Symmetry in nature | etwinning-2015-2016

Many people believe that symmetry is beauty.

Symmetry is considered by many as a reflection of beauty and perfection. A symmetric object is an object where we can place a line such that the images on both sides of the line mirror each other.

The geometric shapes in the first figure are symmetric. As shown, we can draw a line — in fact more than one line — to show symmetry. The equilateral triangle has three lines of symmetry, the square has four, and the circle has infinitely many.

We can see symmetry everywhere. Butterflies, lobsters, and some leaves, and flowers are symmetric. Many structures, buildings, and houses are symmetric. The human face and body are symmetric. In fact, because of its ubiquity, people are trying to relate it to many things. For example, some claim that facial symmetry is an “indicator of freedom from disease and worthiness of mating.”

If someone claims that symmetry is a symbol of love, I would probably agree with him. In the 16th century, Shah Jahan, emperor of the Mughal Empire, was so grief-stricken that he spent about 32 million rupees for the construction of the Taj Mahal. The Taj Mahal was constructed as a mausoleum for his wife’s tomb. To date, it is still one of the most beautiful, symmetric structures ever created.

In school mathematics, we study symmetry in geometry and also along with transformation. In the coordinate plane, we know that if a figure with the y-axis as its line of symmetry, if this figure contains a point with coordinates , then its corresponding point has coordinates , a reflection over the y-axis. We have also learned that the reflected image is congruent to its pre-image and that if we connect their corresponding points with a segment, the segment is perpendicular to the line of symmetry.

Nature is full of symmetric objects. There are many man-made structures that are also symmetric. In this post, we are going to discuss some of the basic mathematical properties of symmetric objects. We will limit our discussion to line symmetric objects.

Line Symmetry

A figure is line symmetrical when it can be folded along a straight line such that the folded shapes fit exactly on top of each other. The fold line is called the line of symmetry.

When a symmetric figure is folded along its line of symmetry, the parts that are on top of each other are called the corresponding parts. In the polygon below with line of symmetry AB, points C and D are corresponding points, segments GB and HB are corresponding sides, and angle G and angle H are corresponding angles. Since the folded shapes fit exactly on top of each other, the corresponding angles are congruent and their corresponding sides are also congruent.

Exercise:

(1) What is the corresponding point of point G? of segment DF? of angle D?
(2) Identify two pairs of congruent angles and two pairs of congruent sides.

If you connect the corresponding points of a line symmetric figure, the line segment connecting them is perpendicular to the line of symmetry (Can you see why?). This means that their distances from the line of symmetry to the corresponding points are equal. In the figure below, the distance from point P to point C is equal to the distance from point P to point D. That is,PC=PA.

In the coordinate plane, if the line of symmetry is the y-axis, then the corresponding point of (2,2) is (-2,2). Since they are on the same horizontal line, their y-coordinates are equal. In addition, since they are on the opposite sides of the y-axis and equidistant from it, their x-coordinates have the same absolute value but the signs are opposite. For example, if a figure contains a point whose x-coordinate is 3, the x-coordinate of the corresponding point is -3. As we can see |3| = |-3| = 3. From this, we can generalize that if a point (a,b) is a part of a figure whose line of symmetry is the y-axis, then its corresponding point is (-a,b).

Using the same reasoning above, if the line of symmetry is the x-axis, then the points are on the opposite side of the x-axis. This means that the y-coordinates of the corresponding points have opposite signs. That is, corresponding point of the point with coordinates (a,b), is the point with coordinates (a, – b).

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"Why do we human beings delight in seeing perfectly round planets through the lens of a telescope and six-sided snowflakes on a cold winter day? The answer must be partly psychological. I would claim that symmetry represents order, and we crave order in this strange universe we find ourselves in. The search for symmetry, and the emotional pleasure we derive when we find it, must help us make sense of the world around us, just as we find satisfaction in the repetition of the seasons and the reliability of friendships. Symmetry is also economy. Symmetry is simplicity. Symmetry is elegance Why do we human beings delight in seeing perfectly round planets through the lens of a telescope and six-sided snowflakes on a cold winter day? The answer must be partly psychological. I would claim that symmetry represents order, and we crave order in this strange universe we find ourselves in. "

The search for symmetry, and

the emotional pleasure we derive when we find it,

must help us make sense of the world around us,

just as we find satisfaction in the repetition of the seasons

and the reliability of friendships.

Symmetry is also economy.

Symmetry is simplicity.

Symmetry is elegance.

Lightman