Do you know The Taj Mahal, situated at the Yamuna river banks, is considered the largest symmetrical building in the world? The beauty of “Taj” lies in its symmetry. Symmetry is often used in context with beauty, harmonious, eye-pleasing, praised by artists, but it is also a base for important mathematical and geometrical concepts. We will understand symmetry from a mathematical perspective, which will help you calculate the maths of hidden symmetry in objects.
Disclaimer: Visualise geometry to get a good grasp on the subject. Visualization will take you to those unanswered questions enabling real learning. As mentioned by Stephen Hawking,: “Equations are just the boring part of mathematics. I attempt to see things in terms of geometry.”
What is Symmetry and Lines of Symmetry in a Rectangle?
Two or more objects resembling after flipping, turning, or sliding them on each other, are called “Symmetric.” Any line that splits an item into identical parts is called an “Axes of symmetry”. The resultant parts are called symmetrical to each other. We will learn about symmetry, its properties, and dive deeper into the axes of symmetry of rectangles in this article.
A rectangle is a 4-sided polygon with two equal and parallel opposite sides having 90-degree angles at all sides. There are two axes of symmetry possible for a rectangular object. Straight lines cutting the rectangle straight into two equal parts can be drawn along the length or width. This line is called “Axes of symmetry” in the rectangle. The dissected parts superimposing each other perfectly explains the concept of “symmetric objects”.
How Many Types Of Symmetry Are There In A Rectangle?
- Linear symmetry: Linear symmetry is also known as reflection symmetry. Linear symmetry can be determined by cutting the object through the middle point to produce two parts. These two parts are then superimposed and checked to find out if they are identical or not. A rectangle can be cut along its breadth or height from the middle points of its sides. The parts produced after cutting rectangles along a line of symmetry are identical and thus called “symmetrical” to each other. Talking in terms of reflection symmetry, by drawing a line from the midpoint, two parts are produced. These should be a mirror image of each other to satisfy the symmetry conditions. The line drawn is known as the “axes of reflection”. Since we can have only two lines that could cut the rectangle to produce identical objects, the linear symmetry is of order 2 for the rectangle.
- Rotational Symmetry: Rotational symmetry is determined in an object by rotating the object by some angle (A1) and then comparing the original object with a rotated object. The key point here is that the rotation should not alter the original object’s shape or size at all. If original and rotated objects superimpose each other, then the object is said to have rotational symmetry at the (A1) angle. Rectangle has a symmetry of order 2 when rotated at 180 and 360 degrees. This means the Rectangle is rotated around X and Y axes, by 180 and 360 degrees angles, the rotated and original objects are superimposed on each other. It produces identical objects even after rotation.
Important Note: Rotations follow reflections. That means if an object has reflection symmetry, then it would necessarily have rotational symmetry corresponding to reflections. Any object can not just have reflection symmetry and not rotational symmetry. At the same time, vice versa is not true. Objects can have rotational symmetry but not reflection / Linear symmetry. For example, some flowers have rotational symmetry, but not linear symmetry.
Let’s compare the symmetry of rectangle with other quadrilaterals:
- Rectangle v/s Square: Square has a linear symmetry of order 4 while a rectangle has a linear symmetry of order 2. The main difference between the Rectangle and Square is that the rectangle has two equal opposite sides while the square has all sides equal. This is the reason that when diagonals are drawn in a square, 4 symmetric parts are obtained and when the diagonals are drawn in a rectangle then 4 non-symmetrical parts are obtained. The length of vertical and horizontal sides is not equal in length which changes the symmetry in the rectangle. If we talk about rotational symmetry then a square has 4, while a rectangle has 2 orders of symmetry. Square has rotational symmetry at angles 90, 180, 270, and 360 degrees, while a rectangle has it at angles 180 and 360 respectively.
- Rectangle v/s Circle: Circle has an infinite symmetry against 2 for a rectangle. If any line is drawn from the center of a circle, that line will produce symmetrical parts. Since one can draw an infinite number of lines passing through the center of a circle, infinite symmetrical parts can be produced in a circle. Considering rotational symmetry, if the circle is rotated at any degree and kept on the original circle, it will completely superimpose the original one. Hence the circle has rotational symmetry of order infinity against 2 for the rectangle.
- Rectangle v/s Parallelogram: Parallelogram is any quadrilateral with opposite sides equal and parallel. The specification of a parallelogram is satisfied by square, rectangle, and Rhombus. Square has 4, a rectangle has 2, and rhombus has 2 orders of symmetry. By comparing the symmetries of rhombus and rectangle a small difference can be detected. The difference is about the side angles. Rectangles have all sides at an angle of 90 degrees while rhombus does not. The sides of the rhombus are equal and at an angle such that when the equal length diagonals are drawn, 4 symmetric parts are obtained. When straight lines are drawn from the middle on sides of the rhombus then the parts obtained are non-symmetrical since they are aligned at particular angles. The rotational symmetry for the rhombus is of order 2 which is equal to that of the rectangle but at different angles of 90 and 270 degrees. The comparison of symmetry between square and rectangle is already discussed in the previous section.
- Rectangle v/s Triangle: Triangle has the symmetry of 3, 1, or 0 for equilateral, Isosceles, and scalene triangles respectively. It depends upon the length of the triangle sides. An equilateral triangle has all equal sides, Isosceles has 2 equal sides while scalene has none. Triangle does not have either of the sides parallel to each other. It is different from a rectangle in terms of shape and properties related to symmetry. The rectangle would always have equal and parallel opposite sides, while the same is not true for triangles. Hence, rectangles have a fixed symmetry of order 2, while triangles do not have a fixed symmetry.
How To Draw Symmetrical Objects?
- Cutting and Comparing: You can start assessing the symmetry of an object by drawing a line from the middle. The parts produced after cutting are kept over each other to check if they are superimposing. Objects superimposing each other completely are called symmetrical. The same method can be used on rectangular paper where you would discover two axes of symmetry.
- Thread Painting: You can dip a thread into ink or any watercolor and then draw a pattern on the paper with the help of colored thread. Fold the paper in half to cover the thread. Press the folded paper from the top with a big book and pull out the thread. You would see a beautiful symmetrical figure on two sides of the paper.
- Folding and Cutting: Fold the paper in two or more layers and then cut any design out of it. You would find multiple symmetrical objects created. The paper cuts would result in a beautifully symmetric design when the paper is unfolded.
- How many rotational and linear symmetries does a rectangle have?
- There are 2 lines of axes for symmetry in the rectangle. The linear and rotational symmetry of the rectangle is 2.
- Is it possible to have linear and rotational symmetry different?
- Yes! It is possible to have rotational symmetry, while no or less linear symmetry. If you check flowers then the leaves of flowers might be not symmetrical, but all leaves can be of the same shape. If the flower is rotated and superimposed with the previous position of the flower then it might superimpose.
It is interesting to read about symmetry and it becomes more interesting when we relate it with real-life objects. You can find a lot of examples of symmetry in nature like butterflies. Major architectural buildings follow the principle of symmetry. It not only gives a great look at the building, but also provides robust architecture. Many interior decorators, web designers, handicraft workers, content writers use the principle of symmetry for presentable work. No matter whatever your field of expertise is, symmetry will add a golden tinge to it.
We have got some insightful courses and articles to unlock the doors of your symmetrical future. For more such interesting facts on Measurement Tips and Tricks get in touch with Cuemath. Our wide repository of knowledge articles would be a great feast for your curiosity.