Rotational symmetry

[brianvds]

Short and to the point: can a 2D shape have an order of rational symmetry of 1? What kind of shape would that be?

[Rigil Kent]

Apparently not, as explained here.

[brianvds]

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. :-)

[Mefiante]

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

'Luthon64

[brianvds]

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

'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. :-)

[Mefiante]

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

[BoogieMonster]

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.

[brianvds]

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. :-)

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.

[BoogieMonster]

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.