Create Plane Figures w/ Non-Isomorphic 12 Element Symmetry Groups

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In summary, the conversation discussed drawing two plane figures with 12-element groups of symmetries that are not isomorphic. The suggested solution was a regular hexagon and a 12-bladed windmill shape with pronged ends, since the latter is not preserved by a reflection.
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Homework Statement


Draw two plane figures, each having a 12 element group of symmetries, such that the two groups are NOT isomorphic. Demonstrate that they are not isomorphic.

Homework Equations


I know that every finite group of isometries of the plane is isomorphic to either Z_n or to the dihedral group D_n.

The Attempt at a Solution


I drew a regular hexagon (D_6) but now I am stuck as to what to draw for a figure to represent Z_12. Would a 12 bladed windmill (pinwheel) type shape with pronged ends work?
 
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Yes, this will work, since it is not preserved by a reflection.
 

1. What are non-isomorphic 12 element symmetry groups?

Non-isomorphic 12 element symmetry groups are groups of symmetries that cannot be mapped onto each other by a symmetry-preserving transformation. In other words, they are distinct groups with different structures, even though they may have the same number of elements.

2. How do you create plane figures with non-isomorphic 12 element symmetry groups?

To create plane figures with non-isomorphic 12 element symmetry groups, you can start by identifying the different possible symmetry groups and their corresponding symmetries. Then, you can use these symmetries to construct the figures by repeating them in a pattern. It is important to note that not all non-isomorphic 12 element symmetry groups can be realized as plane figures.

3. Can you give an example of a non-isomorphic 12 element symmetry group?

One example of a non-isomorphic 12 element symmetry group is the dihedral group of order 12, also known as D12. This group has 12 elements and is commonly represented as a regular hexagon with rotations and reflections as its symmetries.

4. What is the significance of studying non-isomorphic 12 element symmetry groups?

Studying non-isomorphic 12 element symmetry groups is important in the field of mathematics as it helps us understand the different ways in which symmetries can be organized. It also has applications in other areas such as crystallography, chemistry, and physics, where symmetry plays a crucial role.

5. Are there any real-life examples of non-isomorphic 12 element symmetry groups?

Yes, there are many real-life examples of non-isomorphic 12 element symmetry groups. Some examples include the faces of a soccer ball, the shape of a snowflake, and the design of a kaleidoscope. These objects exhibit symmetries that can be represented by non-isomorphic 12 element symmetry groups.

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