  • Corresponding points are equidistant from the center of symmetry away.
  • Connecting lines between corresponding points run through the center of symmetry.
  • Corresponding straight lines (as well as lines and their extensions) run parallel to one another.
  • The direction of rotation of both figures is the same, corresponding corners follow one another either clockwise or counter-clockwise in both figures.

Point-symmetrical figures can also be understood as pairs of point-symmetrical figures whose surfaces partially overlap.