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 Post subject: 6 different symmetries
PostPosted: Sat Dec 15, 2012 6:25 am 
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Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Hi,

this is another batch of 6 slot-fillers.
After finishing these there are currently 4 items missing for the completed series.

What do we have here? Six simple truncations of a 3x3x3. In all six cases I used the same truncation scheme, a completely asymmetrical one: The cutting depth of the corners are 345, compared with 444 for the HalfTruncatedCube.
All six variants represent a subgroup of Oh, which is the group of octahedral point symmetries.
The variants from left to right are:
- The trivial subgroup. No symmetry at all.
- Point symmetry at the cubes central.
- Mirroring at a plane parallel to opposing faces.
- Mirroring at a plane going through opposing edges.
- Rotational symmetry (of 180°) along an axis going through two opposing face pieces.
- Rotational symmetry (of 180°) along an axis going through two opposing edges.

Does anybody understand what I want to describe here? :?

Andreas

EDIT: Corrected some minor mistakes


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Last edited by Andreas Nortmann on Sun Dec 16, 2012 5:02 am, edited 1 time in total.
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 Post subject: Re: 6 different symmetries
PostPosted: Sat Dec 15, 2012 12:18 pm 
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We are missing cubes with the same truncations on corners with connected by 1 edge.

In this case you have a subgroup with 90 degrees rotational symmetry along an axis going through two opposing edges also opp. face pieces.


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 Post subject: Re: 6 different symmetries
PostPosted: Sat Dec 15, 2012 7:42 pm 
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Joined: Sat Apr 21, 2007 11:21 pm
Location: Marin, CA
Andreas- Thanks for these slot fillers! I love seeing the series you create. Do you mean there are 4 symmetries needed to complete the series? Symmetries around vertices?

Nice work!

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 Post subject: Re: 6 different symmetries
PostPosted: Sun Dec 16, 2012 5:18 am 
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Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Hi Jason,

I mean this:
The group of octahedral symmetry has 48 transformations. These are enumerated in the conjugacy classes in that article, although spread along two sections.

The whole series will contain all 33 subgroups known to the octahedral symmetry.
Please note that the wikipedia-article mentions less than 33 subgroups because some are left out because of isomorphism. The to mirrorings I mentioned and presented are treated as one subgroup.

@Claus:
The variant you describe here has a higher degree of symmetry than the six I implemented here.
But coincidentially you are right. That variant is missing but I am already working on it.


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