On Thursday, February 17, 2022 at 9:45:10 PM UTC+1, tita...@gmail.com wrote:

"Extending complex number"

You are aware of quaternions and octonions?

https://en.wikipedia.org/wiki/Quaternion
https://en.wikipedia.org/wiki/Octonion

On 3/15/2022 2:16 PM, Fritz Feldhase wrote:
> On Thursday, February 17, 2022 at 9:45:10 PM UTC+1, tita...@gmail.com wrote:
>
> "Extending complex number"
>
> You are aware of quaternions and octonions?
>
> https://en.wikipedia.org/wiki/Quaternion
> https://en.wikipedia.org/wiki/Octonion

Probably but those are 4- and 8-dimensional respectively.
Plus the first component is different since it is like real numbers
while the other components are different are more like the imaginary
component of complex numbers.

What PengKuan apparently wants to do is create a 3 dimensional system,
corresponding to 3D space along x,y,z axes, like complex numbers to a
plane and x,y axes. Even that isn't quite right since the real and
imaginary parts can't be interchanged by swapping the x, y axes.

Perhaps quaternions if ignoring the first/real component? Or forcing it
to be 0?

On Tuesday, March 15, 2022 at 4:33:11 PM UTC-4, Michael Moroney wrote:
> On 3/15/2022 2:16 PM, Fritz Feldhase wrote:
> > On Thursday, February 17, 2022 at 9:45:10 PM UTC+1, tita...@gmail.com wrote:
> >
> > "Extending complex number"
> >
> > You are aware of quaternions and octonions?
> >
> > https://en.wikipedia.org/wiki/Quaternion
> > https://en.wikipedia.org/wiki/Octonion
> Probably but those are 4- and 8-dimensional respectively.
> Plus the first component is different since it is like real numbers
> while the other components are different are more like the imaginary
> component of complex numbers.
>
> What PengKuan apparently wants to do is create a 3 dimensional system,
> corresponding to 3D space along x,y,z axes, like complex numbers to a
> plane and x,y axes. Even that isn't quite right since the real and
> imaginary parts can't be interchanged by swapping the x, y axes.
>
> Perhaps quaternions if ignoring the first/real component? Or forcing it
> to be 0?

I think Olariu had something like KP's stuff. Didn't he have a cosexp function?
I don't think he had the oddities of KP though. Then again, the angular purity is quite good here. Why hasn't anybody done it before? It seems very sensible. All the angles carry in the exponential form. The generalization holds, yeah?

e to the ( a1 i + a2 i i + a3 i i i ... )

to me would be the pure form. I'm not so sure what KP will say, though.

Binary truth is somewhat a misnomer. The continuum upon the two-fold versus the continuum upon the three-fold puts the two-fold in a compromised position.

On Tuesday, March 15, 2022 at 9:33:11 PM UTC+1, Michael Moroney wrote:

> What PengKuan apparently wants to do is create a 3 dimensional system,
> corresponding to 3D space along x,y,z axes [etc,]
>
> Perhaps quaternions if ignoring the first/real component? Or forcing it to be 0?

"__Quaternions and the space geometry__

The vector part of a quaternion can be interpreted as a coordinate vector inIR^3 therefore, the algebraic operations of the quaternions reflect the geometry of IR^3, Operations such as the vector dot and cross products can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise. A useful application of quaternions has been to interpolate the orientations of key-frames in computer graphics." (Wikipedia, Quaternions)

Another source (actually, a quite interesting article from July 05, 1998):

"Implementing a floating camera that tracks and rotates about a real-time 3D character is just one of the many reasons you should understand the importance and implementation of quaternions. [...]

Another common use for quaternions is in military and commercial flight simulators. Instead of manipulating a plane's orientation using three angles (roll, pitch, and yaw) representing rotations about the x, y, and z axes, respectively, it is much simpler to use a single quaternion.

Many games coming out this year will also feature real-world physics, allowing amazing game play and immersion. If you store orientations as quaternions, it is computationally less expensive to add angular velocities to quaternions than to matrices."

