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Schoenflies Notation

 
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polymorphic



Joined: 17 Feb 2008
Posts: 26

PostPosted: Thu Apr 03, 2008 11:59 am    Post subject: Schoenflies Notation Reply with quote

Hi Rob,

Since this came up over at the Antiprism forum I thought I'd put it here. Other Stella users might want to comment on showing Schoenflies notation symbols for symmetry.

For others reading this, the reason this came up is that I was confuse because of the use of term "Pyramidal Symmetry" when a horizontal reflection was happening. For those interested there is model in this zip file (C2_symmetry.off which are likely mis-named) that exhibit this. The model is the same when flipped top to bottom but is still pyramidal and I think of pyramidal having a base and a top.

http://www.interocitors.com/tmp/sym.zip

I found it curious that when searching on Pyramidal Symmetry in Google that it isn't widely used in terms of geometry.

Roger
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robertw
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PostPosted: Thu Apr 03, 2008 1:43 pm    Post subject: Re: Schoenflies Notation Reply with quote

polymorphic wrote:
Other Stella users might want to comment on showing Schoenflies notation symbols for symmetry.


It's a good idea, after our other discussion I was already thinking I should add them, although the labels are not terribly user-friendly and a drop-down list showing all the subsymmetries would be pretty long!

It actually makes a lot of sense to split the symmetry group into a rotational part and a reflective part, as I do in Stella, but it doesn't seem to be widely done this way, which is why there's some confusion with established terms.

Quote:
the reason this came up is that I was confuse because of the use of term "Pyramidal Symmetry" when a horizontal reflection was happening... The model is the same when flipped top to bottom but is still pyramidal and I think of pyramidal having a base and a top.


Should I rename it from Pyramidal to Cyclic perhaps? I always thought cyclic symmetry was a silly phrase since all symmetry groups contain cycles.

Hmm, I see wikipedia refers to pyramidal symmetry:
http://en.wikipedia.org/wiki/Point_groups_in_three_dimensions#The_seven_infinite_series

As used in Stella, it means having the same rotational symmetry as a pyrmid, which is true of your model. Your model also has an additional reflection symmetry, which is represented separately with the reflection symmetry drop-down list in Stella.

Regarding rotation-reflection symmetry, I've been experimenting with a way to represent this visually. Currently, when you display reflection planes in Stella, rotation-reflection is not displayed (for those wondering what I'm talking about, this is a symmetry where both a rotation AND a reflection must be applied, even though neither is a symmetry of the model on its own). What do you think of these?


The first is your model, which truly has rotation-reflection symmetry. The second is a subsymmetry of the 24-gonal prism (its full symmetry group has more symmetries than this).

For simple reflections I display a circle in the reflection plane. Here I use a wavy circle to indicate that the reflection only applies in conjunction with a rotation. What do you think?

Rob.
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polymorphic



Joined: 17 Feb 2008
Posts: 26

PostPosted: Thu Apr 03, 2008 2:56 pm    Post subject: Re: Schoenflies Notation Reply with quote

robertw wrote:


It's a good idea, after our other discussion I was already thinking I should add them, although the labels are not terribly user-friendly and a drop-down list showing all the subsymmetries would be pretty long!

It actually makes a lot of sense to split the symmetry group into a rotational part and a reflective part, as I do in Stella, but it doesn't seem to be widely done this way, which is why there's some confusion with established terms.


Actually I like the way you split the two. It seems like a logical way to group the sub-symmetries.

Stella is such an all encompassing application that there aren't many people (IMHO) that use all parts. I'm not a model maker nor am I adept at 4D (yet), but I am interested in geometric qualities and such. To me, an additional drop down that would contain the Schoenflies notation would be very helpful, even if it contained a lot of entries (that would make it even more interesting to me!).

Then we could see how your break down of symmetries corresponds to the widely used symbols.

robertw wrote:
Should I rename it from Pyramidal to Cyclic perhaps? I always thought cyclic symmetry was a silly phrase since all symmetry groups contain cycles.


Well I was actually thinking this entire thought myself! When the system was devised they just used C for the generic Cyclic and then go about showing there are two different ones, Cnh and Cnv. (In a way they are doing what you do is to break down the symmetry and mirror plane direction).

They further describe Cnh as Bilateral symmetry and Cnv as Biradial symmetry noting that these are "biological terms". I don't know why bilogical is and dis-merit, simply because they are not geologist terms! Why should crystalography have a monopoly on geometry?!

EDIT: These terms were mention specifically for 2-fold cases (i.e. Bi-). In general use they'd have to be lateral and radial I suppose. END EDIT.

I once used the term bilateral in a forum and was not chided for it and in fact was told that was the only way to describe what I was talking about.

As for Sn symmetry it gets interesting since the word "Spiegel", German for mirror, doesn't have a common word in English other than I suppose Mirror itself. "Rotoreflection" is an interesting word to have come up with!

I like your graphic depiction of rotation reflection. I'd have to see it in a number of cases (i.e. put it in and see what the response is). I am wondering if it is proper to show it as a continuous line but wouldn't argue for anything else.

Hoping to hear other responses to this thread.

Roger
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robertw
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PostPosted: Fri Apr 04, 2008 6:41 am    Post subject: Re: Schoenflies Notation Reply with quote

polymorphic wrote:
To me, an additional drop down that would contain the Schoenflies notation would be very helpful, even if it contained a lot of entries (that would make it even more interesting to me!).


Well, believe it or not, I just finished adding this! I now have a drop-down showing the standard symmetry symbol, with all sub-symmetries listed when you open the list. The old drop-down lists are still there too. They stay in sync with each other and you may select a sub-symmetry from any of the lists.

I'm currently working towards the next major release though, so I'm afraid you may not see this new feature for a little while Crying or Very sad

Quote:
I like your graphic depiction of rotation reflection. I'd have to see it in a number of cases (i.e. put it in and see what the response is). I am wondering if it is proper to show it as a continuous line but wouldn't argue for anything else.


I don't think there's a "proper" way. A broken or unbroken line can both represent this symmetry by having that symmetry themselves. I like the curvy line though Smile One problem though is that the line itself actually has dihedral symmetry! Maybe I should use a more sawtooth shaped wave?

Meanwhile, this leaves just one symmetry with no visual representation, central inversion, so I've come up with something for that too:

It now shows a 3D icon at the centre of the model, with several double-headed arrows through it. That was the best representation I could come up with, indicating that elements from one side are reflected through the centre to the other side.

Rob.
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Anglica



Joined: 15 Dec 2012
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PostPosted: Sat Dec 15, 2012 5:34 am    Post subject: Reply with quote

Excellent post. I want to thank you for this informative read, I really appreciate sharing this great post. Keep up your work. Thanks for this very useful
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robertw
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PostPosted: Sat Dec 15, 2012 5:50 am    Post subject: Reply with quote

Thanks. Been a while since I wrote that. I wrote that I was working towards the next major release back in 2008 and didn't finally get around to releasing it till this year 2012! So now finally those new representations of unusual symmetry groups are available (though I wonder how many people will ever come across them!).

Rob.
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