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Hi, i d like to propose a simple ideas to help in the way to obtain genuine 3d mandelbroot.

In short, to work with a analogue of complex number,but for a 3-signed arithmetic ,too it use 3  unities, 3 dimensions ( six real dimensions), it use a 3-signed arithmetic instead of the usual 2-signed arithmetic, and the 3 unities work in mod 9 instead of mod 4.The 3-signed arithmetic is like the einsenstein integers ( http://en.wikipedia.org/wiki/Eisenstein_integer) but with a important change of views. Well, here is my paper (in spanish):

- http://vixra.org/pdf/0911.0034v1.pdf  **pag 75

If there is interest i ll explain in simple way.

Regards

I'm interested but my Spanish is even worse than my French...

well, we begin, the n-signed arithmetics work with mods instead of signs. in mod 2, or 2 signs, it s  the 2-signed arithmetic, where if it s congruent to 1(mod 2) is negative, and if congruent to 0 (mod 2) if positive. The numbers with the same sign-values (or values of the mod ) are aditioned. In the substraction, the number with the same sign-value is aditioned, and later it is subtract to all the piles the minor number between the piles:

we abreviate   a mod b like [a¬b], but in the case of mod 2, we ll omit the 2

..1+(-5)+2+(-1)=3+(-6)=(-3)

equal to:

..[0]1+[1]5+[0]2+[1]1=[0]3+[1]6=[0](3-3)+[1](6-3)=[1]3

oh, I see...

you know, multi-signed algebra already was tried.
Maybe your investigation leads to more interesting stuff but the results 'till now where rather not overly stunning...

(just as an algebra, I silently discussed with David Making, where I just redefined i to be: i³=-1. That might actually be not that much away from a multi-sign algebra and it actually did give similarish results...)

But who knows... multi-signed algebra could hold some potential not yet being found. So, let's go on anyway :)

i forgot, in [a¬b] , a is the sign-value and b is the quantity of signs

Like you can see, it more easy with signs, but before, we extend it to 3-signs.

-In the adition: they are aditioned in piles the numbers with same sign-value and in the substraction , all the numbers with same sign-value are aditioned in piles, and after, the minor number between the piles is substract to all piles

example:

again we omit in [a¬3], but now for the 3

where  ..[0]=+ , [1]=$ , [2]=#
...

ps: kram, in the case of 3-signed arithmetic is a plane, with triangular tillings, where, every sign, is a cubic root of 1 (in the complex) , but the 3-signed algebra and generalized, n-signed algebra, it dont go by the way like extension( in the real dimension sense exactly).... too is a isomorphism of complex numbers,i ll try explain to my way..

then example:

($3)+1+(#7)+3+($6)+(#5)=4+($9)+(#12)=(4-4)+$(9-4)+#(12-4)=$5+#8

equal to say:

[1]3+[0]1+[2]7+[0]3+[1]6+[2]5=[0]4+[1]9+[2]12=[0](4-4)+[1](9-4)+[2](12-4)=[1]5+[2]8

of general way to the substraction in 3-signed arithmetic:

a+#b+$c=(a-m)+#(b-m)+$(c-m)      where m is the min between a,b y c

if as a result of the substraction there are 2 number like limit, that is well because of the 3-signed arithmetic is represented in the plane, if a=b=c, then it result 0. Every sign is in the vertex of a unitary triangle.
.......

to the product, the signe-values are aditioned, example:

#3 x $4 = 12 equal to say:  [2]3 x [1]4 = [(2+1)mod3] 12 = ..[0]12

in the case of n-signed arithmetic, in the substraction, simply it s generalize the substraction for n, and for the product follow being correct, but with mod=n, where [0¬n] is the positive sign.
in the n-signed arithmetic, the geometry, is like n axes beginning of zero, and cross the unitary vertex of regular simplex in dimension n-1

then, now we can to advance to the tool can be posible create a kind of mandelbrot 3d:

the 3-signed arithmetic too it has posible analogs to the complex, but with 3 basis, where every number can be represented in a plane, then result a total of 6 real dimensions, with a,b,c  belong to 3-signed arithmetic:

L=a+bq+cp     where:  p^1=p         p^2=#q       p^3=#1
                                      p^4=#p      p^5=$q        p^6=$1
                                      p^7=$p       p^8=q          p^9=1

con   pq=qp=1

from notable product analog    (a+b+c) (a+#b+$c) (a+$b+#c) = a^3+b^3+c^3-3abc
we obtain the norma of this algebra     r^3= a^3+#b^3+$c^3-3abc

I wait this algebra can be a posibility to generate some kind of 3d mandelbrot.
some question, critique, failure or advise,??

regards

Well, other way of posible mandelbrot is to use a 4-signed algebra, in its graphic, every sign is in a vertex of regular unitary tetrahedron, it s in mod 4, and the positive axis has like sign-value = 0. It s of form:

 i]a + [j]b + [k]c= M      con i,j,k belong to (0,1,2,3)  and  i different of j different of k

regards

Kujonal, this is not a multi signed arithmetic this is an operator arithmetic with some newly defined operators. Try to be a bit more explicit about the map from the arithmetic to the 2d and 3d representation. eg how does the 2d mandelbrot formula look?

hello jehovaja, really the positive and negative signs are simply arithmetic operators, or if not they dont work with numbers, my proposition, like you said it, it s a operator generalize ¨of a way¨ the concept of signs with the help of mod operator, where, the 2 signs is a particular case (mod =2), giving de option of to work with numbers or signs depending when it will be more easy to use.

I think that using numbers at least in 3d, for obtain in a simple way some kind of 3d fractal, giving 2 options:

the last that i propose, it was to use a 4-signed arithmetic that is represented by a tetrahedral tilling in 3d ( in mod 4), and it is of the form  ..i]a + [j]b + [k]c = M, 3 coordenates because of it is 3d space. and later to work it to generate fractals.

the first that i propose, it was use a 3-signed arithmetic, that is representated by a triangular tilling in 2d, of a form ..i]a + [j]b = S ( that result to be a isomorphism if complex plane), and later to extend it to obtain a of their posible analog complex( but for 3 signs , in six real dimensions ) and later to work it with this algebra to generate fractals ( iterating)

some other questions?

regards

A diagram would help here. I have quickly glanced through tour pdf and it is most interestingand along the lines i thought.

my brief summary is that inthe set of transforms there are unary operators and binary operators , in general algorithms. These algorithms are procedures for iteration processes in general. The binary operators are culturally determined iterations and represent developments of the counting iteration which i refer to as "+1". +1 is the procedure or operation or algorithm. The iteration is a repeated binary: region to be transformed +1 object gives transformed region which becomes region to be transformed.

Graphically, which means also geometrically and symbolically i can reduce the above algorithm to

0 + 1=> 0^t=0

Now 0^t is normally valued in 2 main ways graphically  by a shape say @ or | or phonemically by a name say Uno, Wahed , etc. There are other ways to value it but culturally we choose or associate names from a namespace or marks from a numeralspace.

By this iteration we culturally construct the natural or counting numbers so called.  I will use to refer to those.

Now later mathematicians agreed a construction  of the integers using a unary operator called sign and a new numeral 0. Again Geometrical thinking was used to make sense of this unary operator as well as to give an intuitive feel for its effects. Sign allowed the number line and axes and vectors to develop naturally and laid the ground for the union of geomtry and algebra through the coordinate system. However as i have already pointed out geometrical thinking lay behind the construction of the integers from . The unary operation sign takes a natural numeral and transforms it to ±that natural numeral,

graphically/symbolically  

n  sign=> ±n.

We then us e these to construct by seperating all the + and counting them, and all the - and counting them. Next we place 0 in the centre of the two piles/ strands/ groups etc and rule that if we subtract from the + strand we must pass through 0 to get to the - strand. The number line follows naturally from this by iteration.

The next so called advance in number was the operator i. This is a unary operator and it act on the natural numbers to do what? Again geometrical thinking made sense of it , but for the same reason to call it an extension to the number systems and thus a number lead to many problems and generated a rich vein of marhematical invention.

