So how does graphing in more than three dimensions work?

[TJ McCaustland]

In short, I am curious about how this even works and I want to know because I enjoy math, so I decided to ask people with a higher education than mine and who definitely know more about math than I do.

 

So my understanding of graphing in three dimensions to give you an idea of my level of understanding of the concept so you can explain it better is essentially this:

 

In order to graph in more than two dimensions we must simply add a third axis: Z. Z represents the depth axis and intersects X and Y at the origin vertically relative to the plane that X and Y form. The Z axis forms a plane with the Y axis vertically and the X axis horizontally. Graphing a function in three dimensions is somewhat similar to graphing a function in two dimensions except that at least from the way I have been exploring it you have to have a chain of functions or relations: X has to be a function of Y, and Z has to be a function of X or Y or some combination of the three. 

For example (probably incorrect notation) x²=y³=z² is an exponential function that is always y positive because of the odd exponent that y has, and thus the output of Y's function is always in the domain positive integers while X and Z's are simply integers. I'm not sure if this is correct since I have not yet run into 3-D graphing past a cursory level in my education but it should give you enough information to explain how graphing in more than three dimensions works in terms that I can understand. 

For reference, this is the graph of x²=y³=z²

So, probably going to get this totally wrong, but would then a four dimensional function not look something like this where we simply add an axis? X=Y=Z=A? Or some other combination of axis names? And the ordered pairs would look like (X,Y,Z,A) just like with 3D graphing? Probably getting ahead of myself, but what would intervals look like with that many dimensions? 

 

Also, is anyone willing to work with me on 3D vectors? I am mostly self taught since I have not come across this in my degree classes and I am interested. 

 

 

Edited August 31 by TJ McCaustland
Correction of a mistake in defining what Y's domain is

[MigL]

There is no 'graphing' involved.
You need n variables to describe the position of a point, in orthogonal n-dimensional space.

[TJ McCaustland]

2 minutes ago, MigL said:

There is no 'graphing' involved.
You need n variables to describe the position of a point, in orthogonal n-dimensional space.

Got it. So it is much simpler than I initially thought and the ordered pair example I gave was correct, in that for n dimensions there will be n variables. Neat. Thanks. 

[studiot]

1 hour ago, MigL said:

There is no 'graphing' involved.
You need n variables to describe the position of a point, in orthogonal n-dimensional space.

But that isn't quite 'graphing', that describes the coordinate system not the 'graph'.

So yes the coordinate of a point is described by n variables in n space.

But as n increases the types and nature of availbale 'graphs' increase.

 

In  1 dimension n = 1 and you can't reall describe much. just different length segments of the same infinite line.

In two dimensions you can play about with lines and get straight and (plane) curved lines as graphs. The idea of a function makes sense as does an independent variale (usually x) and the dependent variable (usually y).

In three dimensions you get more scope and can describe surfaces, solids and so on. One type uf graph using 3 dimensions is a countour graph such as is used on maps.
This is really a sequence of plane curved sections, ie 2D sections.
Recently it has become very import in the technology of '3D fabrication by printing' of objects.

You can also have knots and other fancy constructs in 3D that have no counterpart in 2D.

Many of these constructs dissapear with the step up to 4 D.

 

How are we doing are you following so far ?

3 hours ago, TJ McCaustland said:

 

Also, is anyone willing to work with me on 3D vectors? I am mostly self taught since I have not come across this in my degree classes and I am interested. 

 

Keep asking and we will cover these.

Edited August 31 by studiot

[TJ McCaustland]

27 minutes ago, studiot said:

But that isn't quite 'graphing', that describes the coordinate system not the 'graph'.

So yes the coordinate of a point is described by n variables in n space.

But as n increases the types and nature of availbale 'graphs' increase.

 

In  1 dimension n = 1 and you can't reall describe much. just different length segments of the same infinite line.

In two dimensions you can play about with lines and get straight and (plane) curved lines as graphs. The idea of a function makes sense as does an independent variale (usually x) and the dependent variable (usually y).

