symmetry in a compass and straightedge

[shahar]

There is a way to prove any symmetry of figures by using compass and straightedge methods?
If it is possible, can I use a robot that use these tools to do it?

 

[Dr.Peterson]

I'm not sure what you mean. A compass can't do a proof for you; all it can do is to draw!

And if you are asked about symmetry of a figure, you need to look at it, not to draw it for yourself.

But maybe some of the theorems used to prove constructions can also be used in a proof of symmetry ...

Can you give an example of such a problem, and how you want to answer it?

 

[shahar]

i ask about problem that asks about symmetry that I can "formulate" it as:
"Show "by straightedge and compass" that a butterfly has symmetric body".
As you say: There is a lack of information to solve the problem".
So, we (the solvers of such kind of problems) can't go further/deeper to do it (in advance (???)), Right?
My original question was about clues that helps to show it. Cause that using compass and straightedge are only to draw. Or I am wrong?

 

[Dr.Peterson]

Assuming you have presented the problem fairly accurately (precise words matter!), I would say that it is not asking for a proof, just for a way to convince yourself that a particular image of a butterfly is approximately symmetric across its midline. (You can't prove anything this way, particularly about all butterflies; and no living thing has perfect symmetry!)

A compass and straightedge construction can find the line of symmetry between two opposite points on wings, and then find the reflection of any given point on one side, to show that that corresponds to a point on the other side. Enough such points can build evidence for the conclusion -- not a deductive argument, but sort of an inductive one.

 

[shahar]

Nice explanation.

 

[shahar]

The next Question that appears in my mind:
"Why is there no prefect symmetry in butterfly body?"
"Where is there a prefect symmetry in nature"? Why there is? Why not?

 

[Dr.Peterson]

Are you perfectly symmetrical? Why not?

Now, it happens that I initially wrote that "nothing in nature" has perfect symmetry, but changed that to "no living thing". Why? Because I thought of crystals. Even they are probably never exactly what they "ought" to be, though. Why? Randomness.