Source: https://www.gamedeveloper.com/programming/rotating-objects-using-quaternions

Le mardi 15 mars 2022 à 18:17:18 UTC+1, timba...@gmail.com a écrit :
> Another quick thought and it is just my opinion, but you are roaming back and forth between (h,i,j) in R and R1 + R2i + R3j, and this you may be consistent at, but may be annoying particularly to people of the quaternion variety, which is likely a large part of your readership future or present. I do see the amount of work you have into your document and I wouldn't rush it in, but rather think it over and try to optimize notation. Possibly you are correct to stay where you are. In polysign the ordered form does work though the values are not signed at all. So a value like (1.2,2.3,3.4) is unambiguously a value in P3 which is equivalent to -1.2+2.3*3.4, or a value *1.2-2.3+3.4, depending on where the identity product goes. These sorts of notational difficulties should be repeatedly addressed consistently so that confusion cannot arise. I don't need the ordered form very often and there is actually a need of a zero sign '@' in polysign so that the identity sign does not shift around, at which point the identity position is (1,0,0,...) as in:
> (1,0,0,0,...) z = z
> or:
> ( @ 1 ) z = z
> In P2 this zero sign '@' is equal to '+' with two strokes to draw the sign and modulo two these are the same thing yet in P3 the '*' sign with three strokes to draw it is the same as the zero sign. If polysign were ever seriously accepted this zero sign would need to be introduced, yet it can coexist with the traditional real value's usage of '+', which is really merely coincidence. And of course our usage of '+' to mean summation has to be carefully dealt with in polysign; the '@' symbol is unambiguously the sum in Pn using what is left of traditional notation.
>
> This idea that notation matters: to some they will cast it off, yet our usage of notation with a sum requiring a symbol yet the product requiring mere juxtaposition is a matter of convention only. At some level that we communicate here in a singular series of characters could be an impediment to such things as general dimensional notions as we are working on. The Cartesian product as well proves to be problematic. Functional analysis relies heavily on a Cartesian product to define sum and product within the modern math curriculum. Also these are defined as binary operators whereas the n-ary form would be more general. Strangely enough in the n-ary form we see the singleton show its face as well. Translating the usual binary operator into n-ary would suggest that a sum of three values is a three dimensional problem in say RxRxR for real values. I assure you that this is not the case, and that even the binary form is a fraud. Closure as a mathematical concept requires just one set to work in. For instance if I offer you a sum of a and b yielding c where is the need of a Cartesian product?
> a+b=c ; a in R, b in R, c in R.
> It's one set only.
>
> Ooh, it seems somebody has been at this page:
> https://en.wikipedia.org/wiki/Ring_(mathematics)
> fixing it up to hide this problem.
>
> Still, here we see the old problem:
> https://en.wikipedia.org/wiki/Binary_operation
>
> and the earlier page does reference the binary operator still. I know these concepts are subtle, but if there is room for reinterpretation at some fundamental level then possibly the upset will unshuffle the deck, so to speak. The accumulation in mathematics is extremely annoying and I certainly prefer to distance myself from it as you do, though we can't help but bump our heads straight back into it. Possibly the upside is that it is a tolerant place, and academia needs room enough for so many PhDs now that this tolerance could be obscenely permissive. On the downside claims of perfection within mathematics are definitely false. Escapism is more the truth of the matter. Escape into your specialty mostly with no regard for physical correspondence. For those who want physics to be born this cannot be good. Still as individuals room for everybody is established.
>
> I don't think this tension will go away until we have a TOE or some unification and even then the idea that such a theory will carry parallel interpretations is entirely plausible. So a necessary tension between mimicry and freedom somehow will lead us into a progression, but if an invalid assumption lays beneath our works than all are flawed. I do see the Cartesian product as a candidate for this analysis. Since when are you allowed two copies of the real value? since the Cartesian product, really. So then what are two independent systems doing tied together at zero and perfectly perpendicular to each other? Is this really a natural occurrence? And here polysign do form an entirely different means of achieving the general dimensional situation. Indeed, the word 'dimension' as tied to the real line has to be carefully noted as old language. We can still use it, but the nonorthogonal coordinate systems that polysign natively develops are based on a balance amongst signs.

“Another quick thought and it is just my opinion, but you are roaming back and forth between (h,i,j) in R and R1 + R2i + R3j, and this you may be consistent at, but may be annoying particularly to people of the quaternion variety, which is likely a large part of your readership future or present. I do see the amount of work you have into your document and I wouldn't rush it in, but rather think it over and try to optimize notation. Possibly you are correct to stay where you are. ”

You are always sharp in your thinking. Indeed, it is bad to be annoying and go back and forth when presenting my work. This means that I have not well adapted my thinking to the way the readers think. After reading your comment, I have written an explanation for you and the others to make clear the main idea and concept of my multidimensional complex system, which is in fact based on multiplication rather than on addition. The text is here for download.
https://drive.google.com/file/d/15ATCrPo8c9PCXVbWZDOCDcdPCITByhuq/view?usp=sharing

The main idea is to represent a 3D vector as the product of two 2D complexes. For example, A is a 2D complex in the horizontal plane, B a complex in the vertical plane. Then, the 3D complex that is defined in the 3D space with the horizontal and vertical planes is the product AB.
A=e^(i*theta), B=e^(j*psi), then, AB=e^(i*theta+j*psi)

This way, 3D complex can be multiplied. In the same way, nD complex is the product
e^(a1) e^(a2) e^(a3)… e^(an).
Such construction makes the nD complex number apt to be multiplied.

