Explain mathematically and illustrate/show/draw/copy-and-paste a cultural artefact in which reflection is used to develop the pattern or artistic effect involved.

Question: Explain mathematically and illustrate/show/draw/copy-and-paste a cultural artefact in which reflection is used to develop the pattern or artistic effect involved.

Reflection is a type of symmetry that occurs when an object or a pattern is flipped over a line or a point. The flipped object or pattern is called the reflection or the mirror image of the original one. The line or the point that divides the original and the reflected object or pattern is called the axis of symmetry or the center of symmetry. Mathematically, reflection can be described as a transformation that maps each point (x, y) to a new point (x’, y’) such that:

If the reflection is over the x-axis, then x’ = x and y’ = -y.

If the reflection is over the y-axis, then x’ = -x and y’ = y.

If the reflection is over the origin, then x’ = -x and y’ = -y.

If the reflection is over a line y = mx + b, then x’ = (x + 2my - 2bm) / (1 + m^2) and y’ = (y - 2mx + 2b) / (1 + m^2).

One example of a cultural artifact that uses reflection to create a symmetrical pattern is the Rangoli. Rangoli is a traditional art form from India, where colorful patterns are drawn on the floor using rice flour, sand, or flower petals. Rangoli is usually made during festivals or special occasions to welcome guests and deities. Rangoli designs often have reflection symmetry along one or more axes. Here is an example of a Rangoli design that has reflection symmetry along four axes:

You can see that each quadrant of the design is a reflection of the adjacent quadrant over a diagonal line. The center of the design is also a reflection of itself over both the horizontal and vertical lines. This Rangoli design has four axes of symmetry: y = x, y = -x, x = 0, and y = 0.

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