Is Symmetry Limited in Geometry?

[Islam Hassan]

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

 

[chiro]

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.

 

[Islam Hassan]

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

 

[M Quack]

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.

 

[blue_raver22]

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.