As for the vector in space like R1 + R2i + R3j, it is an expression for AB=e^(i*theta+j*psi). Because the angular form is suitable for multiplication and the vector form for addition, we have no choice when doing multiplication and addition, but to use the appropriate form.

The angular form and the vector form represent the same and unique point in space. I have put the clear explanation in the link above.

“I understand that your math comes out of the angular form and I've sort of been pushing on the vector form. At some level the ultimate form is the single variable form z so that we can simply discuss z1z2 for instance as the product and z1+z2 as the sum. We don't generally expect any surprises in the sum but the product always seems to be of interest. For instance division as the reverse operator of the product and its mechanics can get difficult.”

In fact, a geometrical point in space is one thing and a mathematical representation or expression is another. e^(i*theta) represents a point say A, and cos(theta)+ sin(theta)i represents a point B. It happens that A=B, but not necessarily always the case. This is a construction by human, not a natural object like rock or tree.

As e^(i*theta) and cos(theta)+ sin(theta)i are not the same algebraic thing, they do not necessarily bear the same operation, such as z1z2 or z1+z2. So, z3(z1+z2) is not the same thing as z3z1+ z3z2. However, in geometry, they could represent the same point in space because human has defined so.

“This certainly is true of polysign. Only just recently did division get an algorithm, and it is not as if it is easy. The old familiar behavior:
| z1 z2 | = | z1 | | z2 |
does not hold in polysign and there are nonzero products yielding zero such as:
( - 1 + 1 )( + 1 # 1 ) = 0 [P4]”

My system of multidimensional complex deals well division because
A=e^(i*theta), B=e^(j*psi), A/B= e^(i*theta) e^(-j*psi)

“Supposedly Dedekind proved that associative ( z1(z2z3) = (z1z2)z3 ) systems will always portray images of R and C as RxC (Cartesian product), CxCxR, CxCxC, etc., these being higher dimensional forms.”
My system of multidimensional complex is also well associative for multiplication because
e^(a1) e^(a2) e^(a3)=[ e^(a1) e^(a2) ] e^(a3)= e^(a1) [e^(a2) e^(a3) ]

“It turns out that the interpretation of the continuum as if resolved by the rational value is problematic according to my own analysis.”
It seems that you are into research in continuum. Does this mean that you are interested in the definition that the line of real is a continuum and that the set of all the real numbers is the continuum? From my research, I have got the conclusion that the set of all the real numbers is not a continuum but a discrete set.

“…who cares to question a value such as
2.01
and the mathematician's version of this value versus the engineer's version..”
Because the set of all the real numbers is discrete, mathematician will agree with engineer about 2.01

Le mardi 15 mars 2022 à 19:16:27 UTC+1, Fritz Feldhase a écrit :
> On Thursday, February 17, 2022 at 9:45:10 PM UTC+1, tita...@gmail.com wrote:
>
> "Extending complex number"
>
> You are aware of quaternions and octonions?
>
> https://en.wikipedia.org/wiki/Quaternion
> https://en.wikipedia.org/wiki/Octonion
Yes. I cited them in the original post.

Quaternion has multiplication table, ij, jk, ki ijk. And the multiplication of 2 Quaternion does not equal the multiplication of two 4D complex.

Le mardi 15 mars 2022 à 21:33:11 UTC+1, Michael Moroney a écrit :
> On 3/15/2022 2:16 PM, Fritz Feldhase wrote:
> > On Thursday, February 17, 2022 at 9:45:10 PM UTC+1, tita...@gmail.com wrote:
> >
> > "Extending complex number"
> >
> > You are aware of quaternions and octonions?
> >
> > https://en.wikipedia.org/wiki/Quaternion
> > https://en.wikipedia.org/wiki/Octonion
> Probably but those are 4- and 8-dimensional respectively.
> Plus the first component is different since it is like real numbers
> while the other components are different are more like the imaginary
> component of complex numbers.
>
> What PengKuan apparently wants to do is create a 3 dimensional system,
> corresponding to 3D space along x,y,z axes, like complex numbers to a
> plane and x,y axes. Even that isn't quite right since the real and
> imaginary parts can't be interchanged by swapping the x, y axes.
>
> Perhaps quaternions if ignoring the first/real component? Or forcing it
> to be 0?
Thanks.

