Is Symmetry Limited in Geometry?

In summary, the complete classification of finite simple groups allows us to determine the number of conceivable 2D/3D symmetrical geometric objects/arrangements. However, this may vary depending on the cardinality of the sets involved in the groups. Additionally, the answer is known for periodic structures, such as in crystallography, where a "unit cell" is repeated to form an infinite 2D or 3D lattice. The sets involved in this context may be both finite and infinite, as one category of finite simple groups can have an infinite number of elements.
  • #1
Islam Hassan
237
5
Given the complete classification of finite simple groups, can one say that the number of all conceivable 2D/3D symmetrical geometric objects/arrangement is limited?

Is spatial symmetry limited in our 3D world?


IH
 
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  • #2
Islam Hassan said:
Given the complete classification of finite simple groups, can one say that the number of all conceivable 2D/3D symmetrical geometric objects/arrangement is limited?

Is spatial symmetry limited in our 3D world?

IH

Hello Islam Hassan and welcome to the forums.

The question that I have to ask that comes to mind is what the cardinality of the sets themselves involved in the groups you are talking about?

I don't know that much about group theory (finite or otherwise) but I do know that the sets relating to the associated groups you are referring to are going to have some kind of relation to your answer.
 
  • #3
chiro said:
Hello Islam Hassan and welcome to the forums.

The question that I have to ask that comes to mind is what the cardinality of the sets themselves involved in the groups you are talking about?

I don't know that much about group theory (finite or otherwise) but I do know that the sets relating to the associated groups you are referring to are going to have some kind of relation to your answer.


I believe that the sets involved may be both finite and infinite; one category of finite simple group in the classification theorem are simple groups of Lie type which, if I recall correctly, can have an infinite number of elements.


IH
 
  • #4
The answer is known for periodic structures where a "unit cell" is repeated to form an infinite 2D or 3D lattice.

The application in physics is crystallography.
 
  • #5
onestly, it is difficult to definitively answer whether symmetry is limited in geometry or in our 3D world. On one hand, the classification of finite simple groups does provide a comprehensive list of all possible symmetries in finite geometric structures. This suggests that there may be a limit to the number of conceivable 2D/3D symmetrical geometric objects or arrangements.

However, it is important to note that this classification only applies to finite structures and does not take into account infinite or continuous symmetries. Additionally, there may be symmetries that have not yet been discovered or fully understood, further complicating the idea of a limit to symmetry in geometry.

Furthermore, in our 3D world, symmetry can be found in a wide range of scales and contexts, from microscopic atomic structures to large-scale celestial bodies. It is difficult to say whether there is a finite limit to the number of symmetrical arrangements in our 3D world, as our understanding of the universe is constantly evolving and expanding.

In conclusion, while the classification of finite simple groups may suggest a limit to symmetry in geometry, the concept of symmetry is complex and multifaceted, making it difficult to definitively say whether it is limited in our 3D world. It is an ongoing area of research and discovery in the scientific community.
 

FAQ: Is Symmetry Limited in Geometry?

What is symmetry in geometry?

Symmetry in geometry refers to the balanced distribution of shapes, angles, lines, and/or points within a figure. It is a visual property that occurs when one part of a figure is a reflection or rotation of another part.

How is symmetry limited in geometry?

Symmetry is limited in geometry because not all figures or shapes have symmetry. For example, a circle has infinite lines of symmetry, while a triangle only has three. Additionally, some figures may not have any lines of symmetry at all. Therefore, symmetry is limited to certain types of figures and cannot be applied to all geometric shapes.

What are the different types of symmetry in geometry?

There are three main types of symmetry in geometry: reflection, rotation, and translation. Reflection symmetry, also known as mirror symmetry, occurs when a figure can be divided into two equal parts that are mirror images of each other. Rotation symmetry occurs when a figure can be rotated around a fixed point and still look the same. Translation symmetry, also known as slide symmetry, occurs when a figure can be moved or translated and still look the same.

Is symmetry important in geometry?

Symmetry is important in geometry because it helps us understand and analyze different shapes and figures. It also allows us to identify patterns and relationships between different parts of a figure. Additionally, symmetry is often used in art and design to create aesthetically pleasing and balanced compositions.

How is symmetry used in real-life applications?

Symmetry is used in various real-life applications, including architecture, engineering, and design. For example, symmetry is often used in the design of buildings and bridges to ensure structural stability and balance. In manufacturing, symmetry is important for creating precise and symmetrical products. Additionally, symmetry is also used in nature, such as in the symmetry of butterfly wings or the structure of crystals.

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