Determining isomorphic summetry groups
I have three shapes, the group for each shape is a symmetry group. I need to determine which two groups are isomorphic.
The shapes are a triangle, a square, and a four pointed star (which is essentially a square with a triangle stuck on each side).
Visually the square and the four pointed star appear to be isomorphic, but how to I show this?
The shapes are a triangle, a square, and a four pointed star (which is essentially a square with a triangle stuck on each side).
Visually the square and the four pointed star appear to be isomorphic, but how to I show this?
ASKER
I mean symmetrical operations performed on the shape - so the points of the shape are numbered, and then transformed through various axis of symmetry.
The star and the square have the same axis of symmetry so I think they are isomorphic, but I don't know what would constitute an acceptable proof for this.
The star and the square have the same axis of symmetry so I think they are isomorphic, but I don't know what would constitute an acceptable proof for this.
Just list the symmetrical operations required.
The square and the star are both D4
ASKER
I don't know what D4 is.
Is listing symmetrical operations a proof of isomorphism? The examples I have seen of showing two groups are isomorphic is to map each element from one group to the other group.
If listing symmetrical operations (e.g. reflections) is a good proof, the way I have seen symmetrical operations denoted is to mark them (e.g. reflections) on a diagram and label them with letters.
If this was done on both shapes, would mapping the letters of different operations be an acceptable way of showing isomorphism?
Is listing symmetrical operations a proof of isomorphism? The examples I have seen of showing two groups are isomorphic is to map each element from one group to the other group.
If listing symmetrical operations (e.g. reflections) is a good proof, the way I have seen symmetrical operations denoted is to mark them (e.g. reflections) on a diagram and label them with letters.
If this was done on both shapes, would mapping the letters of different operations be an acceptable way of showing isomorphism?
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From http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Nazarewicz/7210_final_2/7210_Project/index.html
So since both objects have the same symmetry, they should be isomorphic. But this is an observation, not a proof, and if you haven't been given that as a tool, then you probably shouldn't use it (assuming this is an academic excercise).
An object with D4 symmetry would have four rotations, each of 90 degrees, and four reflection mirrors, with each angle between them being 45 degrees.
So since both objects have the same symmetry, they should be isomorphic. But this is an observation, not a proof, and if you haven't been given that as a tool, then you probably shouldn't use it (assuming this is an academic excercise).
If the proportions or angles are different they are not.