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Subject: "distance vs orthogonality vs choice"     Previous Topic | Next Topic
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gaespes
Member since Feb-1-11
Feb-01-11, 04:15 PM (EST)
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"distance vs orthogonality vs choice"
 
   Since the mid-seventies, during my first university studies, I was unsatisfied with the way of introducing the metric in the usual treatments of the complex plane and the traditional cartesian plane as a numerical plane R.
The Pythagorean theorem was pulled out of a synthetic environment and placed in the geometric-analytical context in a way which seemed very unnatural to the same analytical framework.
The same happens when a scalar product in introduced in a vector space environment (Dieudonn style).
When, after my graduation, I was more free to deepen elementary questions, I formulated a systematic theory of the complex plane that does not rely on the unnatural (read "external") modulus definition:
https://w3.romascuola.net/gspes/materiali/num_compl_e_trasf_geom.pdf .
Herein a "modulus" is defined as an invariant for orthogonal symmetries (page 28, theorem THE.2), so that the metric depends essentially on orthogonality, which is defined through the relation of "oriented orthonormality" which relates the real and the imaginary unities as well as the complex numbers a+bi and -b+ai.
Basically, |u|=|v| ⇔ (u+v)⊥(u-v), so that:
(a+b)+(c+d)=(a+c,b+d) ⊥ (a,b)-(c,d)=(a-c,b-d) ⇔ (a+c)(a-c)+(b+d)(b-d)=0 ⇔ a-c+b-d=0 ⇔ a+b=c+d,
... and when d=0 we obtain the pythagorean equality.

Many years later I turned the previous theory in an axiomatic introduction of the complex plane (ie not based on a previous introduction of the real field): https://w3.romascuola.net/gspes/c/
With this approach the basic concept of orthogonality itself becomes a relative concept, in agreement with the fact that in a vector space with a fixed scalar product one can change the basic bilinear (positive defined)form with infinite possible others, all being equivalent through non-degenerate linear transformations.
The choice of one of these scalar products is merely conventional, and the choice of one of these is the same thing of choosing a vectorial independent base as prototype for orthogonality.
So all the structure of the complex plane C is based on a triple of non-collinear points (0, 1, i), where "non collinearity" simply means that the non-zero (ie different from the point 0) point "i" doesn't belong neither to an axiomatically introduced set "R " (the "positive verse"), which intuitively is the open ray of positive real numbers, nor to its opposite.
So the "usual" feeling of orthonormality can be subverted by changing the position of the imaginary unit (move the point "c" in the following applet: https://w3.romascuola.net/gspes/obliq.html ), and the same process leads from any "ordered independent triple" (taken as an "orthonormal base") to an its own
implementation of circles
(move point "i" in the applets of: https://w3.romascuola.net/gspes/0_1_i_modulo.html ).

gaespes


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  Subject     Author     Message Date     ID  
distance vs orthogonality vs choice gaespes Feb-01-11 TOP
  RE: distance vs orthogonality vs choice alexb Feb-02-11 1
     RE: distance vs orthogonality vs choice gaespes Feb-09-11 2
         RE: distance vs orthogonality vs choice alexb Feb-09-11 5
             RE: distance vs orthogonality vs choice gaespes Feb-11-11 6
                 RE: distance vs orthogonality vs choice alexb Feb-12-11 7
                     RE: distance vs orthogonality vs choice gaespes Feb-13-11 8
                         RE: distance vs orthogonality vs choice gaespes Feb-14-11 9
                             RE: distance vs orthogonality vs choice gaespes Feb-18-11 12
     RE: distance vs orthogonality vs choice gaespes Feb-09-11 3
         RE: distance vs orthogonality vs choice alexb Feb-09-11 4
             RE: distance vs orthogonality vs choice gaespes Feb-14-11 10
                 RE: distance vs orthogonality vs choice alexb Feb-15-11 11

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alexb
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2769 posts
Feb-02-11, 11:56 AM (EST)
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1. "RE: distance vs orthogonality vs choice"
In response to message #0
 
   Thank you for sharing.

