Pinwheel Symmetry Formulas Exposed

[mathjam0990]

Question:

How many different pinwheels with 8 identically shaped pins are there if each pin can be colored one of 4 different colors?

My answer:

4^8 because each pin has 4 choices Is this correct? I think this only includes rotational symmetries. Do I need to consider reflectional symmetries? If so, would reflectional symmetries = 4^5?

Thank you for your help!

 

[Evgeny.Makarov]

I'm not a specialist in combinatorics, but Wikipedia says the number is ${1\over 8}\sum_{d\mid 8}\varphi(d)4^{8/d}$.

By the way, how can a pinwheels spin if it has several pins? Do you mean 8 curls?

 

[mathjam0990]

This is actually for an abstract algebra class. Not sure why they are teaching us combinatorics :( Also, I stated the question quote from our textbook so I am not sure. Thanks for the info though, I appreciate it.

 

[Deveno]

This is actually for an abstract algebra class. Not sure why they are teaching us combinatorics :( Also, I stated the question quote from our textbook so I am not sure. Thanks for the info though, I appreciate it.

Because Burnside's lemma involves the use of group (action) theory.

Usually, we talk about "distinct colorings"; for example, if we have for the "pins":

1 = red
2 = yellow
3 = green
4 = blue
5 = red
6 = yellow
7 = green
8 = blue

this is regarded as the same coloring as:

1 = yellow
2 = green
3 = blue
4 = red
5 = yellow
6 = green
7 = blue
8 = red

since the $8$-cycle $(1\ 2\ 3\ 4\ 5\ 6\ 7\ 8)$ takes the first to the second (that particular $8$-cycle generates the rotation group).

$4^8$ is the size of the total "configuration space" $X$ of all possible colorings where we keep track of which "pin" got painted what color. The number of "distinct colorings" is much smaller, since we might rotate one configuration into another, with no way to tell them apart (this assumes all eight "pins" are identical).

I daresay a pinwheel is being used here to avoid the harder question of including reflection symmetries (like you would have using a necklace, with beads).

 

[mathjam0990]

Thank you Deveno for your help. The question in the book definitely stated quote pinwheel. I think it's a poorly written book and a bit ambiguous at times which doesn't help someone like me who is already not too great at this stuff. Anyway, thanks for the info!

 

[johng1]

Evgeny Markov gave you a link to the correct formulas. I searched the web for a proof of these formulas, but couldn't find such. Deveno gave you a link to Burnside's lemma which is instrumental to a proof. Mainly, I wrote the following for myself, but thought you or other readers might be interested.

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