complex not so complex

[jehovajah]

Very nice work puntopunto!

Analogous thinking has been an essential part of human proportioning since records were kept. Of course, your approach will be unfamiliar to some, but may be a godsend to others.

Just remember it is Analogous and hopefully will be more intuitive, less arcane in terminology.
Carry forward the principles of the Erlangen Programme and make Klein proud!

[s31415]

hehehe.  Found that a long time ago. 

<Quoted Image Removed>

This is a good example of what happpens when using a non-conformal map. The small scale patterns are stretched and unrecognizable.

[kram1032]

I wonder if there would be a way to have the small scale patterns stretch in ways that don't cause whipped cream (infinite stretch) or perfectly recognizable patterns (as happens in the MSet where, if you zoom in further and further on a certain point, it will just become a more and more defined shape, looking more and more like the Julia set corresponding to that point), but rather where the smaller scale structures are visually different, yet still full of details.
That, then, wouldn't be a fractal in one of the stricter senses of repeating (almost) the same thing over and over but the component of indefinite amounts of details would still happen.

[puntopunto]

 "i" is essentially a construct that simplifies a ton of things, even if the initial construction doesn't do it justice and even is based on a contradiction of previously learned.

You're sure?
Take the ordered pair of real numbers. Use points with there coordinates and/or the 2D vectors as representation. Define addition, multiplication and the scalar multiplication on the ordered pairs. Justify it by parallelogram construction and rotation. Show (0,1)^2=(-1,0). From there you can have expression as p^2+(2,3)p+(0,1)=0 and |p-(0,1)|=0 (thinking in points) or the same with a v(thinking in vectors) or with z for compatibility  with the most used notation.
Let's do a calculation as an example (a+(0,1)b)² (a,b complex, points, vectors) =a²+2(0,1)b+((0,1)b)²=a²+2(0,1)b+(0,1)²b²))=a²+(0,2)b+(-1,0)b²=a²+(0,2)b-(1,0)b². Try it for yourself, there is not so much difference with the i notation. In the beginning there is some struggle, as always if you change a notation system.

About that, there is a superb chapter in Escher, Gödel, Bach. There, Hofstadter introduce a collection of objects, I think he names it an alphabet and defines a few relations on it with unusual names and symbols. Then he ask you to do some calculations. I did, found it not really easy and made several mistakes. After that he shows that what I had done was essentially the same as adding and subtracting natural numbers. I found it superb because I absolutely did not have the idea I was doing so.

Take a page with some complex function theory. How much "i" is there? After introduction it's all  about log(z), exp(z), z² etc. And that won't alter if you consequently use (0,1) and not i.
You certainly can avoid i without much harm. But I am not promoting that. For continuity you have to use i. Practically everything that has been written uses it.

The introduction of i with i=square root (-1) is not so much based on a contradiction. It is a problem of order. It should be the principal square root of (0,1) is (-1,0). We abbreviate (0,1) to i and (-1,0) to -1. Then the square root of -1 still doesn't exist on the reals, but it does on the complex numbers.

@Sam
Explanation very clear. Thanks.

[puntopunto]

This picture is a bit strange to me. I expect near the "border" where the white/grey parts are to enter the Julia set. To me it looks like the iteration process is changing. The small blue points are obvious in. Can somebody tell  what is happening?
(julia_general_folding with one up_fold)

[puntopunto]

Oeps! I was going through the zooms of Pauldebrot and noticed that (approximately) many fold symmetries suddenly changes in 4, 6, 8 fold symmetries.

[kram1032]

Puntopunto, "normal" vector algebra doesn't even have a vector product defined. I mean, there are the scalar and the cross products, but the former gives you a scalar and the later only works in 3D.
If you define all those extra things, you essentially already do complex numbers. Whether you take two orthonormal unit vectors x,y or a real and imaginary direction really doesn't change anything anymore. You're using clifford algebra while forcing the notation of normal vector algebra onto it.

When I said, you can do the same things with vector algebra, I meant without a concept of multiplying vectors together. You'll have to go for other tricks to rotate or reflect your vectors.

So yes, I'm sure. (0,1)² = (-1,0) is just overly bloated for something that can just as well be written as i²=-1.

[M Benesi]

This is a good example of what happpens when using a non-conformal map. The small scale patterns are stretched and unrecognizable.

  No they aren't, just like the "circles" in the 2d Mset aren't unrecognizable (although they aren't perfect circles, except that one period bulb).  Just zoom in (well- not that image, you need to zoom into the fractal with a decent 3d fractal generator). 

  The same pattern is repeated throughout.  You just zoom in.  Everything repeats, with some variation- things have more "spokes" when you zoom into certain areas (similar to 2d p/q type relations).

[jehovajah]

For anyone not quite sure what is the mathematical differences here, just realise that it is over which notation or terms to use to describe exactly the same behaviours. So it is like one saying it sounds sweeter in Italian, while another is saying it is clearer in German.

The intuition that gets lost in this kind of debate is what drives innovation, new solutions, different viewpoints.

The thing here is to go with the flow and see where it takes us. We will go over old ground as well as new, but hopefully with our eyes and ears and intuition apprehending things slightly differently, which may lead us to the goal we seek, the holy grail.

Kram1032 this reminds me so much of your work on conformal actions, or maybe it was Tglad's.
I will look it up.

Its Tglad's
http://www.fractalforums.com/new-theories-and-research/new-type-of-fractal/

[s31415]

  No they aren't, just like the "circles" in the 2d Mset aren't unrecognizable (although they aren't perfect circles, except that one period bulb).  Just zoom in (well- not that image, you need to zoom into the fractal with a decent 3d fractal generator). 

