Symmetry

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Symmetry

Postby moonlord » Tue Mar 28, 2006 4:58 pm

It seems that some n-dimensional bodies have m-dimensional symmetries (m<n). For example, a equal-sided triangle has a line (3 in fact) of symmetry, but not a point of symmetry. A paralelogramme (not sure if spelt correctly) has a point, but not a line of symmetry. A cilinder has a point of symmetry, a line of symmetry but an infinite number or planes of symmetry.

Anyone knows some rules for these properties? Thanks in advance.
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Postby pat » Tue Mar 28, 2006 5:31 pm

As usually defined, a symmetry is a continuous, distance-preserving function that maps a shape onto itself. So, if your shape is S, then for any point s in S, your function f maps s onto a point in S.

By virtue of being distance-preserving, the function must be one-to-one. If a and b are two points in S, then |a-b| = |f(a) - f(b)|. The distance is only zero if a = b.

What you are calling a symmetry are the fixed points of what I'm calling a symmetry. The fixed points of a symmetry f are all points s in S such that f(s) = s.

Your concern is with the dimension of the set of fixed points or with the number of unique sets of fixed points at each dimension.

It is clear from the above that the fixed points are a subset of the shape. So, automatically, we cannot have that m bigger than n.

In the above, the trivial symmetry is also possible where f(s) = s for all S. This is the identity map on the shape. For the identity map m = n.

For some shapes, there are symmetries where m = 0. For example, rotating an annulus in its plane around its center leaves no fixed points.
The annulus has an infinite number of symmetries for m = 1 (where you flip the annulus over a line through its center).

Maybe more later... it's lunchtime now...
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Postby bo198214 » Thu Mar 30, 2006 11:18 am

@moonlord: for me it became not clear, whether you always mean mirroring by symmetry. So would in your sense a pentagram have a point of symmetry in the center?

pat wrote: It is clear from the above that the fixed points are a subset of the shape. So, automatically, we cannot have that m bigger than n.

That is absolutely not clear. The fixed points of a f:R<sup>n</sup> -> R<sup>n</sup> can of course lie outside a selfmapped shape S. And indeed a line of symmetry exceeds every restricted shape. And indeed we can extend the point of symmetry of a 1-d-Interval (i.e. its center) in 3 dim space to a plane of symmetry. So 2=m>n=1.
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Postby moonlord » Thu Mar 30, 2006 6:37 pm

Not quite. A nD shape has a (n-1)D space of symmetry whether by mirroring across that space keeps its original shape, NOT ROTATED. So I see mirroring as a particular symmetry. In my opinion, yes, a pentagram has a point of symmetry in its center.

By the way, why did you choose a pentagram? Are you a satanist? :p nevermind.
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Postby bo198214 » Thu Mar 30, 2006 7:18 pm

By the way, why did you choose a pentagram? Are you a satanist? :p nevermind.
Because triangle was too simple. And now I see that my question was redundant and your answer is contradictive because you already stated:
a equal-sided triangle has a line (3 in fact) of symmetry, but not a point of symmetry
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Postby pat » Thu Mar 30, 2006 7:43 pm

bo198214 wrote:
pat wrote: It is clear from the above that the fixed points are a subset of the shape. So, automatically, we cannot have that m bigger than n.

That is absolutely not clear. The fixed points of a f:R<sup>n</sup> -> R<sup>n</sup> can of course lie outside a selfmapped shape S.


I'm sorry... I wasn't considering f:R<sup>n</sup> -> R<sup>n</sup>. I was considering f:S -> S. If you prefer to consider f:R<sup>n</sup> -> R<sup>n</sup>, then my comments only apply for R<sup>n</sup> intersect S.
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Postby pat » Thu Mar 30, 2006 7:48 pm

Of course, I suppose in so-doing, I cut out one of the symmetries of the annulus that the original post would have probably included.... the point at the center.

But, if one wishes to consider fixed points of maps on the embedding space, then one has to carefully define the embedding space. An equilateral triangle has only three reflection symmetries and two rotation symmetries and the identity symmetry if it's embedded in R<sup>2</sup>. It has a whole slew (a continuum) of other symmetries if you embed it in R<sup>4</sup>.
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Postby moonlord » Fri Mar 31, 2006 1:44 pm

bo198214 wrote:
By the way, why did you choose a pentagram? Are you a satanist? :p nevermind.
Because triangle was too simple. And now I see that my question was redundant and your answer is contradictive because you already stated:
a equal-sided triangle has a line (3 in fact) of symmetry, but not a point of symmetry


Oh, where did my mind go!?

A equal-sided triangle has a point of symmetry. I was reffering to a triangle with two sides equal, and a third with different length (how do you call it in english, anyway?)...
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Postby pat » Fri Mar 31, 2006 2:39 pm

Triangle with all sides equal ... equilateral.
Triangle with only two sides equal ... isosceles.
Triangle with no sides equal ... scalene.
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Postby moonlord » Fri Mar 31, 2006 3:26 pm

Thanks ;)
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