Is Physical Symmetry Limited Due To The Classification Theorem?

[Islam Hassan]

Given that the Classification Theorem says that every finite simple group is isomorphic to one of 4 broad categories of (specific) finite simple groups, does this mean that any conceivable symmetric:

i) 2D form; or
ii) 3D object

is also isomorphic to one of these 4 categories. Otherwise said, is physical symmetry limited in the universe (not to mention n-dimensional spaces)?

If so, i find this extremely surprising given that:

i) Numbering is an infinite proposition; and
ii) The fact that one can conceive of a infinity of forms and objects in the physical world.

Such limited nature of symmetry is to my mind the single most counter-intuitive and amazing result in all of mathematics. I truly find it incredible!IH

 

[chiro]

Given that the Classification Theorem says that every finite simple group is isomorphic to one of 4 broad categories of (specific) finite simple groups, does this mean that any conceivable symmetric:

i) 2D form; or
ii) 3D object

is also isomorphic to one of these 4 categories. Otherwise said, is physical symmetry limited in the universe (not to mention n-dimensional spaces)?

If so, i find this extremely surprising given that:

i) Numbering is an infinite proposition; and
ii) The fact that one can conceive of a infinity of forms and objects in the physical world.

Such limited nature of symmetry is to my mind the single most counter-intuitive and amazing result in all of mathematics. I truly find it incredible!

IH

Hey Islam Hassan.

When you talk about finite groups does that mean you do not include Lie Groups?

In terms of the actual group operator (I'm assuming the object is just a 2D or 3D shape), is it just some finite cyclic group? The thing is if its not like this then assuming our operator just does something like 'rotate' an object about its centre in a way that preserves geometric symmetry of some sort, then if you have a pure circle then there is infinite symmetry if you have something like a Lie group.

It would help if you described the actual symmetry that you are talking about because I feel I am getting the wrong idea of what you call symmetry and what I call symmetry.

 

[Islam Hassan]

Hey Islam Hassan.

When you talk about finite groups does that mean you do not include Lie Groups?

In terms of the actual group operator (I'm assuming the object is just a 2D or 3D shape), is it just some finite cyclic group? The thing is if its not like this then assuming our operator just does something like 'rotate' an object about its centre in a way that preserves geometric symmetry of some sort, then if you have a pure circle then there is infinite symmetry if you have something like a Lie group.

It would help if you described the actual symmetry that you are talking about because I feel I am getting the wrong idea of what you call symmetry and what I call symmetry.

I do include Lie Groups as identified in the Classification Theorem (quote from Wikipedia):

i) the classical Lie groups, namely the groups of projective special linear, unitary, symplectic, or orthogonal transformations over a finite field; and

ii) the exceptional and twisted groups of Lie type (including the Tits group which is not strictly a group of Lie type).

In terms of the type of symmetry, it would be those symmetries applicable in our 3D world, namely reflection, rotation and translation. Perhaps to be more clear, when I talk of limited symmetries, I mean types of symmetries and not individual symmetries of an specified form/object. Regarding the group represented by the points on a circle under rotation/reflection about its center for example, it would count as one type of physical symmetry despite the fact that it has an infinite number of individual, form-specific symmetries.


IH