The operator i is a unary operator that transforms geometrically on the plane by rotating it  anticlockwise .

the ooperator sign is a unary operator that transforms geometrically on the plane by rotating it anticlocwise .

So there are a whole class of unary operators that transform geometrically on the plane by fractions of  .

I believe that what you are describing using mod arithmetic as a set definer are these other unary operators. Using them you have to define binary operators and algorithms for transforming them to our established frameworks. However you should be able to write a programme to do this in principle.

Now the big question? How useful are they for the topics you have explored? Are they more useful than matrices or in matrices? Do they form a Lie group and thus have a use in string theory? In fact do they form a ring or group or field in combination with their binary operators?

hello jehovaja, you caught quick the idea of the notation, we follow :

-i dont unsderstand the penultimate paragraph very well because of my english, but which binary operator do i have to define?  :hmh:
-i dont know their usefulness, i hope they have. :D
-i think more useful in matrix, a strange question  :-X
-what s a lie group?  :o
-well, if that s it i think, i see conmutativity and asocitivity in they, at least, intuitively , or, how would the no-conmutativity be obtained without to add some more?

regards

ps: the complex number ( in mod 4)  can to be seen like 4 axis, every axis with a value from 0 until 3, but the aditional rule, where, the number of the form 2n (0,2) is substract between their, equal the number of the form 2n+1 (1,3) between their, that is, 4 axis, opposites 2 to 2 (2-signed). ( complex number and analogous complex of the 3-signed aritmetic could belong to the group of n^2)...

good nights
 

Thinking about whether axes are fundamental to unary operators. I do not think so. I think axes are fundamental to geometry and even though euclid did not particularly mention them they are implicit in what he was describing. Their implicit nature in geometry was made explicit by algebra. This is because algebra is about the syntax and syntax forms of any ARITHMETIC and the syntax of the outcome of any manipulation / transform of that arithmetic. Because of algebra we can communicate alternative arithmetics and introduce and define new transforms as well as describe in revealing ways old arithmetic schemes.

So mod(0)  define as {0,1,2,3,....} : addition and subtraction (multiple addition and multiple subtraction methods) as in everyday early centuries arithmetic.

mod(1)   define as {0} : addition and subtraction etc as in modern definition of nullity

mod(2)    define as {0,1} : addition and subtraction as in binary arithmetics and boolean algebras and modifications.

mod(3)    define as {0,1,2} : http://en.wikipedia.org/wiki/Modular_arithmetic

mod(4)    define as {0,1,2,3}

mod(5)    define as {0,1,2,3,4}                       Just in passing these sets may be useful for describing truncated iterations.

mod(6)    define as {0,1,2,3,4,5}

mod(n)    define as {0,1,2,3,.......,n-1}

All of these are algebraic definitions from which the arithmetics are prescribed for integers.

Of course we can extend this to the real numbers and then we would have formal partitioning of the reals into equivalence sets. This may be useful in describing fractal relationships at different levels, and may be what Musean numbers are attempting to describe.
Now the term number can be seen as a hangover from the historic arithmetic connection, but when these structures are aplied outside of a quantitative scheme the term number becomes misleading. So i generalise it to numeral.  

I also generalise the notion of sign to signal. IT IS EXACTLY THE SAME AS  using superscription subscription index tilde etc to distinguish a different but related denotation/ definition. In kujonai method he uses the mod set numerals as signals placing them in [], amongst other notations.

So in the case of mod(2)

define addition   0 + 0 = 0                             define multiplication      0 * 0 = 0
                       0 + 1 = 1                                                              0 * 1 = 0
                       1 + 0 = 1                                                              1 * 0 = 0  These used for logic applications.
                       1 + 1 = 0                                                              1 * 1 = 1

I will use the add definition for the signal manipulation.

write u₂, v₂    {0,1} for the general unary operator from mod(2).

 u₂ therefore has the numeral values 0 or 1.