I could not follow your links, as I do not read Italian. You may need a broader audience in the hope to elicit more interest in your ideas. Perhaps, https://groups.google.com/group/sci.math/topics would be a better address.


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gaespes
Member since Feb-1-11
Feb-09-11, 07:01 PM (EST)
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2. "RE: distance vs orthogonality vs choice"
In response to message #1
 
   Just a little "add on" to better explain in a formal way:
if we define an "orthonormality" ω as an operator on C (i.e. C→C) with the following properties:
a) ω( ω( 1 ) ) = -1
b) ω( xu + yv ) = xω(u) + yω(v)
and such that C is generated by 1 and i := ω(1), then we obtain the following facts:
1) ω(1) is not real; otherwise, setting ω(1)=x, we would derive -1=ω(ω(1)=ω(x)=ω(x1)=xω(1)=x
2) ω( x+yi ) = ω( x1 + yi ) = xω(1) + yω(i) = -y + xi
3) ω(0)=0 and ω(-u)=-ω(u)
4) ω is surjective, being ω(-ω(v)) = -ω(ω(v)) = -(-v)=v
5) -ω is also an orthonormality.

For each u in C independent from 1 (so that C is spanned by 1 and u), the mapping x+yu → -y+xu is an orthonormality which maps 1 to u.

After the introduction of such an operator ω, and having defined i:=ω(1), and the conjugate of x+yi as conj(x+yi):=x-yi,
a roto-homothety is quickly defined as a map ρ:C→C such that:
I) ρ(i) is orthonormal to ρ(1), i.e. ρ(i)=ω(ρ(1)), that ρ(ω(1))=ω(ρ(1))
II) ρ(x+yi)=ρ(x1+yi)=xρ(1)+yρ(i), i.e. the coordinates of a point remain the same (this is the etymological meaning of "homothety")
and ρ is defined to be a rotation if it maps conj(ρ(1)) to 1 (at the same time that it maps 1 to ρ(1)).
So all the goniometrical theory can be derived from here.

gaespes


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alexb
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2769 posts
Feb-09-11, 07:15 PM (EST)
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5. "RE: distance vs orthogonality vs choice"
In response to message #2
 
   I believe I understand all that, but how ω is different from a rotation by 90°?

Aha, I see. You develop an axiomatic theory for which the common complex numbers serve a model. You give a fine definition of rotation.

Your claim is that this is a more natural way of looking at complex numbers. Is that right?


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gaespes
Member since Feb-1-11
Feb-11-11, 00:30 AM (EST)
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6. "RE: distance vs orthogonality vs choice"
In response to message #5
 
   "Elementary" plane euclidean geometry (I mean the one we usually call "sinthetical" euclidean plane geometry) is an axiomatic theory, whose constructive arithmetical model is R. All the axioms of congruence
(of segments and angles) have to be satisfied in this numerical model by defining a congruence relation, which is usually obtained by means of isometrical transformations in R. To define these functions the classical way is to start from a scalar product in R, or anyway from introducing a metric, the pythagorean
one, carrying it from the synthetical pythagorean theorem. I just observed that isometries can be introduced, instead, by means of a simple definition of an "orthonormal mapping" R → R : (x,y) → (-y,x) (or its opposite), which allows to derive the usual relation between pendences of perpendicular lines and from this also the fundamental euclidean and pythagorical relations about right triangles. The funny fact is that when we interpret R by means of vector drawing (i.e. coming back to a synthetical background, but in an intuitive environment) the two canonical base vectors can also be non-orthogonal to each other in the "optical way" and all the theory slides out well all the way. As a consequence of that, a 90 rotation is still a right angle rotation, because it is just the orthogonality wich is differently implemented (and the 1 - one degree - too). So this comes out to be a skew model (by the way, an infinite set of skew models) af the same former axiomatic theory. That makes me think, once again, about the Goedel's incompleteness theorem.
An axiomatic framework for developing plane analytical geometry can be carried out by means of an "octet" structure (C , + , 0 , - , R+, , 1 , i ) described in the map https://w3.romascuola.net/gspes/c/c.html (the primitive objects and the axioms are in pink color), where the field of real numbers is pulled out from C and not the vice versa.