I have written an explanation to make clear the main idea and concept of my multidimensional complex system, which is in fact based on multiplication rather than on addition. The text is here for download.
https://drive.google.com/file/d/15ATCrPo8c9PCXVbWZDOCDcdPCITByhuq/view?usp=sharing

The main idea is to represent a 3D vector as the product of two 2D complexes. For example, A is a 2D complex in the horizontal plane, B a complex in the vertical plane. Then, the 3D complex that is defined in the 3D space with the horizontal and vertical planes is the product AB.
A=e^(i*theta), B=e^(j*psi), then, AB=e^(i*theta+j*psi)

This way, 3D complex can be multiplied. In the same way, nD complex is the product
e^(a1) e^(a2) e^(a3)… e^(an).
Such construction makes the nD complex number apt to be multiplied.

As for the vector in space like R1 + R2i + R3j, it is an expression for AB=e^(i*theta+j*psi). Because the angular form is suitable for multiplication and the vector form for addition, we have no choice when doing multiplication and addition, but to use the appropriate form.

The angular form and the vector form represent the same and unique point in space. I have put the clear explanation in the link above.

Le mercredi 16 mars 2022 à 01:02:26 UTC+1, timba...@gmail.com a écrit :
> On Tuesday, March 15, 2022 at 4:33:11 PM UTC-4, Michael Moroney wrote:
> I think Olariu had something like KP's stuff. Didn't he have a cosexp function?
> I don't think he had the oddities of KP though. Then again, the angular purity is quite good here. Why hasn't anybody done it before? It seems very sensible. All the angles carry in the exponential form. The generalization holds, yeah?
>
> e to the ( a1 i + a2 i i + a3 i i i ... )
>
> to me would be the pure form. I'm not so sure what KP will say, though.
>
> Binary truth is somewhat a misnomer. The continuum upon the two-fold versus the continuum upon the three-fold puts the two-fold in a compromised position.
"e to the ( a1 i + a2 i i + a3 i i i ... ) "
Yes. I think this is correct. My pure form is e^(ia+jb+kc....)

Le mercredi 16 mars 2022 à 02:31:38 UTC+1, Fritz Feldhase a écrit :
> On Tuesday, March 15, 2022 at 9:33:11 PM UTC+1, Michael Moroney wrote:
>
> > What PengKuan apparently wants to do is create a 3 dimensional system,
> > corresponding to 3D space along x,y,z axes [etc,]
> >
> > Perhaps quaternions if ignoring the first/real component? Or forcing it to be 0?
> "__Quaternions and the space geometry__
>
> The vector part of a quaternion can be interpreted as a coordinate vector inIR^3 therefore, the algebraic operations of the quaternions reflect the geometry of IR^3, Operations such as the vector dot and cross products can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise. A useful application of quaternions has been to interpolate the orientations of key-frames in computer graphics." (Wikipedia, Quaternions)
>
> Another source (actually, a quite interesting article from July 05, 1998):
>
> "Implementing a floating camera that tracks and rotates about a real-time 3D character is just one of the many reasons you should understand the importance and implementation of quaternions. [...]
>
> Another common use for quaternions is in military and commercial flight simulators. Instead of manipulating a plane's orientation using three angles (roll, pitch, and yaw) representing rotations about the x, y, and z axes, respectively, it is much simpler to use a single quaternion.
>
> Many games coming out this year will also feature real-world physics, allowing amazing game play and immersion. If you store orientations as quaternions, it is computationally less expensive to add angular velocities to quaternions than to matrices."
>
> Source: https://www.gamedeveloper.com/programming/rotating-objects-using-quaternions
Quaternions system rotates an object around an axis. This is the use of it in computer graphics and flight simulators. Because it rotates the entire airplane in space.

My system rotate a single point. Like 2D complex, when multiplied, the point rotate.

On Friday, March 18, 2022 at 2:26:19 PM UTC-4, tita...@gmail.com wrote:
> Le mercredi 16 mars 2022 à 01:02:26 UTC+1, timba...@gmail.com a écrit :
> > On Tuesday, March 15, 2022 at 4:33:11 PM UTC-4, Michael Moroney wrote:
> > I think Olariu had something like KP's stuff. Didn't he have a cosexp function?
> > I don't think he had the oddities of KP though. Then again, the angular purity is quite good here. Why hasn't anybody done it before? It seems very sensible. All the angles carry in the exponential form. The generalization holds, yeah?
> >
> > e to the ( a1 i + a2 i i + a3 i i i ... )
> >
> > to me would be the pure form. I'm not so sure what KP will say, though.
> >
> > Binary truth is somewhat a misnomer. The continuum upon the two-fold versus the continuum upon the three-fold puts the two-fold in a compromised position.
> "e to the ( a1 i + a2 i i + a3 i i i ... ) "
> Yes. I think this is correct. My pure form is e^(ia+jb+kc....)