Now for any a,b  the natural numbers    or a,b   lines in  Euclidean space, a || b, a units and b units in length.
 
define u₂a = 0]a or 1]a                  geometrically    u₂a = a or a rotated about a point O
and     u₂0]a =1]a or 0]a                                      u₂a = a rotated or a
and     u₂1]a =0]a  or 1]a...                                  u₂a rotated = a or a rotated

specifically

if u₂a = 0]a then u₂0]a =(u₂ + u₂)a =(0 + 0)]a

if u₂a = 1]a then u₂0]a =(u₂ + v₂)a =(1 + 0)]a

if u₂a = 0]a then u₂1]a =(u₂ + v₂)a =(0 + 1)]a

if u₂a = 1]a then u₂1]a =(u₂ + u₂)a =(1 + 1)]a
Define a binary operation for addition as follows

u₂a + v₂b                                            u₂a + v₂b = length a followed by length b

if u₂ = v₂ then = u₂(a + b)                     lengths added in direction a or in a rotated
if u₂ v₂ then = u₂a + v₂b                   lengths added in opposite directions by moving in direction a followed by a rotated or vice versa

There is a procedure that collects like things together which applies in algebraic manipulation which allows the first type of addition to be applied to collections of the same types reducing any form of addition to the second type .

Kujonai now adds another rule

pick the smallest of {a,b} call it s
 Perform a binary subtraction on the numerals a,b to give a value to u₂a + v₂b

u₂a + v₂b  = u₂(a-s) + v₂(b-s)
                     = either  u₂(a-s) or v₂(b-s)

Define multiplication as follows

u₂a * v₂b  = u₂+ v₂]ab

if u₂ = v₂ then = 0]ab

if u₂ v₂ then = 1]ab,

There is a logarithmic relationship to unary operators acting on each other in that apparent products have indices that are added mod(n).

So the algebraic description does not rely on axes, and so has a wider application. The signals could be tones or radio signals for example.

signals, that is, a posible practical aplications??

So why the minimum rule for subtraction in mod(3)?  And why stop when one pile is zeroed?  Why relate this to the plane as eisenstein numbers are so called complex plane coordinates not natural numerals as per your illustration?

The unary operator mod(3) would rotate an axes through so the signal is different and subtracting or adding to different piles is not the same as in mod(2) , eg 1]3 + 0]5  = 1]3-1]3 + 0]5 + 1]3 = 0]5 + 0]-3 where - is a binary operation.

the substraction stops when a pile of the 3 is zero, because of in the plane are necesary only 2 coordenates, the eisenstein numbers are alternative form of the 3-signed arithmetic, more exactly a kind of polinomial form of 3-signed arithmetic( at least i think that), and is too a particular case (n=3) , the only difference is that the mod-operator is tranfered to the exponent. like it follow:

 0]1+1]1+2]1=0                1+v+v^2=0                   to  remember:  0]1=+1     and v is a complex cube root of unity

(1]1)^1=1]1                     v^1=v

(1]1)^2=2]1=-1]1-1         v^2=-v-1

(1]1)^3=3]1=1                v^3=v^0=1

http://en.wikipedia.org/wiki/Eisenstein_integer

and simply, the 4-level of musean hypernumber w........  w=-v

http://en.wikipedia.org/wiki/Musean_hypernumber

regards

I have seen this rule on the polysigned site http://www.bandtechnology.com/PolySigned/PolySigned.html

But think it is one example of many ways to proceed with this unary operator exploration. It has its origins in geometrical constraints. In general unary operetors are defined for operating on one thing at a time. We must define binary trinary, ... many argument operators to deal with each signals combination with the others. The law of identity so called is such a many argument operator . I am not assuming that unary operators only act on the plane geometrically,and i am not assuming that the axes which may be used to illustrate a signal collection is the only way to arrange a signal collection. The chap at polysign points out the need to be clear what one is doing. So addition and subtraction may make sense within a signal group but across the signals he uses the notion of superposition.

The law of identity is thus placing equal magnitude signals in superposition we can define as eliminating the signals of that magnitude. What superposition is in a concrete example has to be explored for each mod(n).