gaespes


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alexb
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2769 posts
Feb-12-11, 04:35 PM (EST)
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7. "RE: distance vs orthogonality vs choice"
In response to message #6
 
   >So this comes out to be a skew model (by
>the way, an infinite set of skew models) af the same former
>axiomatic theory.

I undertsand so far.

>That makes me think, once again, about the
>Goedel's incompleteness theorem.

Why?

>An axiomatic framework for developing plane analytical
>geometry can be carried out by means of an "octet" structure
>(C , + , 0 , - , R+, , 1 , i )

This I do not understand. I believe your axiomatics made use of the whole R; C was always a pair of real numbers.


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gaespes
Member since Feb-1-11
Feb-13-11, 10:24 AM (EST)
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8. "RE: distance vs orthogonality vs choice"
In response to message #7
 
   >>That makes me think, once again, about the
>>Goedel's incompleteness theorem.

>Why?

The fact that we cannot "catch" the "optical" orthogonality by the geometrical Hilbert set of axioms (neither, of course, from its vectorial counterpart) reminds me the Goedel's incompleteness theorem.

* * *
>>An axiomatic framework for developing plane analytical
>>geometry can be carried out by means of an "octet" structure
>>(C , + , 0 , - , R+, , 1 , i )

>This I do not understand. I believe your axiomatics made use of the whole R; C was always a pair of real numbers.

in
https://w3.romascuola.net/gspes/c/lo/1.html
the set R+ is defined to be as a subset of C (which is not defined by means of R, but as a set to be "axiomatically defined") with the properties given in
https://w3.romascuola.net/gspes/c/lo/3b.html .
Of course these are "almost" the usual "real line" topological and order properties, but stated in a superset (i.e. C) environment.
A remarkable (and well known from technical design) fact is that the drawing of the sum and product of two real numbers cannot be carried out without the help of a point outside from the unidimensional world of R itself, just because parallelograms (for the sum) and proportional triangles (for the product) need two dimensions. So C is a more suitable environment to graphically work on real numbers.
The minimal topological request to built C is to have a subset of C which allows a linear multiplication (i.e. a set of scalar coefficients for product) and which translates the intuitive "zero-one ray" that, afterwards, can be used to build all the real axis (by means of the opposition symmetry), then to "build" (together with the other unit "i") the general product with complex coefficients, and (as a polar ray) to measure angles by means of the other unit "i" and the Euler "archification" of a real number k, i.e. the set {exp(k*i)=lim(1+t*i/n)^n : t varies in <0,k>}:
move c horizontally in (copy and paste this adress on a browser):
w3.romascuola.net/gspes/pug.html?z=80&c=1-i&k=cx&m=exp(k(i))&f=exp((t)k(i))

gaespes


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gaespes
Member since Feb-1-11
Feb-14-11, 04:06 PM (EST)
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9. "RE: distance vs orthogonality vs choice"
In response to message #8
 
   Concerning the last url I posted, the applet PGC (PlaneGraphicCalculator) - which is a free-to-use implementation of the complex plane - can be used in an url directed way, by means of a javascript I wrote.

Example: (copy and paste the following parametrical url)

w3.romascuola.net/gspes/pug.htm?z=80&q=1&c=1-i/4&k=cx&n=10000&m=(1+k*i/n)^n&f=(1+t*k*i/n)^n{_move_c_horizontally_}&g=1+k*i&h=1+t*k*i

For a short explanation of the url-parameters see:

https://w3.romascuola.net/gspes/pug.html?q=1&f={write_math_here}

gaespes


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gaespes
Member since Feb-1-11
Feb-18-11, 07:22 PM (EST)
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12. "RE: distance vs orthogonality vs choice"
In response to message #9
 
   >Concerning the last url I posted, the applet PGC
>(PlaneGraphicCalculator) - which is a free-to-use
>implementation of the complex plane - can be used in an url
>directed way, by means of a javascript I wrote.
>
>Example: (copy and paste the following parametrical url)
>
>w3.romascuola.net/gspes/pug.htm?z=80&q=1&c=1-i/4&k=cx&n=10000&m=(1+k*i/n)^n&f=(1+t*k*i/n)^n{_move_c_horizontally_}&g=1+k*i&h=1+t*k*i
>
>For a short explanation of the url-parameters see:
>
>https://w3.romascuola.net/gspes/pug.html?q=1&f={write_math_here}
________________________________________________________

For Example, my proof of the PT becomes on PUG:

w3.romascuola.net/gspes/pug.htm?x=550&y=570&o=0.6+0.4i&a=1+2i&z=90&t=1000&f=((|a|+ax)(1-i)/2+(t-(floor(200t)=200t)i)(|a|-ax)/2)(ax>0)(ay>0)&g=(t-(floor(200t)=200t)i)ax(ax>0)(ay>0)&h=((|a|+ax)/2+t(|a|-ax)/2-ax(floor(200t)=200t)i)(ax>0)(ay>0)&k=(ax*t+ay*t(floor(200t)=200t)i)(ax>0)(ay>0)&m=(ax+ay/2+ay/2i+(t+(floor(200t)=200t)i)ay/2)(ax>0)(ay>0)&n=((floor(31t)=1)a+(floor(31t)=2)a(1+i)+(floor(31t)=3)a*i-i(floor(31t)=5)|a|+(floor(31t)=6)|a|(1-i)+(floor(31t)=7)|a|+(floor(31t)=8)a+(floor(31t)=9)(ax-|a|i)-(floor(31t)=10)|a|i-(floor(31t)=11)(|a|+ax)i/2+(floor(31t)=12)(|a|-(|a|+ax)i/2)+(floor(31t)=13)(|a|-ax*i)-i(floor(31t)=14)ax+(floor(31t)=16)|a|+(floor(31t)=18)((|a|+ax)/2-|a|i)+((floor(31t)=17)+(floor(31t)=19)+(floor(31t)=21))(a+|a|)/2+(floor(31t)=22)(ax+ay/2i)+(floor(31t)=23)a+(floor(31t)=24)(a+ay)+(floor(31t)=25)(ax+ay)+(floor(31t)=26)ax+(floor(31t)=27)(ax+ay/2i)+(floor(31t)=28)(ax+ay+ay/2i)+(floor(31t)=29)(ax+ay)+(floor(31t)=30)(ax+ay/2)+(floor(31t)=31)(a+ay/2))(ax>0)(ay>0)

(copy and paste in the browser, and then move the point "a")

gaespes


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gaespes
Member since Feb-1-11
Feb-09-11, 07:01 PM (EST)
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3. "RE: distance vs orthogonality vs choice"
In response to message #1
 
   In my last message all "+" signs have been dropped off!!!

gaespes


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alexb
Charter Member
2769 posts
Feb-09-11, 07:02 PM (EST)
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4. "RE: distance vs orthogonality vs choice"
In response to message #3
 
   No, they are all in.


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gaespes
Member since Feb-1-11
Feb-14-11, 04:06 PM (EST)
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10. "RE: distance vs orthogonality vs choice"
In response to message #4
 
   So, maybe the + symbols are not shown just when the user sees his post visualized in the "wait-for-confirmation" view.

gaespes


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alexb
Charter Member
2769 posts
Feb-15-11, 04:09 PM (EST)
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11. "RE: distance vs orthogonality vs choice"
In response to message #10
 
   Some times it disappears in the message itself. There is a bug in the forum software which I am incapable of fixing.


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