Learning About Isometry From Serge Lang's Basic Math; Very Confused

Question

I'm trying to self learn Serge Lang's "Basic Mathematics". I'm currently in the "Isometries" section of the book, but I'm extremely confused, by the notations and the concepts Lang uses.

• Lang first states that by mapping a plane into itself, he means an association, which to each point of the plane associates another point of the plane.

• But I have no idea what this means. First, what is association? I don’t get what he means by points on a plane association to another point of the plane. Next, what does mapping a plane into itself mean?

• Lang then introduces this notation P↦P’. He says that point P’ corresponds to P under the mapping, or that P is mapped out on P’.

• Again, but what exactly does P’ mean? What does corresponding to P under the mapping mean?

• Lang goes on to say “Just as we used letters to denote numbers, it is useful to use letters to denote mappings. Thus if F is a mapping of
  the plane into itself, we denote the value of F at P by the symbols F(P). We shall also say that the value F(P) of F at P is the image of P under F. If F(P)=P', then we also say that F maps P on P'.

• Again, what is a mapping of a plane into itself, and how does it relate to F?
• Second, what is an image of P under F?
• Lastly, what does F(P)=P'. It reminds me of a function. But if it is a function, what is it's input and what is it's output? And how does it relate to everything else?

I'm sorry for writing so much, I'm new to Math StackExchange. And I'm really confused about the concepts I'm seeing.

Answer 1

When Lang is talking about a plane he means the area/object you're doing your "math" on. Imagine you're drawing a map, then the "plane" is the piece of paper you're drawing on. Similarly, if you're painting a ball, the ball is your (curved) plane. There can be multiple planes and maps across different planes.

When he says he's mapping a plane into itself, he means that any point he uses will be on that plane (the plane is closed, and no matter what new points he makes, they too will be on that same plane).

Back to the map example:

Lets say you're designing a town with a street down the middle. You want to put houses on both sides of the street, and because you want your new town to be nice and clean you want all the houses to be the same distance from the street.

To accomplish this, you could start in the middle of the street and walk to the left 500 ft. This will be house A. The you go back to the street and walk to the right 500 ft. We'll call this house A' (pronounced "A prime").

As you continue to walk down the street and put houses on either side, you notice for every house on the left, there is corresponding house on the right (which is also 500 ft. from the street). Thus for every house A, there is an associated house, A'. Because the houses are both 500ft from the street, this association is an isometry*. Continuing on you'd have houses B and B', C and C', etc...

We can denote this as an association, $A \mapsto A'$

Or as a function F (let's call it the house planning function): $$F(A)=A'$$

which means that every time I put a house down on the A side of the street, I will build a house 500 ft. on the A' side. All the houses on the left are the domain of our house planning function, and all the houses on the right are the image of the house planning function (the image of F on A)...

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*It's important to remember that isometries preserve distances and angles, so not only do the houses have to be the same distance from the street, but they also can't be at different angles to the street (say one being perpendicular and one being diagonal).

Answer 2

He's trying to introduce the concept of function but he wants you to approach it without and pre-assumptions or misconceptions. The most important misconception he wants to disabuse you of is the a function must have some defined rule involved to it. It does not.

An "association" or "mapping" means exactly what it sounds like. Michael is a associated to English muffins if by referencing Michael we always get English muffins. The point $(2, 3)$ "associated" to the point $(-31, 7)$ if I say "everytime you point to the $(2,3)$ on your copy of the plane I'll point to $(-31,7)$ on my copy of the plane."

Now a mapping would be if we associate every point with a point "If you point to $(0,0)$ I'll point to $(53,9)$. If you point to $(0, 0.0000001)$ I'll point to $(3,17)$. If you point to $(0, 0.0000002)$ I'll point to $(-19.5, \sqrt \pi)$. If you point to ....."

So

You || Me

$(2,3)\mapsto (-31, 7)$

$(0,0)\mapsto (53,9)$

$(0, 0.0000001)\mapsto (3,17)$

$(0, 0.0000002)\mapsto (-19.5, \sqrt \pi)$

etc. Is a mapping from your copy of $\mathbb R^2$ to my copy of $\mathbb R^2$.

Hopefully that addresses point 1,2 and 3.

What does $P'$ mean? Just that that is the label we chose to give to the point that is associated to $P$. It just shorthand. $(2,3)\mapsto (-31, 7)$ so $(2,3)' = (-31,7)$. It's just a way to refer to the points that are mapped to if we don't actually know their specific values.

On to point 5.

What is $F$.

So we have this huge mapping where

$(2,3)\mapsto (-31, 7)$

$(0,0)\mapsto (53,9)$

$(0, 0.0000001)\mapsto (3,17)$

$(0, 0.0000002)\mapsto (-19.5, \sqrt \pi)$

And for any possible $(x,y)$ there is some $(x,y)'$ so that $(x,y) \mapsto (x,y)'$.

This collection of all the point-by-point mappings is a collection of some mapping $\mathbb R^2 \to \mathbb R^2$.

We want to refer to this collection by some name. We call it $F$.

So that $(2,3)\mapsto (-31, 7)$. So we say $F((2,3)) = (-31, 7)$

And $(0, 0.0000002)\mapsto (-19.5, \sqrt \pi)$ so we say $F((0, 0.0000002)=(-19.5, \sqrt \pi)$

And for every $P = (x,y)$ for any point $P=(x,y)\mapsto (x,y)' = P'$ and we say $P((x,y)) =F(P) = P'=(x,y)'$.

It reminds me of a function.

It's funny you should say that ....

It is a function.

But if it is a function, what is it's input and what is it's output?

The points of the plane are its input and the points of the plane are its output.

And how does it relate to everything else?

We don't know. This is just the abstract idea of a function. We haven't actually defined any specific relations and values.

Well, okay, for this example I chose 4 arbitrary points as input and mapped them to 4 arbitrary points as output.

But that was just to make an example. There are an infinite number of functions and mappings we could have. Lang is just trying to explain what the idea of a function is in the abstract.

...

And a function simply is: a collective mapping from one set $X$, to another set $Y$ where each individual element of the set $X$ is mapped or associated with a specific element of $Y$.

And that is all he is talking about. It's just that both $X$ and $Y$ are the same thing. The points of a plane.