Well, as you form shifts into an indexed form of some sort I think you'll have a better general dimensional solution.
Whether it works all the way down or not: it seems you'll bottom out in 2D.
The real number as a 1D by definition concept is always sitting there under the hood.
That the general form will develop into a polynomial is entirely consistent with rather a lot of mathematics.
Most would wrap i^N around to unity where N is near your max dimension.
I don't know what happened to Olariu's work and his cosexp function. I can't find it anymore.
It definitely was on arxiv.org back in time. Possibly I have a typo here. It was awfully close to your stuff.
Ah, I've found a paper that references a book of his heavily: https://arxiv..org/pdf/1801.03773.pdf

In response to your other reply on the engineer's versus the mathematician's interpretation of a value like 2.01:
when an engineer sets a voltage on a power supply, say, to 2.01 Volts DC they are not claiming perfection. They have worked only to a a certain resolution and know better than to chase any finer.

The mathematician on the other hand sees the value as pristine and perfectly defined and without any error. These are starkly differing interpretations of the continuum. One version exposes the gray nature of the number whereas the other denies the existence of such. The notion of epsilon/delta refinement has only gone on within the value 2.01 a few digits. There is a sort of refusal of mathematicians to admit that chasing digits is what epsilon/delta does, yet if we were to literally carry out the computation of say the square root of two this is exactly as it will play out. Of course the inane nature of such a practice while unity is so arbitrarily specified on any continuum further exposes the mathematician's absurdity.
Any ability to work with such a fine value on the continuum or computationally is badly compromised.

Strangest of all, upon realizing that the rational value as composed of values and operators (the division operation which arguably is not even fundamental) we see that the value 2.01 can in fact take a natural representation, that being 201, and of course the decimal point does require some augmentation to that but it is certainly marking a new form of unity; a secondary unity, so to speak. So by doing away with the rational analysis which here has two strikes against it we land in a simplification that leaves us with our radix ten values fully intact and hardly any change from the natural value has occurred. Certainly we will not claim that two such in division magically breed the continuum. We could have stayed within the first radix but instead your real value as developed by mathematicians which is at the heart of your own number system (and mine, in the past) has reradixed into some sort of escapist cloud of complexity shrouded only by the irrational value, which we are supposed to stuff down next. In hindsight the mixture of operators and values as if they are pure values is to blame, and you will not find any mathematician here who will confront this analysis. This is the third strike. Sadly, the game is rigged. Well, of course you should take this as rhetoric and content. Still, the content is simply exposed.
Is 3/5 a value or is it two values and an operator?
On and on it can go and into subjects such as abstract algebra where operators are very carefully defined. Instances such as
a0 + a1X
with real coefficients (forgoing that the real number is under attack already) exposes two operators which do not obey their own closure requirements. The X you see is somehow an undefined thing within abstract algebra. Somehow the polynomial form as the number is evident, yet as to which came first we'd have to give the number that status.

Anyway, it seems that the physicist happily rests his head on the mathematician's real value, but I'm pretty sure what's there is different than the great defense that real analysis provides. And of course these subjects are upheld under the threat of failure. Mimicry is heavily enforced within academia. Here we are outside of that system.