With regard to complex nos or the set mod(4) should define the unary operator i in the plane (rotating by ) each time, but we still have to define a ninary operator for the superposition of signals and the structure of transformation of (a, b)

(http://bandtechnology.com/PolySigned/MiniS4SphereSquared.jpg)

As I see it Kujonai may still be developing his format.
I would pose the question:
   Are you attempting to generalize sign?
If so then I believe it is important to first accept that the real number is two-signed and that its elemental format is
   s x
where s is one of two signs; either - or +, and x is a magnitude. Thus a three-signed number can reuse this format with the addition of a new sign. I have chosen to remain consistent with the real number, though in higher sign systems the meaning of '+' then changes. Still, the mnemonic meaning of the signs is consistent, with the number of strokes it takes to draw each sign consistent with its modulo mechanics under product:
   -, +, *, #, ...
with one stroke, then two strokes, then three strokes, then four strokes, and then I run out of symbols, but this is acceptable, since upon getting that far it becomes apparent that a general component form can take over:
   ( 1.1, 1.2, 1.45, 2.34, 0.23 )
being a five-signed number in its most concrete instance. The position of the components are the sign positions. Heading back down to the real number we see that:
   ( 1, 1 ) = 0
which is simply stating that
   - 1 + 1 = 0
and it is this property; the balance of the signs; which yields dimension since the next form (P3, the three-signed numbers) takes the behavior
   ( 1, 1, 1 ) = 0
such that a simple expression in P3 as
   - 1.2 + 2.3
does not any longer cancel as did a real valued( two-signed; P2) expression. Hence the three-sign numbers are two-dimensional.
The four-signed numbers are three dimensional. P3 are the complex numbers, entirely consistent with the definition of the reals, merely up one in sign. Someone who comprehends this should be able to appreciate polysign, even just in the context of definition of real and complex numbers. They are united through polysign. The simplex coordinate system is natural to polysign. There is no need of any cartesian product construction. The generalization of sign immediatley yields dimensional behavior.

I see Kujonai somewhat taking a more complicated approach which may have some validity as the embedding of say a two-sign system within a three-sign system is as appropriate as embedding a line in a plane. Still, I would argue that the fundamental layout that I've presented lays beneath such an approach and will aid in the construction of that higher level concept.

 - Tim
 

Hi Tim. I would like to say that I really like your polysign construction, and like the idea of basing the coordinates on the simplex, of which there's one in every dimension.
You may also have noticed that P3 lacks some of the oddities of complex numbers. The +i and -i are completely interchangeable in complex numbers so the + means something very different to + on the reals. Also, unlike the reals, the complex units 1,-1,+i,-i are not equidistant, whereas the polysign units are always so.

However, I think the problem is the product function. I can't comment for >4 signs but the 4-signed numbers are basically just being complex numbers on a plane. That is why they produce the extruded mandelbrot here: http://www.fractalforums.com/theory/polysigned-mandelbulbs/?action=post;quote=9059;num_replies=4;sesc=cf68cf6d4b24e76361b129afd4de4d84

I think a probable better product rule would fold the 3d space onto itself, as show in this video http://www.fractalforums.com/3d-fractal-generation/a-new-3d-mandelbrot-like-fractal/?action=post;quote=9787;num_replies=10;sesc=cf68cf6d4b24e76361b129afd4de4d84
I'm not sure exactly the math but, the basic cases would be roughly like this:
(+)(+) = +
(-)(+) = -
(*)(+) = *
(#)(+) = #

(-)(-) = +
(*)(*) = +
(#)(#) = +

(-)(*) = #
(*)(#) = -
(#)(+) = *

the 4-signed and complex use mod 4, i tinhk the differences are in the substraction rules,
in the equation x^n-1

the complex see the answer in the regular poligons
the poly-signes see the answer in the  regular simplex

regards   :)

In the mod-2 system we see that '+' as sign=2
makes sense even within the product.

