Exploring Symmetry Group of a Greek Vase

In summary, the symmetry operations for an en greek vase build up a symmetry group. The conditions for this to happen are (a\cdot b)\cdot c=a\cdot (b\cdot c), a\cdot e=a, and a\cdot a^{-1}=e. To verify these conditions, the vase must have a multiplication table and closure.
  • #1
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Homework Statement


Show that symmetry operations for en greek vase build up a symmetry group.





Homework Equations



For en greek vase we have
[tex] \Gamma=[e, C_{2},\sigma, \sigma^{'}][/tex]
And there are 3 conditions which must be fullfilled so that the elements will create a symmetry group [tex]
1) (a\cdot b)\cdot c= a\cdot (b\cdot c)[/tex]
[tex] 2) a\cdot e= a[/tex]
[tex] 3) a\cdot a^{-1}=e[/tex]



The Attempt at a Solution


So we know that the vase is invariant under [tex]180^{0}[/tex] so it is of [tex]C_{2} type[/tex]
do I understand correctly[tex]C_{2}\cdot C^{-1}_{2}=e[/tex] rotation [tex]180^{0}[/tex] and another one [tex]180^{0}[/tex] in the opposite direction
second condition-([tex]a\cdot e=a[/tex])
can we write then [tex]C_{2}\cdot e=C_{2}[/tex]?

How will it work for the condition 1?[tex](a\cdot b)\cdot c=a(b\cdot c)[/tex]
Can we show it in this way? [tex](C_{2}\cdot e)\cdot C^{-1}_{2}=C_{2}\cdot (e\cdot C^{-1}_{2})\rightarrow e=e[/tex]
How can we show it with using other symmetry elements? [tex][\sigma, \sigma{'}, e][/tex]

for example[tex] (C_{2}\cdot \sigma)\cdot e=C_{2}\cdot(\sigma\cdot e)[/tex]?
 
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  • #2
There's actually a condition that you must verify before considering the ones that you've listed. It is closure, that the product of two elements of [tex]\Gamma[/tex] is also an element belonging to [tex]\Gamma[/tex]. Therefore, the first thing you want to do is build the multiplication table for the elements of [tex]\Gamma[/tex].

When you know the rules for multiplication, verifying associativity will be fairly simple.
 
  • #3
you mean I have to calculate
[tex] C_{2}\cdot e=[/tex]
[tex] C_{2}\cdot C_{2}=[/tex]
[tex] C_{2}\cdot \sigma=[/tex]
[tex] C_{2}\cdot \sigma^{'}=[/tex]
[tex] \sigma^{'}\cdot \sigma{'}=[/tex]
[tex] e\cdot e=[/tex]
[tex] e\cdot C_{2}=[/tex]
[tex] e\cdot \sigma=[/tex]
[tex] e\cdot \sigma^{'}=[/tex]

and so on?

I thought that these 3 conditions had to be fullfilled to call these elements as a symmetry group
 
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  • #4
Like I said, the requirement that the product of two elements of a set is another element in the set is also a requirement to have a group. When you write

[tex](a\cdot b)\cdot c= a\cdot (b\cdot c),[/tex]

you're assuming that [tex](a\cdot b)\cdot c[/tex] is actually in [tex]\Gamma[/tex].

However, it's not just that you have to verify closure. It's also that knowing the multiplication table is necessary to verify condition 1 anyway. How could you say that

[tex] (C_2 \cdot \sigma)\cdot \sigma' = C_2 \cdot ( \sigma\cdot \sigma')[/tex]

if you don't know what [tex] C_2 \cdot \sigma[/tex] is equal to?
 
  • #5
yes you are right, thank you, but here I meet another problem. I do not know what I will get when for example
[tex] C_{2}\cdot \sigma=[/tex] or
[tex] C_{2}\cdot \sigma^{'}=[/tex]

first I rotate the vase and then mirror reflection...
 
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  • #6
You'll want to draw some pictures to work it out. Some of those combinations will just give the identity, others are equivalent to a reflection.
 

1. What is symmetry in the context of a Greek vase?

Symmetry in the context of a Greek vase refers to the balance and visual harmony created by the arrangement of patterns, motifs, and figures on the vase's surface. This can include both bilateral and radial symmetry.

2. How can you identify the symmetry group of a Greek vase?

The symmetry group of a Greek vase can be identified by analyzing the vase's shape, the placement of handles and other decorative elements, and the repetition and arrangement of patterns and figures. It can also be determined by using mathematical principles such as reflections, rotations, and translations.

3. Why is exploring the symmetry group of a Greek vase important?

Exploring the symmetry group of a Greek vase can provide insight into the cultural and artistic influences of the time period in which it was created. It can also help in understanding the techniques and methods used by ancient Greek artists to create visually appealing and balanced designs.

4. Can the symmetry group of a Greek vase be used to date the vase?

While the symmetry group can provide clues about the time period in which a Greek vase was created, it should not be solely relied upon for dating purposes. Other factors such as the style, subject matter, and archaeological evidence should also be considered.

5. How is the exploration of symmetry in a Greek vase relevant to modern science?

The study of symmetry in a Greek vase can provide insights into the principles of geometry and symmetry that are still used in various fields of modern science, including mathematics, architecture, and design. It also highlights the timeless beauty and influence of ancient Greek art on contemporary aesthetics.

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