Title: Why z²+c and not a²+b ?
Post by: Chillheimer on February 13, 2016, 01:48:31 PM
Hi!
I was just wondering, is there a special reason why Mandelbrot used z and c as his variables. Do they stand for anything specific? Or did he just choose these 'randomly'?
cheers!
I was just wondering, is there a special reason why Mandelbrot used z and c as his variables. Do they stand for anything specific? Or did he just choose these 'randomly'?
cheers!
Title: Re: Why z²+c and not a²+b ?
Post by: claude on February 13, 2016, 03:29:30 PM
The key paper seems to be:
FRACTAL ASPECTS OF THE ITERATION OF z → Λz(1- z) FOR COMPLEX Λ AND z
Benoit B. Mandelbrot
Annals of the New York Academy of Sciences
Volume 357, Nonlinear Dynamics pages 249–259, December 1980
http://onlinelibrary.wiley.com/doi/10.1111/j.1749-6632.1980.tb29690.x/abstract
(but full paper is pricy so I didn't read it)
So originally it was
, possibly named after the logistic map.
z is common in complex analysis, presumably because you need a variable name and
is the next in the alphabet, also common is 
.
It turns out that all quadratic polynomials are equivalent to the z²+c form (conjugate is a keyword to search for), so the exact form doesn't matter so much when you get down to abstract mathematical properties.
FRACTAL ASPECTS OF THE ITERATION OF z → Λz(1- z) FOR COMPLEX Λ AND z
Benoit B. Mandelbrot
Annals of the New York Academy of Sciences
Volume 357, Nonlinear Dynamics pages 249–259, December 1980
http://onlinelibrary.wiley.com/doi/10.1111/j.1749-6632.1980.tb29690.x/abstract
(but full paper is pricy so I didn't read it)
So originally it was
z is common in complex analysis, presumably because you need a variable name and
It turns out that all quadratic polynomials are equivalent to the z²+c form (conjugate is a keyword to search for), so the exact form doesn't matter so much when you get down to abstract mathematical properties.
Title: Re: Why z²+c and not a²+b ?
Post by: Chillheimer on February 13, 2016, 04:24:22 PM
thank you claude..
so it doesn't really matter and it wouldn't be wrong to explain it as a->a²+b to people with even less math background than I have?
I noticed that some newbies are a bit confused by z & c and why they are important. so if you take a & b to explain the same thing it might be a bit easier..
so it doesn't really matter and it wouldn't be wrong to explain it as a->a²+b to people with even less math background than I have?
I noticed that some newbies are a bit confused by z & c and why they are important. so if you take a & b to explain the same thing it might be a bit easier..
Title: Re: Why z²+c and not a²+b ?
Post by: claude on February 13, 2016, 04:33:53 PM
exactly, as long as the symbols are consistent it doesn't make a difference, it's more about the shape of the equation. you could even write it :fiery:→ :fiery:² + :angel1:
conjugacy is a slightly more advanced concept, it's discussed here (but it doesn't seem very clear language) https://en.wikipedia.org/wiki/Complex_quadratic_polynomial
conjugacy is a slightly more advanced concept, it's discussed here (but it doesn't seem very clear language) https://en.wikipedia.org/wiki/Complex_quadratic_polynomial
Title: Re: Why z²+c and not a²+b ?
Post by: hobold on February 13, 2016, 06:42:11 PM
Some more math trivia/history regarding notation of Mandelbrot's famous formula.
TL;DR: the subsitution "z := a + b*i" for a complex number is older than Mandelbrot, and will keep showing up in the literature. Likewise "c" is a relatively common convention to hint that a number is constant within some context.
1.
Back when complex numbers were new (nope, that was not in ancient Greece, but more like renaissance Europe or shortly before), mathematicians (theoretical physicists, actually) were not yet sure about the best concept. Some thought that only the "imaginary unit", i.e. the square root of -1, was really new. They were happily writing sums of the form "a + b*i" (where i is shorthand for square root of minus one) and kept using the algebraic rules for real numbers.
Some other faction of mathematicians noticed that one could "hide" most occurrences of i by substituting "z := a + b*i" and then handling the placeholder z with only one or two additional rules. In other words, it was a decision between two minor inconveniences: either you had to drag around several extra "i" symbols, or you had to add a small number of algebraic rules.
