In geometry, why the invariant properties that matter?

[mafendee]

Dear all,

i'm trying to understand geometry by studying the subject myself. i came across idea that I'm very much confuse of. it say's that 'geometry is a studies of geometric properties that is invariant under transformation' such as distance for euclidean geometry.

my question is: why do we studies invariant properties, why not study variant properties instead? why are we so interested in invariant properties? what role does it plays in the study of geometry?

i do hope that someone will be able to enlighten this to me. some textbooks that I'm reading only stop at telling the geometry is a studies of invariants prop, w/o telling why.

million of thanks.

 

[lurflurf]

We study variant properties too. Basically invariant properties are what matter and variant properties are what don't matter. This varies from problem to problem and in different areas of geometry. Sometimes we care about the variant properties and we use the transformation to split the problem into multiple parts and solve them then recombine them. For example if we have a triangle we might not care what side is up, what way it faces, or where it is. So we want to study isometry invariant properties. Even if we do care we can learn from the isometry invariant properties and throw the isometry variant properties in latter.

 

[mafendee]

thanks for your reply and sorry for late response. I did a lot of reading on this. My further question would be, is it safe to say that the invariant properties is very much related to the isometric properties, and in case of euclidean geometry such that: f(ab)=f(a).f(b) where a and b are distance (that is preserve)?