Cheers PK

Le dimanche 20 mars 2022 à 21:16:59 UTC+1, timba...@gmail.com a écrit :
> On Friday, March 18, 2022 at 2:26:19 PM UTC-4, tita...@gmail.com wrote:
> Well, as you form shifts into an indexed form of some sort I think you'll have a better general dimensional solution.
> Whether it works all the way down or not: it seems you'll bottom out in 2D.
> The real number as a 1D by definition concept is always sitting there under the hood.
> That the general form will develop into a polynomial is entirely consistent with rather a lot of mathematics.
> Most would wrap i^N around to unity where N is near your max dimension.
> I don't know what happened to Olariu's work and his cosexp function. I can't find it anymore.
> It definitely was on arxiv.org back in time. Possibly I have a typo here. It was awfully close to your stuff.
> Ah, I've found a paper that references a book of his heavily: https://arxiv.org/pdf/1801.03773.pdf
>
> In response to your other reply on the engineer's versus the mathematician's interpretation of a value like 2.01:
> when an engineer sets a voltage on a power supply, say, to 2.01 Volts DC they are not claiming perfection. They have worked only to a a certain resolution and know better than to chase any finer.
>
> The mathematician on the other hand sees the value as pristine and perfectly defined and without any error. These are starkly differing interpretations of the continuum. One version exposes the gray nature of the number whereas the other denies the existence of such. The notion of epsilon/delta refinement has only gone on within the value 2.01 a few digits. There is a sort of refusal of mathematicians to admit that chasing digits is what epsilon/delta does, yet if we were to literally carry out the computation of say the square root of two this is exactly as it will play out. Of course the inane nature of such a practice while unity is so arbitrarily specified on any continuum further exposes the mathematician's absurdity.
> Any ability to work with such a fine value on the continuum or computationally is badly compromised.
>
> Strangest of all, upon realizing that the rational value as composed of values and operators (the division operation which arguably is not even fundamental) we see that the value 2.01 can in fact take a natural representation, that being 201, and of course the decimal point does require some augmentation to that but it is certainly marking a new form of unity; a secondary unity, so to speak. So by doing away with the rational analysis which here has two strikes against it we land in a simplification that leaves us with our radix ten values fully intact and hardly any change from the natural value has occurred. Certainly we will not claim that two such in division magically breed the continuum. We could have stayed within the first radix but instead your real value as developed by mathematicians which is at the heart of your own number system (and mine, in the past) has reradixed into some sort of escapist cloud of complexity shrouded only by the irrational value, which we are supposed to stuff down next. In hindsight the mixture of operators and values as if they are pure values is to blame, and you will not find any mathematician here who will confront this analysis. This is the third strike. Sadly, the game is rigged. Well, of course you should take this as rhetoric and content. Still, the content is simply exposed.
> Is 3/5 a value or is it two values and an operator?
> On and on it can go and into subjects such as abstract algebra where operators are very carefully defined. Instances such as
> a0 + a1X
> with real coefficients (forgoing that the real number is under attack already) exposes two operators which do not obey their own closure requirements. The X you see is somehow an undefined thing within abstract algebra. Somehow the polynomial form as the number is evident, yet as to which came first we'd have to give the number that status.
>
> Anyway, it seems that the physicist happily rests his head on the mathematician's real value, but I'm pretty sure what's there is different than the great defense that real analysis provides. And of course these subjects are upheld under the threat of failure. Mimicry is heavily enforced within academia. Here we are outside of that system.
>
> Cheers PK

Cheers Timothy Golden
But I really do not understand what you say. You must be very frustrated when discussing with people.

Le vendredi 25 février 2022 à 03:24:19 UTC+1, Chris M. Thomasson a écrit :
> On 2/17/2022 12:42 PM, PengKuan Em wrote:
> Post it over on https://fractalforums.org. There are a lot of people who
> will try to create fractals with it. I do not have the time to give it a
> really deep dive right now. Fwiw, here is some of my work:
>
> https://fractalforums.org/index.php?action=gallery;sa=view;id=6714

In Fractalforum, Alef has done a great job in computing a 3D Mandelbrot fractal with my system. See here
https://fractalforums.org/index.php?topic=4675.msg33509#msg33509
Beautiful image.

On Wednesday, March 23, 2022 at 3:21:17 PM UTC-4, tita...@gmail.com wrote:
> Le dimanche 20 mars 2022 à 21:16:59 UTC+1, timba...@gmail.com a écrit :
> > On Friday, March 18, 2022 at 2:26:19 PM UTC-4, tita...@gmail.com wrote:
> > Well, as you form shifts into an indexed form of some sort I think you'll have a better general dimensional solution.
> > Whether it works all the way down or not: it seems you'll bottom out in 2D.
> > The real number as a 1D by definition concept is always sitting there under the hood.
> > That the general form will develop into a polynomial is entirely consistent with rather a lot of mathematics.
> > Most would wrap i^N around to unity where N is near your max dimension.
> > I don't know what happened to Olariu's work and his cosexp function. I can't find it anymore.
> > It definitely was on arxiv.org back in time. Possibly I have a typo here. It was awfully close to your stuff.
> > Ah, I've found a paper that references a book of his heavily: https://arxiv.org/pdf/1801.03773.pdf
> >
> > In response to your other reply on the engineer's versus the mathematician's interpretation of a value like 2.01:
> > when an engineer sets a voltage on a power supply, say, to 2.01 Volts DC they are not claiming perfection. They have worked only to a a certain resolution and know better than to chase any finer.
> >
> > The mathematician on the other hand sees the value as pristine and perfectly defined and without any error. These are starkly differing interpretations of the continuum. One version exposes the gray nature of the number whereas the other denies the existence of such. The notion of epsilon/delta refinement has only gone on within the value 2.01 a few digits. There is a sort of refusal of mathematicians to admit that chasing digits is what epsilon/delta does, yet if we were to literally carry out the computation of say the square root of two this is exactly as it will play out. Of course the inane nature of such a practice while unity is so arbitrarily specified on any continuum further exposes the mathematician's absurdity.
> > Any ability to work with such a fine value on the continuum or computationally is badly compromised.
> >
> > Strangest of all, upon realizing that the rational value as composed of values and operators (the division operation which arguably is not even fundamental) we see that the value 2.01 can in fact take a natural representation, that being 201, and of course the decimal point does require some augmentation to that but it is certainly marking a new form of unity; a secondary unity, so to speak. So by doing away with the rational analysis which here has two strikes against it we land in a simplification that leaves us with our radix ten values fully intact and hardly any change from the natural value has occurred. Certainly we will not claim that two such in division magically breed the continuum. We could have stayed within the first radix but instead your real value as developed by mathematicians which is at the heart of your own number system (and mine, in the past) has reradixed into some sort of escapist cloud of complexity shrouded only by the irrational value, which we are supposed to stuff down next. In hindsight the mixture of operators and values as if they are pure values is to blame, and you will not find any mathematician here who will confront this analysis. This is the third strike. Sadly, the game is rigged. Well, of course you should take this as rhetoric and content. Still, the content is simply exposed.
> > Is 3/5 a value or is it two values and an operator?
> > On and on it can go and into subjects such as abstract algebra where operators are very carefully defined. Instances such as
> > a0 + a1X
> > with real coefficients (forgoing that the real number is under attack already) exposes two operators which do not obey their own closure requirements. The X you see is somehow an undefined thing within abstract algebra. Somehow the polynomial form as the number is evident, yet as to which came first we'd have to give the number that status.
> >
> > Anyway, it seems that the physicist happily rests his head on the mathematician's real value, but I'm pretty sure what's there is different than the great defense that real analysis provides. And of course these subjects are upheld under the threat of failure. Mimicry is heavily enforced within academia. Here we are outside of that system.
> >
> > Cheers PK
> Cheers Timothy Golden
> But I really do not understand what you say. You must be very frustrated when discussing with people.

It is true that my position has diverge greatly from the mainstream and somehow continues to develop. One old question that I had avoided in the past of polysign: If you generalize the sign of the real number; essentially redefining it, then is it necessary as well to redefine the magnitude of the real value? The part that I had worked on was the s in sx, where s is sign and x is (unsigned) magnitude. Clearly the structure of the real value contains at least these two parts. The sign was a binary system within the real value with its modulo two behaviors under product, and I found out that it can support a three-signed system with a modulo three behavior under product, and in four signs and so forth. The product is not nearly so mysterious as the sum since the real valued behavior (P2) -z+z=0 in P3 becomes -z+z*z=0. Lo and behold the graphical requirement of the real line as a line with its one dimensional naming becomes two dimensional; yet the coordinate system is not orthogonal. We see through polysign that the ray is more fundamental than the line; and indeed back in P2 the system is composed of two rays. It is profound that P3 are the complex numbers in a new format since the rules which beget them are identical to the rules which beget P2. Of course we can discuss Pn in these terms, and most famously (in perpetuity perhaps) will be P1: the singleton whose zero dimensional nature is upheld by not making exceptions:
P1 : - z = 0
P2: - z + z = 0
P3: - z + z * z = 0
P4: - z + z * z # z = 0
P5: (z,z,z,z,z) = 0 (shifting to series notation here)
Pn: (z,z,z,...z) = 0.
So that while algebra is available in P1 its geometry does not render dimensionally. This is just as time behaves. The unidirectional system which we have no ability to freely move an object about in is enough to claim time as zero dimensional; after all, it is just this which allows us to claim the three dimensions of space. This does go in contradiction to relativity theory. Polysign does go in contradiction to all who bow to the real value as fundamental. Recently I have labeled these troglodytes two-signed morons (TSMs). Of course this is in fun, but really is half serious too. The inability of the practicing mathematician to confront polysign and its gains indicates a problem with humanity. Emergent spacetime awaits from an arithmetic basis and you look the other way.