In three dimensions you get more scope and can describe surfaces, solids and so on. One type uf graph using 3 dimensions is a countour graph such as is used on maps.
This is really a sequence of plane curved sections, ie 2D sections.
Recently it has become very import in the technology of '3D fabrication by printing' of objects.

You can also have knots and other fancy constructs in 3D that have no counterpart in 2D.

Many of these constructs dissapear with the step up to 4 D.

 

How are we doing are you following so far ?

 

Keep asking and we will cover these.

You're doing great! I am following everything so far, and I imagine you lose these constructs with the step up to the 4 D because of "self folding" nature of some 4 D geometric shapes such as tesseracts because of the way translation occurs on 4 axis' instead of 3? Eh probably got that wrong but still that is the only part I do not get

[KJW]

How are functions represented?

Suppose one has an equation of the form:

[math]x_{n} = f(x_{1}, x_{2}, ..., x_{n-2}, x_{n-1})[/math]

In this case, one has [math]n[/math] variables parameterizing an [math]n[/math]-dimensional space in which the equation is describing an embedded [math](n-1)[/math]-dimensional space. Such an equation could also be expressed in implicit form:

[math]g(x_{1}, x_{2}, ..., x_{n-1}, x_{n}) = 0[/math]

However, suppose one has [math]m[/math] independent simultaneous equations:

[math]g_{1}(x_{1}, x_{2}, ..., x_{n-1}, x_{n}) = 0[/math]
[math]g_{2}(x_{1}, x_{2}, ..., x_{n-1}, x_{n}) = 0[/math]
[math]...[/math]
[math]g_{m-1}(x_{1}, x_{2}, ..., x_{n-1}, x_{n}) = 0[/math]
[math]g_{m}(x_{1}, x_{2}, ..., x_{n-1}, x_{n}) = 0[/math]

Then in this case, the [math]n[/math]-dimensional space contains an embedded [math](n-m)[/math]-dimensional space. That is, the number of dimensions of the embedded space described by the set of equations is equal to the total number of variables minus the total number of equations (assumed to be independent). Each equation is a constraint on the values of the variables that satisfy the equations, and each constraint reduces by one the dimension of the space formed by the values of the variables that satisfy the equations.

Another way to describe a [math]d[/math]-dimensional space embedded in an [math]n[/math]-dimensional space is to introduce [math]d[/math] additional variables that parameterize the [math]d[/math]-dimensional space. Then the [math]d[/math]-dimensional space is described by the [math]n[/math] equations:

[math]x_{1} = h_{1}(t_{1}, t_{2}, ..., t_{d-1}, t_{d})[/math]
[math]x_{2} = h_{2}(t_{1}, t_{2}, ..., t_{d-1}, t_{d})[/math]
[math]...[/math]
[math]x_{n-1} = h_{n-1}(t_{1}, t_{2}, ..., t_{d-1}, t_{d})[/math]
[math]x_{n} = h_{n}(t_{1}, t_{2}, ..., t_{d-1}, t_{d})[/math]

Note that there are [math]n+d[/math] variables and [math]n[/math] equations describing a [math]d[/math]-dimensional space.

In the case of a one-dimensional curve in an [math]n[/math]-dimensional space, one has:

[math]x_{1} = h_{1}(t)[/math]
[math]x_{2} = h_{2}(t)[/math]
[math]...[/math]
[math]x_{n-1} = h_{n-1}(t)[/math]
[math]x_{n} = h_{n}(t)[/math]

Therefore:

[math]t = h^{-1}_{n}(x_{n})[/math]

And eliminating [math]t[/math] from the remaining [math]n-1[/math] equations:

[math]x_{1} = h_{1}(h^{-1}_{n}(x_{n}))[/math]
[math]x_{2} = h_{2}(h^{-1}_{n}(x_{n}))[/math]
[math]...[/math]
[math]x_{n-2} = h_{n-2}(h^{-1}_{n}(x_{n}))[/math]
[math]x_{n-1} = h_{n-1}(h^{-1}_{n}(x_{n}))[/math]

Note also that:

