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Rotational symmetry

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brianvds
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« on: February 19, 2015, 04:52:47 AM »

Short and to the point: can a 2D shape have an order of rational symmetry of 1? What kind of shape would that be?
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Rigil Kent
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« Reply #1 on: February 19, 2015, 06:35:42 AM »

Apparently not, as explained here.
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brianvds
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« Reply #2 on: February 19, 2015, 15:06:40 PM »

And I assume then that an asymmetrical shape would have a symmetry order of zero?

I just wanted to make sure the stuff in the grade 6 textbook is correct. :-)
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Mefiante
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« Reply #3 on: February 19, 2015, 16:01:13 PM »

Actually, it would have a rotational symmetry of order 1 because there is only one orientation in which its outline matches… Evil

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brianvds
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« Reply #4 on: February 20, 2015, 04:14:03 AM »

Actually, it would have a rotational symmetry of order 1 because there is only one orientation in which its outline matches… Evil

'Luthon64

Well, that is my whole problem. In the textbook's "teacher's guide," which contains answers to the exercises, such a shape is claimed to have a symmetry order of zero. Now I still can't work out whether that is correct or not, because your smiley face indicates that your reply above may or may not be entirely serious.

If something like a propeller has an order of 2, i.e. we count the original position (or the 360 degree rotation, which comes to the same thing), then logic would suggest that in a shape lacking rotational symmetry, we should still count the original position as 1. I can't work out whether this is what is in fact done because of conflicting reports from various authorities. :-)
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Mefiante
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« Reply #5 on: February 20, 2015, 07:34:27 AM »

By convention, it would have a rotational symmetry order of 0, i.e. no rotational symmetry.  This is a holdover from reflectional symmetry where the number of reflection axes/planes are counted (0 would be none, i.e. no reflectional symmetry).  Symmetry gets rapidly trickier as the dimensionality of the mathematical space grows.

It’s a bit like the question whether 1 is a prime number or not.  While convention dictates that it is not, some authors on occasion validly consider 1 to be prime.

'Luthon64
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BoogieMonster
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« Reply #6 on: February 20, 2015, 09:01:06 AM »

Some people say atheism is a belief, and some say it's a lack of belief.

So some may say the object is has a rotational symmetry of 1, some may say it's asymmetrical: lacks symmetry.
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brianvds
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« Reply #7 on: February 20, 2015, 14:34:26 PM »

By convention, it would have a rotational symmetry order of 0, i.e. no rotational symmetry.  This is a holdover from reflectional symmetry where the number of reflection axes/planes are counted (0 would be none, i.e. no reflectional symmetry).  Symmetry gets rapidly trickier as the dimensionality of the mathematical space grows.

I can just imagine. :-)

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It’s a bit like the question whether 1 is a prime number or not.  While convention dictates that it is not, some authors on occasion validly consider 1 to be prime.

I.e. the orders of rotational symmetry are 0, 2, 3 etc. Weird, but I can live with it.
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BoogieMonster
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« Reply #8 on: February 20, 2015, 15:11:16 PM »

You could remove this weirdness by defining it to be the number of additional times it self-resembles the original position. (iow, don't count the original position)

Then you'd have a nice 0, 1, 2... sequence.
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