2.
So why the letters a, b, and z? The true reason for a and b has not been traded on through history, but chances are they were picked because they are simply the first two letters of the alphabet, and mathematicians often start there just for convenience.
With z, it was a bit of a different story. The last letter of the alphabet also happens to be the first letter of the German word "Zahl" which means "number" in English. The two factions of differing notation that I mentioned earlier were, simply speaking, disciples of different influential mathematicians of their time. The ones who favoured the idea that complex numbers should be regarded as standalone units (as opposed to an aggregate of two real numbers and the imaginary unit) were educated in the tradition of German mathematicians, and so they picked z to signify that a complex number is one single "Zahl", i.e. a single number.
3.
Finally we get to "+ c". Initially, Mandelbrot studied Julia sets, which are based on the known "z_n+1 := z_n + c" formula. But with Julia sets, the value for c is constant for any given set. It is a relatively common convention among mathematicians to use the letter "c" or "k" for a number that is constant within some context (but does vary in a bigger picture).
4.
With the Mandelbrot set, the value of c isn't actually constant. But Mandelbrot hat written down many results from his research into Julia sets. In mathematical papers, you don't change notations on a whim - you don't want to confuse the readers. So the existing body of Mandelbrot's work was setting a precedent.
So, summing up: Mandelbrot was probably following professional habits when he picked a formal notation for the iteration of the Mandelbrot set. There is no special design or enlightenment to this particular formula. However, people with education in higher math will recognize a detail or two (even the "bug" with c not actually being a constant :-).
TL;DR: the subsitution "z := a + b*i" for a complex number is older than Mandelbrot, and will keep showing up in the literature. Likewise "c" is a relatively common convention to hint that a number is constant within some context.
1.
Back when complex numbers were new (nope, that was not in ancient Greece, but more like renaissance Europe or shortly before), mathematicians (theoretical physicists, actually) were not yet sure about the best concept. Some thought that only the "imaginary unit", i.e. the square root of -1, was really new. They were happily writing sums of the form "a + b*i" (where i is shorthand for square root of minus one) and kept using the algebraic rules for real numbers.
Some other faction of mathematicians noticed that one could "hide" most occurrences of i by substituting "z := a + b*i" and then handling the placeholder z with only one or two additional rules. In other words, it was a decision between two minor inconveniences: either you had to drag around several extra "i" symbols, or you had to add a small number of algebraic rules.
2.
So why the letters a, b, and z? The true reason for a and b has not been traded on through history, but chances are they were picked because they are simply the first two letters of the alphabet, and mathematicians often start there just for convenience.
With z, it was a bit of a different story. The last letter of the alphabet also happens to be the first letter of the German word "Zahl" which means "number" in English. The two factions of differing notation that I mentioned earlier were, simply speaking, disciples of different influential mathematicians of their time. The ones who favoured the idea that complex numbers should be regarded as standalone units (as opposed to an aggregate of two real numbers and the imaginary unit) were educated in the tradition of German mathematicians, and so they picked z to signify that a complex number is one single "Zahl", i.e. a single number.
3.
Finally we get to "+ c". Initially, Mandelbrot studied Julia sets, which are based on the known "z_n+1 := z_n + c" formula. But with Julia sets, the value for c is constant for any given set. It is a relatively common convention among mathematicians to use the letter "c" or "k" for a number that is constant within some context (but does vary in a bigger picture).
4.
With the Mandelbrot set, the value of c isn't actually constant. But Mandelbrot hat written down many results from his research into Julia sets. In mathematical papers, you don't change notations on a whim - you don't want to confuse the readers. So the existing body of Mandelbrot's work was setting a precedent.
So, summing up: Mandelbrot was probably following professional habits when he picked a formal notation for the iteration of the Mandelbrot set. There is no special design or enlightenment to this particular formula. However, people with education in higher math will recognize a detail or two (even the "bug" with c not actually being a constant :-).
Title: Re: Why z²+c and not a²+b ?
Post by: Chillheimer on February 14, 2016, 12:30:10 AM
Thank you hobold. Worthy answer for an encyclopedia.. :)
Anyways, I think it's obvious that we should agree to use this in the future:
Anyways, I think it's obvious that we should agree to use this in the future:
:fiery:→ :fiery:² + :angel1: