The Mysterious Connections Between Irrational Numbers - e, pi, and phi

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In summary, the conversation discusses the significance of irrational numbers such as e, pi, and phi, and their relation to each other and different systems. The concept of irrational numbers and their representation is also explored, with the idea that they cannot be accurately represented using natural number notations. The accuracy of irrational numbers is debated, with some arguing that they can be accurately represented through infinite series, while others question this definition of accuracy. The conversation also touches on the idea of the real line and the representation of numbers on it.
  • #36
(a xor b) and (a xor b) was the same as a and b, which it isn't,
I don't understand what do you want to say.

Please choose a xor b:

a) ((a xor b) and (a xor b)) = (a and b)

b) ((a xor b) and (a xor b)) not= (a and b)
 
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  • #37
you're the one who needs to explain how to recover boolean logic from your system because you've not done so as yet. moreover as you've not defined what you mean by uncertainty_and_redeundancy we can't do as you ask.


the second part: you said that these diagrams are the whole collection of somethings between a and b and a xor b

then you drew the cases:

Code: 
  b   b 
    #   #    
    a   a     
    .   .   
    |   |   
    |&__|_   
    | 
    
    [B]a   b     
    .   .   
    |   |  <--- (Standard Math logical system fundamental building-block) 
    |#__|   
    |[/B]

implying this is boolean logic, and one of these is xor the other and, i#m pointing out that neither of these is an any sense "and".

and everything in there must be boolean so you aren't allowed to cite any other kind of logic.

By the way, you always misuse connectives so why on Earth can you expect anyone to take your things seriously?
 
  • #38
My reasoning on this is this:

Let # be xor.

Let & be and.

f=false

t=true

u=uncertainty

r=redundancy

By (f # t) I mean that some single result can be found through a probability of 1:2 .


the complementary logical representation of this probability can be expressed in this way:
((f # t)&(f # t)) where all this expresion is under this "cloude of probebility"

Code: 
<--r--> ^ 
 t   t  |
 #   #  u
 f   f  |
 |   |  v
 |&__|_
 |

So as we see, ((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t), which is definitely not a Boolean Logic state.


Only (f # t) is an excluded-middle f/t locial state with no probebility, after we find our single result.
 
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  • #39
but you said it was boolean, that it was a two valued logic of ordinary maths. so were you wrong or lying? seeing as it is supposed to be proper maths then probabilities cannot lies between 1 and 2, unless it's 1 obviously.
 
  • #41
so why did you say that both digrams were part of two valued logic? and then this contradicts you assertion that it runs from a and b to a xor b. where's "and" gone then?
 
  • #42
And you lose through this generalization (it is trivialization through my point of view) very interesting included-middle ordered Logical states.

No, you don't. Any statement is either in a given "included-middle ordered Logical state" or it is not; a binary fact.
 
  • #43
Really?
No, you don't. Any statement is either in a given "included-middle ordered Logical state" or it is not; a binary fact.
(excluded-middle --> a binary fact) XOR (included-middle --> not a binary fact)

An example of a non-binary system:

Let # be xor.

Let & be and.

f=false

t=true

u=uncertainty

r=redundancy

By (f # t) I mean that some single result can be found through a probability of 1:2 .

the complementary logical representation of this probability can be expressed in this way:
((f # t)&(f # t)) where all this expresion is under this "cloude of probebility"

Code: 
<--r--> ^ 
 t   t  |
 #   #  u
 f   f  |
 |   |  v
 |&__|_
 |
So as we see, ((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t), which is definitely not a Boolean Logic state.


Only (f # t) is an excluded-middle f/t locial state with no probebility, after we find our single result.

---------------------------------------------------------------------------------------------------
Now you can say that:

(excluded-middle --> a binary fact) XOR (included-middle --> not a binary fact) in general is a binary fact.

So what. it is a trivial and non-interesting information.
---------------------------------------------------------------------------------------------------

More than that, for example:

f=(excluded-middle --> a binary fact)

t=(included-middle --> not a binary fact)

Let # be xor.

Let & be and.

u=uncertainty

r=redundancy

By (f # t) I mean that some single result can be found through a probability of 1:2 .

the complementary logical representation of this probability can be expressed in this way:
((f # t)&(f # t)) where all this expresion is under this "cloude of probebility"

Code: 
<--r--> ^ 
 t   t  |
 #   #  u
 f   f  |
 |   |  v
 |&__|_
 |
So as we see, ((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t), which is definitely not a Boolean Logic state.
 
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  • #44
So as we see, ((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t), which is definitely not a Boolean Logic state.

This sounds like a statement using binary logic (about your multi-valued logic).
 
  • #46
I said:

"((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t)" is a boolean statement.

Are you saying that

"((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t)" is simultaneously "(f & f)_(t & f)_(f & t)_(t & t)"?

And even if you are, isn't this new statement of yours true? (according to you)
 
  • #47
WWW said:
Please Prove that Complementary Logic ( https://www.physicsforums.com/showpost.php?p=192318&postcount=25 ) can be reduced to a false/true logic.

Your complementary logic system is a proposition that can be proved, P , or not-proved, ~P .

Some excellent ideas regarding symmetry though.

A__~A___A_V_~A

T___F_______T

F___T_______T


A truth table tautology is very much like a symmetry, it is invariant.

If your complementary logic is context dependent, it still must have an invariant structure that gives a meaningful interpretation?
 
  • #48
Hurkyl said:
I said:

"((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t)" is a boolean statement.

Are you saying that

"((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t)" is simultaneously "(f & f)_(t & f)_(f & t)_(t & t)"?

And even if you are, isn't this new statement of yours true? (according to you)

You are mixing between the existence of some system and it’s logical reasoning.

Any consistent system is limited (incomplete) by definition, otherwise it is inconsistent.

Because my system is consistent by it’s internal structure it is also limited by these structures.

The new thing here, if we compare it to the standard excluded-middle system, is that it is naturally using probability right from it’s first-order level.

For example:

Let us examine Schrodinger's Cat experiment.

f=dead cat

t=live cat

Let # be xor.

Let & be and.

u=uncertainty

r=redundancy

By (f # t) I mean that some single result can be found through a probability of 1:2 .

the complementary logical representation of this probability can be expressed in this way:
((f # t)&(f # t)) where all this expression is under this "cloud of probability"

Code: 
<--r--> ^ 
 t   t  |
 #   #  u
 f   f  |
 |   |  v
 |&__|_
 |
So as we see, ((f # t)&(f # t)) is simultaneously (f & f)_(t & f)_(f & t)_(t & t), which is definitely not a Boolean Logic state.

Please show us (t & f) as a valid(=1=existing) state in an excluded-middle system.

Also through my system the meaning of probability is not some accurate value between 0 and 1 (as we can find in Fuzzy Logic, for example) but an ordered simultaneous associations between redundancy_AND_uncertainty ,which creates “clouds of vagueness” from the most vagueness to the least vagueness, when n > 1 is given.

Shortly speaking, Complementary Logic is based on ordered levels of symmetry breaking, right from its first-order level.
Russell E. Rierson said:
If your complementary logic is context dependent, it still must have an invariant structure that gives a meaningful interpretation?
Yes, because it is consistent it is also incomplete and context depended, but unlike an excluded-middle logical system, it is not looking at vagueness as an enemy that we have to distinct by more and more accurate definitions.

Complementary Logic reasoning is to save and explore the associations between information forms at any given degree of vagueness, where the dynamic process of any research and the explored/explorer interactions are naturally included.
 
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  • #49
But that makes no sense, doron, unless you tell us what f xor t means. what's xor, your # above? and &? we can only interpret in boolean terms because that's all they are.

what is u, what is r, and for that matter what is v?

and you can't have probabilties between 1 and 2 (unless it is 1).
 
  • #50
You are mixing between the existence of some system and it?s logical reasoning.

What does that even mean?

I'll take my best guess, and respond that you're the one unable to accept that one can use ordinary "excluded-middle binary logic" to reason about multi-valued logical systems, or those without the excluded middle.


IIRC, synthetic differential geometry is developed by presenting a system where the law of excluded middle is not a tautology, and then using ordinary logic (including the law of the excluded middle) to reason about it "externally".


Please show us (t & f) as a valid(=1=existing) state in an excluded-middle system.

Interpreting your symbols according to their ordinary meaning, t & f = f. Simple as that.
 
  • #51
d = dead cat

l = live cat

In Complementary Logic (d & l) is a true statement of dead/live probability (like the wave/particle existence).

Please show what is (d & l) by an excluded-middle logical system.
what is u, what is r, and for that matter what is v?

and you can't have probabilties between 1 and 2 (unless it is 1).
I used 'v' letter as an arrowhead in my diagram.

I mean that we have a probability of 1:2 and not some accurate value between 1 and 2.

By (f # t) I mean that some single result can be found through a probability of 1:2 .
 
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  • #52
so the and and xor symbols you are using aren't the usual and and xor symbols. so you need to define them. (ie what they do)

note it is not correct to say that the cat in shroedinger's experiment is alive and dead but the the state will take some value in some hilbert space with certain probabilities. and we've done that using boolean logic.. Anyway, your and is some other binary connective.

to some extent the answer is dependent on which school of QM thought you adopt. and you don't know the probability that the cat is alive is 1/2. it depends on how the experiment is set up.

i don't understand how you can say that you can't describe QM with boolean logic seeing as without it you would never have learned about it in the first place. all the experiments you know of and theory is done in boolean logic.
 
  • #53
Let us say that you explore, for example, Mandelbrot farctal only by R members, without using Complex numbers.

In both cases you will be able to research the results, but by using C and R numbers, we can get much more interesting information.

Because Complementary Logic is based on included-middle results of interactions between opposite elements (where Boolean or Fuzzy Logics are proper sub-systems of it) we get a much more powerful tool to explore and understand the QM phenomenon.
Matt said:
so the and and xor symbols you are using aren't the usual and and xor symbols. so you need to define them. (ie what they do)
AND and XOR connectives are independed and can be changed according to the "logical environment" that using them.

In Complementary Logic, probability is a first-order property that changing the results of AND and XOR connectives.
 
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  • #54
what on Earth does 'explore the mandelbrot fractal only by R' mean? It doesn't even sound plausible.

you've still not shown that boolean logic is a subsystem of your alleged logic. nor how you would use it in any situation.

for instance what is the truth value, for want of a better phrase, of the proposition: If x, an integer, is divisible by 4, then x is even. I reckon it's true. what does your system say?
 
  • #55
Please refresh screen and read again my previous post.
 
  • #56
The real numbers can model the complex numbers; thus, anything you can do with complex numbers, you can do (in some fashion) with real numbers.

For example, I might consider a pair of real numbers, [itex](c, d)[/itex], and study the pairs of numbers [itex](a_0, b_0)[/itex] such that the following iteration

[tex]\begin{equation*}\begin{split}
a_{n+1} &= a_n^2 - b_n^2 + c \\
b_{n+1} &= 2 a_n b_n + d
\end{split}\end{equation*}[/tex]

does not diverge to infinity.

And, in this way, one can study Julia sets (and thus the Mandelbrot set) without ever mentioning a complex number.
 
  • #57
somehow i doubt that was what he had in mind (the unnecessary ontological commitment of the complex numbers...?) , but then i often have no idea what he means.
 
  • #58
Hurkyl,

Some times simple thinking can help us to understand simple examples.

Matt Grime wrote:
Matt Grime said:
don't understand how you can say that you can't describe QM with boolean logic seeing as without it you would never have learned about it in the first place. all the experiments you know of and theory is done in boolean logic.
My example of Mandelbrot set is this:

If we explore its structures in 1-dim we get 1-dim results.

If we explore its structures in 2-dim we get a 2-dim results and 1-dim results.

Shortly speaking, more dim we have less we are limited in our abilities to explore something.

Because Complementary Logic is naturally included-middle logical system (where Boolean and Fuzzy Logics are proper sub-systems of it) we have the freedom to use its extra logical forms, or not.
Matt said:
so the and and xor symbols you are using aren't the usual and and xor symbols. so you need to define them. (ie what they do)
AND and XOR connectives are independed and can be changed according to the "logical environment" that using them.

In Complementary Logic, probability is a first-order property that changing the results of AND and XOR connectives.


For example:

f=dead cat
t=alive cat
r=redundancy
u=uncertainty

When probability is a first-order property then AND connective is used whenever a no-unique result can be found:
Code: 
<--[B]r[/B]--> ^ 
 t   t  |
 #   #  [B]u[/B]
 f   f  |
 |   |  v
 |&__|_
 |
When probability is a first-order property then XOR connective is used whenever a unique result can be found:
Code: 
 f   t   
 |   |   
 |#__| 
 |

Simple as that.

For example:

Let XOR be #

Let AND be &

Let a,b,c,d stands for uniqueness, therefore logical forms of 4-valued logic is:

Code: 
              Uncertainty
  <-Redundancy->^
    d  d  d  d  |
    #  #  #  #  |
    c  c  c  c  |
    #  #  #  #  |
    b  b  b  b  |
    #  #  #  #  |
   {a, a, a, a} V
    .  .  .  .
    |  |  |  |
    |  |  |  |
    |  |  |  | <--(First 4-valued logical form)
    |  |  |  |
    |  |  |  |
    |&_|&_|&_|_
    |
    ={x,x,x,x}


   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |#_|  |  |
    |     |  | <--(Last 4-valued logical form)
    |#____|  |      
    |        |
    |#_______|
    |
    ={{{{x},x},x},x}

[b]
============>>>

                Uncertainty
  <-Redundancy->^
    d  d  d  d  |          d  d             d  d
    #  #  #  #  |          #  #             #  #        
    c  c  c  c  |          c  c             c  c
    #  #  #  #  |          #  #             #  #   
    b  b  b  b  |    b  b  b  b             b  b       b  b  b  b
    #  #  #  #  |    #  #  #  #             #  #       #  #  #  #   
   {a, a, a, a} V   {a, a, a, a}     {a, b, a, a}     {a, a, a, a}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |&_|_ |  |       |#_|  |  |       |&_|_ |&_|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |&_|&_|&_|_      |&____|&_|_      |&____|&_|_      |&____|____
    |                |                |                |
    {x,x,x,x}        {x,x},x,x}       {{{x},x},x,x}    {{x,x},{x,x}}     
 
                                      c  c  c
                                      #  #  #      
          b  b                        b  b  b          b  b
          #  #                        #  #  #          #  #         
   {a, b, a, a}     {a, b, a, b}     {a, a, a, d}     {a, a, c, d}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |#_|  |&_|_      |#_|  |#_|       |  |  |  |       |&_|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |&_|&_|_ |       |#____|  |
    |     |          |     |          |        |       |        |
    |&____|____      |&____|____      |#_______|       |#_______|
    |                |                |                |
    {{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x}      {{{x,x},x},x} 

   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |#_|  |  |
    |     |  |  
    |#____|  |      
    |        |
    |#_______|
    |    
    {{{{x},x},x},x}
[/b]

A 2-valued logic is:

Code: 
    b   b 
    #   #    
    a   a     
    .   .   
    |   |   
    |&__|_   
    | 
 [b]   
    a   b     
    .   .   
    |   |  <--- (Standard Math logical system fundamental building-block) 
    |#__|   
    |
[/b]
 
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  • #59
still not defined uncertainty and redundancy, non-standard terms.

mandelbrot's set doesn't have integer dimension...

so your logical theory trivially encompasses all others, yet you've not shown it has any other non-extant models.
 
