Symmetry in Objects: Is Every Object Symmetric & How to Prove It?

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Discussion Overview

The discussion revolves around the concept of symmetry in objects, questioning whether every object in space can be considered symmetric about at least one axis. Participants explore definitions, implications, and examples related to symmetry, including mathematical and physical perspectives.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions the formal definition of symmetry and proposes that every object in space might be symmetric about at least one axis, seeking proof for this assertion.
  • Another participant argues against the assertion, suggesting that there are manifolds with trivial isometry groups, indicating that not all objects possess symmetry.
  • A claim is made that certain measurable sets in space can be constructed that have no symmetries under reflection, although the participant expresses uncertainty about this.
  • A participant mentions that a human hand, as a 3D object, lacks symmetries.
  • Another participant draws a parallel between even functions and symmetry, proposing a similar expression for 3D objects.
  • One participant suggests that if a hand were placed infinitely far away, it would converge to a point and become symmetric, raising questions about the nature of symmetry at different scales.
  • A later reply encourages participants to refer to an external article on symmetry to better understand its mathematical treatment.

Areas of Agreement / Disagreement

Participants express differing views on the nature of symmetry, with some asserting that not all objects are symmetric while others propose various definitions and scenarios that challenge or support the initial assertion. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants highlight limitations in definitions and the applicability of symmetry across different contexts, such as mathematical constructs versus physical objects. There is also mention of the need for clarity in the definition of symmetry, particularly regarding isometries.

khotsofalang
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is there a formal definition of symetry? suppose i was to make an assertion that every object in space is symetric at least about one certain axis in space, is this assertion true? why or why not...how would you go about the proof?
 
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khotsofalang said:
is there a formal definition of symetry? suppose i was to make an assertion that every object in space is symetric at least about one certain axis in space, is this assertion true? why or why not...how would you go about the proof?

one definition of symmetry is an isometry. I believe there are manifolds with trivial isometry groups i.e. the only isometry is the identity. Since any manifold can be embedded isometrically in space, you conjecture is false.

In any dimension, it should be easy to construct measurable sets in space that have no symmetries under reflection - but I am not sure
 
Your hand is a 3D object that has no symmetries.
 
Even functions are symmetric with respect to the y-axis. This is expressed in the relation
f(x) = f(-x). For a 3-D object, a similar expression of symmetry might be
f(x,y,z) = f(-x,-y,z)
 
but how do we formally define symetry? is it just in terms of isometry? if you were to place you hand infinite distances away, then your hand gets very tiny and converges to a point in space which is definitely isometric about some point...eventually becoming symetric!
 
Before you get carried away with mindless sophistry, take a little time and peruse the following article:

http://en.wikipedia.org/wiki/Symmetry

I think you can glean from these scribblings how mathematics treats the definition of symmetry.
 

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