Re: What type of object is this?
From: richard miller (richard_at_microscitech.freeserve.co.uk)
Date: 08/19/04
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Date: Thu, 19 Aug 2004 10:38:24 +0100
Having read your question, I admit to not really knowing what sort of answer
you want. Group tables?
However, here is one way to make some geometrical sense...
Take a cube
Label any one vertex O, label the vertex furthest from it O
Sit the cube so that vertex O facing you bottom left, i.e. you are looking
at the face of a cube
+ --- +
| |
| |
O ---- +
Label the vertex on its right S, label the vertex furthest from it S
+ --- +
| |
| |
O ---- S
Keeping the cube as is, you should have O bottom left, S bottom right.
Now label the vertex above S, i.e. top right, M., label the vertex furthest
from it M
+ --- M
| |
| |
O ---- S
Finally, label the vertex above O, i.e. top left, D., label the vertex
furthest from it D.
The cube side facing you should look like
D --- M
| |
| |
O ---- S
The side on the right (if you looked at it face on)
M --- O
| |
| |
S ---- D
The side on the back (if you looked at it face on)
O --- S
| |
| |
D ---- M
The side on the left (if you looked at it face on)
S --- D
| |
| |
M ---- O
You will now see that each of the four faces is, when read anticlockwise
starting from bottom left is
(OSMD)
(SDOM)
(DMSO)
(MODS)
which are the four letters staring each of your rows below.
If you take the side originally facing you, (OSMD), rotate 90 degs about a
horizontal axis through the centre of the face, you will get (DOSM), another
90 deg (MDOS), another 90 deg (SMDO) and then, of course, you are back where
you started. This gives you the four combinations in your first row.
(OSMD)(DOSM)(MDOS)(SMDO)
Take the face on the right, do the same, you get 2nd row.
Etc for rear and left face.
So, you have a small group of rotations for the cube. Of course, it is
merely a sub-group of all the possible rotation.
We have missed the top face. Looking down, this would be (DMOS) and rotating
this would give (SDMO), (OSDM), (SDMO). The bottom face, looking from below,
Note, I tried just a plane square before going to a cube, but couldn't get
the combinations right using rotations and reflections.
The points are labelled such that they are essentially reflections about the
centre point of the cube. It is reflection about a point though, not a plane
mirror.
Reading your question again, I'm flumoxed as to whether this is of any help
to you whatsoever? Gave my brain a morning wake-up call.
Hope it is of some help
Richard Miller
"sisial" <sisial@yahoo.com> wrote in message
news:f8daea2a.0408182159.10bcf465@posting.google.com...
> What type of object is this?
>
> (OSMD)(DOSM)(MDOS)(SMDO)
> (SDOM)(MSDO)(OMSD)(DOMS)
> (DMSO)(ODMS)(SODM)(MSOD)
> (MODS)(SMOD)(DSMO)(ODSM)
>
> The first and third columns use complementary generators (2413) and
> (3142). The second and fourth columns use complementary generators
> (4312) and (3421). The four rows are generated by (4123).
>
> Each of the elements can serve as an identity in defining some
> operation on the set of elements. I need tables applying each of these
> 16 operations and information on the algebraic properties of those
> operations. However, I am no mathematician and am having trouble
> finding the information and tools necessary to do this. Any help would
> be greatly appreciated.
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