Symmetry is one of the fundamental concepts of mathematics. It plays a crucial role not only in various areas of mathematics, but also in nature, art, architecture, physics, biology, and even economics. Symmetry, or the property of symmetry, generally refers to the state in which a shape, pattern, relationship, or object remains constant relative to a given point, line, axis, or transformation. Put simply: if the appearance or structure of an object does not change after a certain operation (such as rotation, reflection, or translation), it is symmetrical .
In this article, we will examine in detail the significance of symmetry in mathematics , the types of symmetry in mathematical figures and relations, and the application of the concept in various areas of mathematics and other sciences.
1. General definition of symmetry in mathematics
In the broadest sense, symmetry is a state in which one part of an object or model is identical or similar to another . However, in mathematics, symmetry has a more precise definition:
In mathematics, a body is said to be symmetrical if it remains unchanged under certain geometric or algebraic transformations.
These translations may include the following:
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reflection
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rotation
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transfer
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linear transformation
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Special tasks and functions
In other words, symmetry is a form of stability . A body is described as symmetrical if it remains constant under transformations.
2. Symmetry in geometry
When most people hear the word “symmetry,” they immediately think of geometric shapes. Geometry was one of the first disciplines to systematically study symmetry. The symmetry of geometric shapes can be divided into several important categories.
2.1 Axial symmetry (mirror symmetry)
If it is possible to fold a figure around a line so that its two halves fit together perfectly, the figure possesses axial symmetry.
For example:
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butterfly
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Human face (partially)
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A square with four axes of symmetry.
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A circle whose axis of symmetry is infinity.
2.2 Central symmetry
If a figure retains its shape after being rotated 180 degrees around a point, then it possesses central symmetry.
For example:
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diamond
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rectangle
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Circle
2.3 Rotational symmetry
If a shape does not change when it is rotated less than 360 degrees around a point, then the shape has rotational symmetry.
For example:
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An equilateral triangle is approximately symmetrical about an angle of 120 degrees (it rotates 3 times per revolution).
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Square → 90-degree symmetry
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Circle → Infinite symmetry
2.4 Translation analogy
In translational symmetry, a shape remains unchanged when we shift it a certain distance in one direction.
Examples from nature:
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Repetition of the material structure
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Mosaic
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Repeating patterns on the walls
3. Symmetry in Algebra and Functions
Symmetry is not only reflected in the form, but is also important for functions , equations and algebraic relationships.
3.1 Even functions
Partner work:
f(−x)=f(x)f(-x) = f(x)
His image is symmetrical about the Y-axis.
For example:
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f(x)=x² f(x)= x²
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f(x)=cos(x)f(x)= \cos(x)
3.2 Singular Functions
An odd function is a function that satisfies the following conditions:
f(−x)=−f(x)f(−x) = −f(x)
His image is symmetrical to the original .
For example:
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f(x)=x³ f(x)= x³
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f(x)=sin(x)f(x)= \sin(x)
3.3 Symmetrical Equations
Some equations remain identical when you substitute the variables.
Example:
x + y = y + x
The equation is symmetric x and y .
In group algebra, symmetry plays a crucial role in understanding the structure of groups.
4. Symmetry in set theory
In modern mathematics, symmetry is closely linked to the concept of sets . Set theory is the formal language for describing symmetry.
This theory states the following:
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The set of transformations that do not change the shape forms a set of transformations.
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This group is called a “symmetric group” or “autosymmetric group”.
For example:
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The set of symmetries of an equilateral triangle consists of 6 transformations: 3 rotations and 3 reflections.
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The set of quadratic symmetry contains 8 transformations.
This perspective is widely applied in physics, chemistry, cryptography, and number theory.
5. Matrix symmetry
a square matrix AA is said to be symmetric if:
^ T
This means that the rows and columns are arranged symmetrically to the main diameter.
Symmetric matrices are very important because:
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They always possess a strong sense of self-worth;
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They are always portrayed as Qataris;
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They are used in problems in physics, structural mechanics, and machine learning.
A covariance matrix, for example, is a type of covariance matrix used in statistics.
6. Similarities in nature and in various scientific fields.
Why is the concept of symmetry so important in mathematics? Because the universe itself exhibits a remarkable symmetry.
6.1 Physics
The fundamental laws of physics are based on symmetry. Noether’s theorem states the following:
Every physical symmetry corresponds to a conserved quantity.
For example:
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Time symmetry → Law of conservation of energy
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Translational symmetry → Law of conservation of momentum
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Rotational symmetry → Law of conservation of angular momentum
6.2 Chemistry
Molecules such as water, methane, and benzene have similar structures.
6.3 Biology
Most living things exhibit symmetry:
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People: Bisexuals
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Starfish: Pentagram symmetry
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Flower: Radial symmetry
7. Application of symmetry in engineering and computer science
7.1 Computer Graphics
In 3D design, textures, face modeling , and symmetry facilitate calculations.
7.2 Artificial Intelligence and Machine Learning
The structure of neural networks is sometimes symmetrical, and symmetry helps to reduce the number of parameters.
7.3 Encryption
Symmetric groups play a crucial role in modern encryption algorithms .
7.4 Civil Engineering
Symmetry in architectural design can provide greater stability and make the distribution of stress more even.
8. Why is symmetry important in mathematics?
Symmetry is extremely important for the following reasons:
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Simplifying complex problems
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Reducing the number of calculations
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Understanding the hidden structure in the equation
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Support in building models of natural phenomena.
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Creating beauty and order in mathematics
Analogies are a universal language that helps us to better understand the world.
9. The relationship between symmetry and aesthetics
It is interesting that even discussions about the beauty of art and architecture are based on symmetry.
For example:
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Islamic architecture and tiles
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Dome of History
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Persian carpets
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Works of art from ancient Greece
Symmetry is visually appealing to humans because the human brain naturally tends towards regular and predictable patterns.
10. Concrete examples of symmetry in mathematics
1. Circle → infinitely symmetric
2. Isosceles triangle → one axis of symmetry
3. Equilateral triangle → six axes of symmetry
4. Square → eight axes of symmetry
x² → y- 6. Function x³ → symmetric 7. → symmetric x² + y² =
11. Conclusion
Symmetry is one of the fundamental concepts of mathematics and one of the most beautiful ideas in geometry, algebra, physics, art, and nature. When we say that something or a relationship is “symmetrical,” we mean that it remains unchanged under certain transformations. This concept not only facilitates the understanding of structures but also simplifies calculations, increases the accuracy of models, and deepens our understanding of the world.
From simple geometric shapes to complex equations and physical laws – everything is closely linked to the concept of symmetry. Therefore, in mathematics, symmetry is not just a simple geometric property, but a powerful tool for understanding the hidden order of the universe.