  • #60
mandelbrot's set doesn't have integer dimension...
I know it, but in 1-dim all you can get is the shadow of what you can find between 1-dim and 2-dim, isn't it?
still not defined uncertainty and redundancy, non-standard terms.
Please explain Why do you think they are not defined?
 
  • #61
Using the concept of "invariance/symmetry" :

T|F = F|T = T

The | represents a "choice" between T or F

Some might question the "equals sign".

Here is someone explaining a type of complementary logic also :eek:

http://users.erols.com/igoddard/gods-law.html
 
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  • #63
http://users.erols.com/igoddard/gods-law.html is very interesting and supports Complementary Logic main point of view.

But in Complementary Logic 100%A is some unique result that we can get out of x1 xor x2 xor x3 xor ...
 
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  • #64
WWW said:
http://users.erols.com/igoddard/gods-law.html is very interesting and supports Complementary Logic main point of view.

But in Complementary Logic 100%A is some unique result that we can get out of x1 xor x2 xor x3 xor ...

Of course! One asks oneself the question "What the heck does it mean for a wave function to collapse?"

According to Einstein, there is no instantaneous action at a distance!
 
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  • #65
What is 'distance' from your point of view?

For me 'distance' is the preventing side of some perevent/complement system for example: http://www.geocities.com/complementarytheory/4BPM.pdf

By Complementary Logic any existing element that can be chaneged, is the result of at least two opposites that simultaneously pereventing/defining each other.
 
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  • #66
Some times simple thinking can help us to understand simple examples.

Take your own advice. :smile:


I demonstrated how, using logic and a "1-dim system", we are able to fully "explore" a "2-dim system".

The lesson I'm trying to demonstrate is:

Even if one system is a special case of another system, the first system can still be just as powerful as the second system.


AND and XOR connectives are independed and can be changed according to the "logical environment" that using them.

Incorrect. The representations may be the same (e.g. they're both called AND and XOR), but your AND and XOR are certainly very different from the AND and XOR from boolean logic.


Please explain Why do you think they are not defined?

Because you have not defined them.
 
  • #67
I demonstrated how, using logic and a "1-dim system", we are able to fully "explore" a "2-dim system".
No you did not, because in 1-dim(=x-dim) universe no point can be found as a result of (x-dim,y-dim) system.

Your (c,d)(a0,b0) example is a (x-dim,y-dim) --> 2-dim system, and only then you can show a
"2-d Math picture" of mandelbrot set (which has a fractal-dim between 1-dim and 2-dim).

Shortly speaking, in a 1-dim universe any y-dim reduced to x-dim.

Therefore d reduced to c and b0 reduced to a0, and you have no 2-dim Math picture of some Julia set.
Even if one system is a special case of another system, the first system can still be just as powerful as the second system.
You did not show it yet.

WWW said:
AND and XOR connectives are independed and can be changed according to the "logical environment" that using them.

Hurkyl said:
Incorrect. The representations may be the same (e.g. they're both called AND and XOR), but your AND and XOR are certainly very different from the AND and XOR from boolean logic.

In Complementary Logic, probability is a first-order property that changing the results of AND and XOR connectives.


For example:

f=dead cat
t=alive cat
r=redundancy
u=uncertainty

When probability is a first-order property then AND connective is used whenever a no-unique result can be found:
Code: 
<--[B]r[/B]--> ^ 
 t   t  |
 #   #  [B]u[/B]
 f   f  |
 |   |  v
 |&__|_
 |
When probability is a first-order property then XOR connective is used whenever a unique result can be found:
Code: 
 f   t   
 |   |   
 |#__| 
 |

Simple as that.