[math]h^{-1}_{1}(x_{1}) = t[/math]
[math]h^{-1}_{2}(x_{2}) = t[/math]
[math]...[/math]
[math]h^{-1}_{n-1}(x_{n-1}) = t[/math]
[math]h^{-1}_{n}(x_{n}) = t[/math]

And therefore:

[math]h^{-1}_{1}(x_{1}) = h^{-1}_{2}(x_{2}) =\ ...\ = h^{-1}_{n-1}(x_{n-1}) = h^{-1}_{n}(x_{n})[/math]

 

Edited September 1 by KJW

[MigL]

In the case of 1 dimension any 'graph' would consist of a point; not very useful.
In the case of 3 dimensions, while we can construct a 'graph', it is quite difficult to extract exact information from it, and various methods of representing the 3rd dimension have varying degrees of success.
Anything of higher dimensionality than 3 is impossible to represent graphically and accurately.

A 2dimensional graph is what any 12 year old is familiar with, and can represent information to arbitrary accuracy.
It is the only truly accurate graphical representation of the information.

Am I mistaken in my opinion that a graph is a visual representation of information ?

[studiot]

5 hours ago, MigL said:

Am I mistaken in my opinion that a graph is a visual representation of information ?

Well I agree with you.

 

21 hours ago, TJ McCaustland said:

You're doing great! I am following everything so far, and I imagine you lose these constructs with the step up to the 4 D because of "self folding" nature of some 4 D geometric shapes such as tesseracts because of the way translation occurs on 4 axis' instead of 3? Eh probably got that wrong but still that is the only part I do not get

Glad you are following, don't forget to ask for clarification if you need it.

 

I think it is exchemist who is fond of quoting " The map is not the territory".

Certainly a good insight always worth bearing in mind.

So let us clarify a few things, especially if you are finding KJW's post rather formal (but correct).

 

It will be necessary for you to distinguish between a graph and a function. They are not the same.
Indeed there are graphs that are not functions and functions that are not graphs.

A function is very tightly defined in maths and one distinguishing characteristic stands out.

You select a particular value as an input from the independant variable or set of values if there are several such.
Applying the function yields one and only one result or output as a value of the dependant variable.

I can't stress the importance of this enough.

On the other hand a graph can show what happens when the may be many possible outputs for a given specific input.

Figs 1 and 2 Show the graph of the square root.

The square root function only has positive values as shown.  The graph allows both the negative and positive.

We will come on to why this distinction is made later.

Fig 3 shows another sort of graph (which actually is a function as it passes the one value result test) but it does not fit with KJW's dimension statements.

It is what is known as an area plot of the function y = x2.
The area of the squares represent the value of y at the numbers 1,2,3, ...etc.

I have drawn this one because it leads into a very important idea you will need when we come to vectors.

 

A much more complicated graph appears in the next figure, where each value of x has many values of y in a branching tree structure.

The are the characteristic curves of a transistor amplifier output.

Superimposed on this are some other lines, marked load line, input and output signals.

The load line is actually a  function, the tree of characteristics is not.

 

 

My final figure, fig 4, shows a piece of paper representing the x axis.
Of course this axis goes to infinity in both directions so I hope you can see that it is not possible to get from region A to region B without crossing the x axis?

Now a piece of paper is 2 dimensional.
If we allow 3 dimensions the we can approach the x axis, move around it in the third dimension (ie leave the paper) and get from A to B without crossing the line.

This introduces the idea of what happens in 4 dimensions and why we can get from inside a 3 D box to outside , without passing through a wall - obviously not possible in 3D alone

Some more terminology. The first 3 examples are all about geometry, the fourth example is about topology.

Back to MigL here.  Graphical representations are usually geometrical.  that is how that started.
But topology, which grew out of geometry and is now a different subject, has an entirely differnt definition of a graph in terms of connectivity.

I'm going to be away for most of next week so take your time to think about these things so here is a recommended book, t6he delightful

 

Things to Make and Do in the Fourth Dimension

by a modern mathematician, Matt Parker.

Just a small Penguin paperback but well worth the read and loads of fun.

 

 

Edited September 1 by studiot

[studiot]

@TJ McCaustland

 

Am I to take it that you are no longer interested in your question ?

Edited September 8 by studiot