The Beauty in Patterns:

Symmetry of the Euclidean Plane

 " The mathematical science particulary exhibit order, symmetry, and limitation, and these are the greatest forms of the beautiful" Aristotle

The human eye identifies beauty and harmony in images that are symmetrical and follow precise and ordered patterns. What are the mathematical concepts behind these patterns? The following images created using the letter "P" will give a brief visual introduction to the four forms of  planar and rigid symmetries of the Euclidean plane: translation, reflection, rotation, and glide reflection.

         Translations                 Reflections

        Rotation                     Glide Reflection

The combination of these four isometries result in the beautiful patterns that have been part of human’s artistic expressions since the beginning of the time.

The Euclidean plane is the  set of all ordered pairs of real numbers in two dimensions endowed with distance as represented by the logic definition below. 

In mathematics a symmetry is a transformation that preserves measurements, and more specifically distances between points in the Euclidean plane