For example:

Let XOR be #

Let AND be &

Let a,b,c,d stands for uniqueness, therefore logical forms of 4-valued logic is:

Code: 
              Uncertainty
  <-Redundancy->^
    d  d  d  d  |
    #  #  #  #  |
    c  c  c  c  |
    #  #  #  #  |
    b  b  b  b  |
    #  #  #  #  |
   {a, a, a, a} V
    .  .  .  .
    |  |  |  |
    |  |  |  |
    |  |  |  | <--(First 4-valued logical form)
    |  |  |  |
    |  |  |  |
    |&_|&_|&_|_
    |
    ={x,x,x,x}


   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |#_|  |  |
    |     |  | <--(Last 4-valued logical form)
    |#____|  |      
    |        |
    |#_______|
    |
    ={{{{x},x},x},x}

[b]
============>>>

                Uncertainty
  <-Redundancy->^
    d  d  d  d  |          d  d             d  d
    #  #  #  #  |          #  #             #  #        
    c  c  c  c  |          c  c             c  c
    #  #  #  #  |          #  #             #  #   
    b  b  b  b  |    b  b  b  b             b  b       b  b  b  b
    #  #  #  #  |    #  #  #  #             #  #       #  #  #  #   
   {a, a, a, a} V   {a, a, a, a}     {a, b, a, a}     {a, a, a, a}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |&_|_ |  |       |#_|  |  |       |&_|_ |&_|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |&_|&_|&_|_      |&____|&_|_      |&____|&_|_      |&____|____
    |                |                |                |
    {x,x,x,x}        {x,x},x,x}       {{{x},x},x,x}    {{x,x},{x,x}}     
 
                                      c  c  c
                                      #  #  #      
          b  b                        b  b  b          b  b
          #  #                        #  #  #          #  #         
   {a, b, a, a}     {a, b, a, b}     {a, a, a, d}     {a, a, c, d}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |#_|  |&_|_      |#_|  |#_|       |  |  |  |       |&_|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |&_|&_|_ |       |#____|  |
    |     |          |     |          |        |       |        |
    |&____|____      |&____|____      |#_______|       |#_______|
    |                |                |                |
    {{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x}      {{{x,x},x},x} 

   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |#_|  |  |
    |     |  |  
    |#____|  |      
    |        |
    |#_______|
    |    
    {{{{x},x},x},x}
[/b]

A 2-valued logic is:

Code: 
    b   b 
    #   #    
    a   a     
    .   .   
    |   |   
    |&__|_   
    | 
 [b]   
    a   b     
    .   .   
    |   |  <--- (Standard Math logical system fundamental building-block) 
    |#__|   
    |
[/b]
 
Last edited:
  • #68
WWW said:
I know it, but in 1-dim all you can get is the shadow of what you can find between 1-dim and 2-dim, isn't it?

Please explain Why do you think they are not defined?

in order the answers are:

that's at best wrong, at worst completely meaningless.

where in this thread have you offered a definition of uncertainty or redundancy? and i don't mean their plain english meanings.

it is a courtesy whenever you introduce non-standard terms to explain them.

you have i believe offered a vague idea of one of them in some other thread, but it didn't explain it fully.
 
  • #69
WWW said:
No you did not, because in 1-dim(=x-dim) universe no point can be found as a result of (x-dim,y-dim) system.

Your (c,d)(a0,b0) example is a (x-dim,y-dim) --> 2-dim system, and only then you can show a
"2-d Math picture" of mandelbrot set (which has a fractal-dim between 1-dim and 2-dim).

Shortly speaking, in a 1-dim universe any y-dim reduced to x-dim.

Therefore d reduced to c and b0 reduced to a0, and you have no 2-dim Math picture of some Julia set.

You did not show it yet.


sentence one ahs no content as far as i can tell.

the mandelbrot set is a subset of R^2, or C, so the second bit is silly too.

third part? nope, nothing there that makes sense either (reduces?)

and finally, show what?
 
  • #70
It is very simple matt,

Any b0 or d in Hurkyl's example is always 0 in a 1-dim universe, and if not then we are no longer in a 1-dim but in a 2-dim universe.

Therefore he gets (a0,0) or (c,0) 1-dim representation, which is definitely not a 2-dim representation of some Julia set.
 
Last edited:

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