   - : sign = 1
   + : sign = 2

To me the ultimate solution is the zero sign:
   @ : sign = 0
This then becomes the universal superposition sign, which will not
change with a change in level succh as going from mod2 to mod3.
Unfortunately the '+' symbol will not maintain its mnemonic meaning in
generalized sign. The '@' sign takes its place, the '@' so chosen
because it is like a zero but cannot be confused with zero. With a zero
sign the sign product definition can be defined as modulo superposition
without the complication of the plus sign as sign=2. To me the sign
symbology should be clean mnemonically. This means
   @ : sign = 0 (like zero)
   - : sign = 1 (one stroke)
   + : sign = 2 (two strokes)
   * : sign = 3 (three strokes)
   # : sign = 4 (four strokes)
Anyhow we run out of decent symbols. There is little value in quibbling
over formats. Many formats are appropriate; from
   a + b i + c j
type of nonorthogonal unit vector to
   ( a, b, c )
type of coordinate format to
   @ a - b + c
polysign format to
   + a $ b # c
in opuesto aritmeticas format.

We have to coexist. In order to do so I suggest that in our
communication we each abandon our signs as much as possible and settle
into the coordinate representation where in the mod-4 system we see the
product
   (1,0,0,0) (z11,z12,z13,z14) = (z11,z12,z13,z14)
   (0,1,0,0) (z11,z12,z13,z14) = (z14,z11,z12,z13)
and sums are straightforwardly defined using the '+' symbol as
summation:
   (1,0,0,0) + (0,1,0,0) = (1,1,0,0)
The system is sealed to the graphical representation via the excellent
upuesto
   (1,1,1,1) = (0,0,0,0).
This is the 3D mathematics that I propose is the generalization of the
complex plane in 3D; the four-signed numbers.

i have mentioned this before, but orientation is often overlooked due to over familiarity with the terms dimensions. This is why i use the notion of unary operator as opposed to just sign and generalise the notion of sign to signal. Using the coordinate system of notation also implies a vector space or preferably a tensor space rather than a number or rather numeral. These are fundamental confusions i am exploring and trying to get round.

Anyway your assertion of a 3d number system requires that the axes are non orthogonal. They have to lie in the four corners of a tetrahedron if they are to be regularly spaced, otherwise the four axes fit the xy plane and one can use sign on both pairs of opposing axes. The plane as it turns out is used by many paired variables to geometrise their locii. 

Hello Kujonai and TimGolden,

I just discovered your previous post of 2009.
I proposed an algebra identical or very near of yours in 2011, we converge !
I thing that formalism is perhaps the best for a true (natural) generalization of complexes numbers a any dimensions, even if the pseudo-fractals calculated by that way are a little desappointing. It's certainly a new concept and representation of multicomplexes MCn.
See my post: "the ABSOLIEN numbers" on fractal forum : fractal maths/general discussion/

My proposition seems more simple because I use only vectors and matrix formalism of positive values unsigned (the sign simply corresponds to the position in the vector), and a very simple multiplication = descrete convolution. But it's sure that Tim Golden discribes the first that new formalism witch is equivalent to mine.
You can see my site on google too, in english or french:  https://sites.google.com/site/yannispicart/

Espero su opinion con mucho intéres.

Tim is about and infrequently active on this forum, but Kujonai i have not heard from for years. I suggest you message Tim if you want him to get back to you.
I have marked your thread as one to read as and when i get time Yannis, but that time is not just yet. Please do not feel ignored, Many of us are very busy and getting on with the outfall after the big Mandelbulb push when we spent so much of our lives collaborating that we nearly lost the balance in the rest of our lives!

This forum is truly of world class importance, the gateway into an understanding of our universe, and yet Chris struggles to keep it online and accessible! IT WAS EVER THUS, I AM AFRAID. THE ENTHUSIASM OF OUR MEMBERS has uncovered the deepest relationships we can know, but this is thought to be inconsequential because no one has figured out how to rob us of it by making big bucks from our findings.

I feel proud to be ablw to say "our" even though i did not originate or formulate the solutions, others did . Nevertheless the wealth of experise and understanding available in this forum is invaluable and i celebrate it and encourage all who can to donate what they can afford to kep it such a fun, creative artistic experimental place.