Here are three orbits,
The gray orbit has four elements, the green orbit has two elements, and the pink one has a single element. Can you identify what attribute affected the number of elements in each orbit? Since the pink orbit has a single square, we are allowed to say that it has the symmetries of D_2 as well.
Conversely, the pink square suggests the action of D_2. The pink square is a sort of an elementary framework. By coloring it in different ways we may eventually get different orbits. In Chemistry, molecules are different colorings of elementary frameworks.
So, now it may become more apparent why symmetry is what you see at a glance ... the elements in the gray orbit, from afar, seem like the pink square, and elusively borrow its symmetries .... until you look close enough!
all with the reflection symmetries of D_2={1,h,v,o}.
The gray orbit has four elements, the green orbit has two elements, and the pink one has a single element. Can you identify what attribute affected the number of elements in each orbit? Since the pink orbit has a single square, we are allowed to say that it has the symmetries of D_2 as well.
Conversely, the pink square suggests the action of D_2. The pink square is a sort of an elementary framework. By coloring it in different ways we may eventually get different orbits. In Chemistry, molecules are different colorings of elementary frameworks.
So, now it may become more apparent why symmetry is what you see at a glance ... the elements in the gray orbit, from afar, seem like the pink square, and elusively borrow its symmetries .... until you look close enough!
Last revised 06/06/2011
These postings are based on "Symmetry Studies An Introduction to the Analysis of Structured Data in Applications" Cambridge Press (2008)
These postings are based on "Symmetry Studies An Introduction to the Analysis of Structured Data in Applications" Cambridge Press (2008)
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