Find order of rotational symmetry for the given shape

[chwala]

Homework Statement

Write down;
1. The order of rotational symmetry.
2. The number of lines of symmetry.

Relevant Equations

symmetry and shapes

Ok for (1) I would say that the order of rotational symmetry is $8$. Would that be correct? What about $4$?

For (2) The number of lines of symmetry is $4$.

And if one would say infinity for both (1) and (2) would that be correct?And if you consider a kite. Would the order of rotational symmetry be $1$ or $0$? Cheers guys!

Note;
I do not have the solutions for this problem.

 

[andrewkirk]

your answer for (2) is correct.
Infinity would be wrong for both.

A line of symmetry is a line such that the reflection of the image in that line is identical to the original image.

The order of rotational symmetry is the number of different angles in $[0,2\pi)$ such that rotation through that angle around any point leaves the image identical, disregarding any necessary translation. In this case there's an obvious point to choose, being the intersection of the vertical and horizontal lines. How many different angles can you rotate the shape by around that point, and get an unchanged result? List all the angles. Remember to include zero (no rotation) in your count.

 

[chwala]

your answer for (2) is correct.
Infinity would be wrong for both.

A line of symmetry is a line such that the reflection of the image in that line is identical to the original image.

The order of rotational symmetry is the number of different angles in $[0,2\pi)$ such that rotation through that angle around any point leaves the image identical, disregarding any necessary translation. In this case there's an obvious point to choose, being the intersection of the vertical and horizontal lines. How many different angles can you rotate the shape by around that point, and get an unchanged result? List all the angles. Remember to include zero (no rotation) in your count.

Then that's $8$ thanks mate...but do note that the dotted lines that are meeting at the centre of circle weren't in the original shape...hope it doesn't make a difference in the outcome.

 

[andrewkirk]

@chwala No 8 is not correct. I presume your angles are multiples of 45 degrees. But look carefully! - a 45 degree rotation does not leave the figure unchanged, whereas a 90 degree rotation does. That is true whether we include the dotted lines or not - as long as we recognise that a dotted line is not the same as a solid line.

 

[chwala]

I think now you may be confusing me a little bit...yes, it remains unchanged at multiples of $45$ degrees...ie the 1st order of rotation in my understanding is equivalent to a 45 degree turn...a complete turn would realize $360$ degrees divide $45$ degrees = $8$

Arrrrgh I see your point...the answer is $4$

 

[andrewkirk]

No, you are not looking closely enough. Rotate the figure 45 degrees and look at the lines inside the inner circle. Are they in the same places? No!

 

[chwala]

And for the kite? i realize different literature may have their view...

 

[Steve4Physics]

And for the kite? i realize different literature may have their view...

A rotation by 360º (or if your prefer, by 0º) always counts when assessing the order of rotational symmetry.

The question you must ask yourself is: can the order of rotational symmetry ever be zero?

Also, you have to be careful about how you are defining 'kite'. You can consider a rhombus or a square to be special cases of a kite - which affects the order of symmetry.

 

[chwala]

A rotation by 360º (or if your prefer, by 0º) always counts when assessing the order of rotational symmetry.

The question you must ask yourself is: can the order of rotational symmetry ever be zero?

Also, you have to be careful about how you are defining 'kite'. You can consider a rhombus or a square to be special cases of a kite - which affects the order of symmetry.

I meant 'kite' in it's literal sense as we know it...I understand that there are different kinds of kite. Having said that,... I am also conversant with your explanation...my only concern is that some literature indicate; no order of rotational symmetry and other literature indicate $1$ as the order of rotational symmetry of a kite...
...no order means what? $1$?

 

[Steve4Physics]

...no order means what? $1$?

If a shape has no rotational symmetry (e.g. a scalene triangle) I would say it's order of symmetry (n) is 1 (in agreement with @andrewkirk's definition in Post #2).

I presume that is the most common convention. (I've not seen others but they may exist - I'm not a mathematician!)

If you are studying for an examination, simply make sure you are using the convention required by the examination board.

If you are reading generally, simply make sure you are aware of which convention the author is using.