Question

Abstract algebra. Ill explain.

Answer 1

today in class we were asked to give a formal definition of symmetry.
the instructor started us off with

def: A symmetry of a figure is a rigid motion that takes the figure to itself.

we were supposed to come up with better words than
figure, rigid motion, takes a figure to itself.

we came up with

A symmetry of a set of points, is a transformation that preserves distance, and angle, that is bijective.

he challenged us to figure out if we needed the function to preserve angle, and im pretty sure it does not, but I dont know how to prove that.

he hinted that "preserving the angle" may be redundant

Answer 2

I also see formal definitions of symmetry as

"a distance preserving function that maps an object onto itself"

I don't understand why in this definition the function is not 1-1.

Answer 3

Why would it need to be one to one?

Answer 4

well I guess the fact that in his definition our function is bijective which implies 1-1

Answer 5

but maybe not needed...

Answer 6

Can you give an example of a very simple symmetry?

Answer 7

rotating a square

Answer 8

So preserving distance means that the determinant (in the case of linear transformation) will be 1?

Answer 9

um your way ahead of me. only determinant i know is dealing with matricies.

Answer 10

matrices*

Answer 11

Linear transformations can be expressed as determinants

Answer 12

This is just something I learned in vector calculus. When we find the Jacobian for change of variables, we're really just finding the distance distortion factor of the transformation.

For linear transformations, it's just a constant: the determinant of the matrix representation.

Answer 13

So how would I go about proving that we get preservation of angle through preservation of distance?

Answer 14

Here is a question: If a transformation perseveres distance, can it not be bijective?

Answer 15

Maybe requiring it to preserve distance also requires it to be bijective.

Answer 16

I think to prove it you have to formally define what you mean by distance preserving and define what you mean by angle preserving.

Answer 17

\[d(a,b)=d(f(a),f(b))\]
so in euclidean space would be the distance function
and
\[g(a,b) = g(f(a),f(b))\]
g would I guess then be
\[g(a,b) = cos^{-1}(\frac{ab}{|a||b|})\]

Answer 18

\[g(a,b) = cos^{-1}(\frac{a\cdot b}{|a||b|})\]

Answer 19

So you have to prove that the first one is sufficient for the second one I guess.

Answer 20

back to your question, what would be a symmetry that is not bijective?

Answer 21

Oh I get what you are saying...

Answer 22

So for example, if \(f(x)\) isn't bijective, is it even possible for it to be distance preserving.

Answer 23

Just sticking with the bijective thing for a second, consider that you had three points, a, b, and c. You're only considering \(d(a,b)\) and \(d(a,c)\). If it's not bijective it's possible that \(f(b)=f(c)\)

Answer 24

Meaning that \(d(f(a), f(b)) = d(f(a), f(c))\)

Answer 25

I suppose it isn't necessaryly the case that \(d(a,b)\neq d(a,c)\) though.

Answer 26

(I was hoping for a simple contradiction, no luck though)

Answer 27

:)

Answer 28

Just from casually browsing through this thread, I gather that you're trying to find a simple definition of symmetry, is that right?

Answer 29

not really, I know the common definition of symmetry. But while thinking of symmetry I notice that angle between points in preserved and thus added that to "my"definition, and I am trying to figure out why that is redundant. I am also trying to figure out why bijective is redundant.

Answer 30

Wait! I found the contradiction!
\(b\neq c\implies d(c,b)\neq 0\) but we know \(f(b)=f(c)\implies d(f(b),f(c))=0\)
Meaning \(d(c,b)\neq d(f(b),f(c))\)

Answer 31

So that is why distance preserving necessarily implies bijective function, I think.

Answer 32

Does that make sense? If so it's a good warm up for angle preservation and distance preservation.

Answer 33

I mean, the fact that your system is identical before and after the transformation is kind of inherent in the definition of symmetry, so I'm not quite sure what the issue is.

Answer 34

@Jemurray3 The issue is showing that angle preservation is redundant for the the definition.

Answer 35

when thinking about bijective function
\[f:A\overset{onto}{\rightarrow} A\]
does this not also imply 1-1?

Answer 36

maybe only if A is finite?

Answer 37

A symmetry of a set of points, is a transformation that preserves distance, and angle, that is bijective.

That's the definition you're working with, right? So my first question is what does it even mean to preserve angle? You're talking about a transformation of a set of points, not vectors, so angle doesn't mean anything.

Answer 38

great job on proving that preserving distance implies one to one

Answer 39

for sure makes sense.

Answer 40

in this world vectors and points can be used synonymously. points are just the tips.

Answer 41

That is absolutely not true.

Answer 42

ok

Answer 43

then just think of the object as the trace of the vectors...

Answer 44

In spherical coordinates, there would be 2 angles. Does angle preserving mean both angles would be preserved?

Answer 45

we are in 2d

Answer 46

So we're only talking about 2d symmetry?

Answer 47

yeah, sorry I should have said that.

Answer 48

what ever the answer here is will im sure extend to n dimensions

Answer 49

Okay, we can come up with a definition of angle preserving that is in the spirit of whatever the class was smoking at that time, then we can show it is redundant.

Answer 50

"A bijective function that maps a set of points to itself" is a sufficient definition of symmetry, in the sense that you're talking about. Do you agree?

Answer 51

no
it must preserve distance...

Answer 52

Why?

Answer 53

in other words think of a smiley face
switching the eyes is not a symmetry, but rotating it is

Answer 54

rotating it around the median axis...

Answer 55

Why is switching the eyes not a symmetry??

Answer 56

we are talking about definitions
one cant prove a definition...

Answer 57

it needs to preserve distance because it is defined that way....

Answer 58

Why do you intuitively think swapping the eyes makes it not a symmetry?

Answer 59

because distance is not preserved.

Answer 60

The definition is coming from an intuitive understanding of symmetry that predates abstract algebra.

Answer 61

Also, I should point out that swapping the eyes does in fact preserve distance.

Answer 62

well this is the way its defined in abstract algebra:)

Answer 63

no it does not...

Answer 64

Of course it does, think about it.

Answer 65

between the eyes, I mean.

Answer 66

ALL POINTS

Answer 67

so the distance from the every point on the right eye is now further from the right ear than the distance of all the points were from the right ear

Answer 68

I dont want to argue over what a symmetry is, because it is defined.

I want to know why preservation of angle is redundant.

Answer 69

Because if the angle wasn't preserved it obviously wouldn't be the same figure, would it?

Answer 70

if you hand that in to my teacher he will kick you out of the class:P

Answer 71

Listen kid, and I say kid because it sounds like you're still in high school -- the world of abstract algebra is a lot more complicated than you appear to understand. I'm trying to lead you down a reasonable path to a correct understanding of symmetry but you're making it pretty tough by assuming that I'm stupid.

Answer 72

Is this why it isn't distance preserving?