  The same pattern is repeated throughout.  You just zoom in.  Everything repeats, with some variation- things have more "spokes" when you zoom into certain areas (similar to 2d p/q type relations).

I don't know what you mean. This fractal has obviously been created by iterating a non-conformal map of the Mandelbulb type. What looks like almost smooth rings are overly stretched versions of the pattern. Of course, this doesn't mean that there are no small scale detail. Just that they are stretched, some of them beyond recognition. This is not what happens in the M-set.

[puntopunto]

@kram1032
You missing my point. I try to discuss the possible introductions of complex numbers. Which notations, representations and naming you can use. And which consequence that has. Not about how you or someone else, should/would do it in the end. That this is important I can show you with your first remark:

Puntopunto, "normal" vector algebra doesn't even have a vector product defined.

But there is, they named it only different. So if I say: "Take the 2D vector space with complex multiplication and addition Then expressions like v²+3v, even log(v)  makes perfectly sense.
And no you don't have to define extra things. If complex numbers are introduced as ordered pairs of reals, there is simply no more to define then as with
the introduction with a+bI In fact there is one thing less, the definition of i.
Starting with ordered pairs, you stay closer to the representation as points/vectors of complex numbers. And there is no need for the mathematical incorrect definition of i with the square root of -1. And there are also disadvantages, you miss the analogy with the notation system we use for the reals.

[puntopunto]

@jehovajah

Your description in your last post describes very well what it is about.

[Roquen]

I'm half agreeing and half disagreeing with what you're say.  To be semi-formal then you could say something like: complex numbers form a 2 dimensional vector space with basis set {1,e} over field F.  axiom 1: e²=-1.  Now you can demonstration the form of the product over an arbitrary field.  We can then limit the field to reals, add the rule for conjugation and then develop the full algebra in terms of geometric operations.  Since 'e' is only used in formulation of the product, we can certainly not look at it again except to translate historic formalism to whatever form individually works.  Personally I likewise use a tuple notion for most things:  I certainly would never write a 3D vector as:  v = x e₀ + y e₁ + z e₂, because it's too verbose and requires slight more work to decipher vs. (x,y,z).  The same for writing a complex as (x,y).  So here we seem to be in agreement.  My attempted point is the system you're describing could be equivalent described in complex and retain the ability of algebraic reasoning (although I admit than fold introduces a problem).

Puntopunto, "normal" vector algebra doesn't even have a vector product defined. I mean, there are the scalar and the cross products, but the former gives you a scalar and the later only works in 3D.

It's true that vectors don't have a product.  The dot product doesn't close, it produces a scalar.  But the cross product also doesn't close it returns a bi-vector/pseudo-vector/etc. (poor physics folks).  And from a vector-centric view point the cross product is only definable in 3D.  But from a complex view-point:  X=(a,b), Y=(c,d), X^*Y = (a,b)^*(c,d) = (a,-b)(c,d) = (ac+bd, ad-bc).  So from a projection standpoint you get the parallel and orthogonal projections and the results are in the expect positions.  Oh!  Hello difference of angles identity!

[kram1032]

I guess I'm fine with switching between notations when ever one is more practical than the other.
In case of (0,1)²=(1,0) vs. i²=1, the later clearly is less verbose.

Though yeah, vectors do not have a true vector multiplication defined on them within vector calculus.
And clifford algebras are exactly defined by how you multiply vectors together. So once you do define multiplication on vectors, you already are doing clifford algebras, unless you choose a very different way of doing it.

As long as you just stay with vectors, doing a tuple notation is fine. But if you go into the details of adding a scalar, some bivectors and possibly even other structures, you'd end up with something like (example 3D space)

a+(x,y,z)+(yz,zx,xy)+xyz

Two three-tuples and two scalars, or alternatively

(a,x,y,z,yz,zx,xy,xyz)
an 8-tuple.

The addition part with mixed tuples and non-tuples is kinda weird and if you go all tuples, the individual meaning of each component is kinda lost in the notation.
And if you don't distinquish vectors from bi-vectors, which is entirely possible, you'll end up with "axial" and "polar" vectors which you have to be really careful with, because they act differently under reflections.

So as long as you only need vectors or only bivectors or something like that, grouping them together is probably a good idea, but if you need to manipulate the full object, you're probably better off with splitting it all into a sum-of-components notation.

And for the 2D case, the complex notation* is actually less verbose than the tuple one:
(x,y) vs x+iy
You save one symbol.
Or at worst, it's of equal verbosity:


Either way, my main point actually wasn't about the notation but rather about the fact that, if you're doing vector products, you're no longer strictly doing vector algebra.

*There's one caveat in that, technically, 2D euclidean space corresponds to 1+x+y+xy.
When you're working with the normal vector part, you only have (x,y), while (1,xy) is like what you'd do in complex numbers. If you define x and y both to square to -1, this actually will give you quaternions 1+i+j+ij = 1+i+j+k, so in a geometric sense, quaternions actually describe a 2D space with i and j being its vectors, while k=ij describes the entire plane, a bivector and also a pseudo scalar.
But that's a different story... As many other simplifications are possible, to my knowledge, all you can do in 2D can be done equally well with complex numbers.
Though I'm not sure on that. Maybe there are some things in 2D which would benefit from full quaternion or other 2D geometric algebra description.
But even then, which notation best to choose would depend on the context.

[puntopunto]

I try to discuss the possible introductions of complex numbers. Which notations, representations and naming you can use. And which consequence that has. Not about how you or someone else, should/would do it in the end. That this is important I can show you with your first remark:

Quote:
Puntopunto, "normal" vector algebra doesn't even have a vector product defined.

But there is, they named it only different.

They named it rotation