Clearly the promise of delivering electromagnetics is here. That the progressive form
P1 P2 P3 | P4 P5 ...
mimics the electromagnetic tensor, but without the redundancy, is telling. Still, it is not done, and the idea that one man should do it all is not necessarily sensible. As to how much to go after here: clearly a TOE is in tow, potentially.

On 3/23/2022 12:23 PM, PengKuan Em wrote:
> Le vendredi 25 février 2022 à 03:24:19 UTC+1, Chris M. Thomasson a écrit :
>> On 2/17/2022 12:42 PM, PengKuan Em wrote:
>> Post it over on https://fractalforums.org. There are a lot of people who
>> will try to create fractals with it. I do not have the time to give it a
>> really deep dive right now. Fwiw, here is some of my work:
>>
>> https://fractalforums.org/index.php?action=gallery;sa=view;id=6714
>
> In Fractalforum, Alef has done a great job in computing a 3D Mandelbrot fractal with my system. See here
> https://fractalforums.org/index.php?topic=4675.msg33509#msg33509
> Beautiful image.

Great! I need to look at the thread in more detail. It would be nice to
implement it over in ShaderToy. Here is a Mandelbulb:

https://www.shadertoy.com/view/MdXSWn

On 3/26/2022 2:58 PM, Chris M. Thomasson wrote:
> On 3/23/2022 12:23 PM, PengKuan Em wrote:
>> Le vendredi 25 février 2022 à 03:24:19 UTC+1, Chris M. Thomasson a
>> écrit :
>>> On 2/17/2022 12:42 PM, PengKuan Em wrote:
>>> Post it over on https://fractalforums.org. There are a lot of people who
>>> will try to create fractals with it. I do not have the time to give it a
>>> really deep dive right now. Fwiw, here is some of my work:
>>>
>>> https://fractalforums.org/index.php?action=gallery;sa=view;id=6714
>>
>> In Fractalforum, Alef has done a great job in computing a 3D
>> Mandelbrot fractal with my system. See here
>> https://fractalforums.org/index.php?topic=4675.msg33509#msg33509
>> Beautiful image.
>
> Great! I need to look at the thread in more detail. It would be nice to
> implement it over in ShaderToy. Here is a Mandelbulb:
>
> https://www.shadertoy.com/view/MdXSWn
>
>

Can you view the shadertoy link?

Le samedi 26 mars 2022 à 23:01:06 UTC+1, Chris M. Thomasson a écrit :
> On 3/26/2022 2:58 PM, Chris M. Thomasson wrote:
> > On 3/23/2022 12:23 PM, PengKuan Em wrote:
> >> Le vendredi 25 février 2022 à 03:24:19 UTC+1, Chris M. Thomasson a
> >> écrit :
> >>> On 2/17/2022 12:42 PM, PengKuan Em wrote:
> >>> Post it over on https://fractalforums.org. There are a lot of people who
> >>> will try to create fractals with it. I do not have the time to give it a
> >>> really deep dive right now. Fwiw, here is some of my work:
> >>>
> >>> https://fractalforums.org/index.php?action=gallery;sa=view;id=6714
> >>
> >> In Fractalforum, Alef has done a great job in computing a 3D
> >> Mandelbrot fractal with my system. See here
> >> https://fractalforums.org/index.php?topic=4675.msg33509#msg33509
> >> Beautiful image.
> >
> > Great! I need to look at the thread in more detail. It would be nice to
> > implement it over in ShaderToy. Here is a Mandelbulb:
> >
> > https://www.shadertoy.com/view/MdXSWn
> >
> >
> Can you view the shadertoy link?
Yes, I see it. This is a power 8 Mandelbrot I think.

PK

On 3/27/2022 9:30 AM, PengKuan Em wrote:
> Le samedi 26 mars 2022 à 23:01:06 UTC+1, Chris M. Thomasson a écrit :
>> On 3/26/2022 2:58 PM, Chris M. Thomasson wrote:
>>> On 3/23/2022 12:23 PM, PengKuan Em wrote:
>>>> Le vendredi 25 février 2022 à 03:24:19 UTC+1, Chris M. Thomasson a
>>>> écrit :
>>>>> On 2/17/2022 12:42 PM, PengKuan Em wrote:
>>>>> Post it over on https://fractalforums.org. There are a lot of people who
>>>>> will try to create fractals with it. I do not have the time to give it a
>>>>> really deep dive right now. Fwiw, here is some of my work:
>>>>>
>>>>> https://fractalforums.org/index.php?action=gallery;sa=view;id=6714
>>>>
>>>> In Fractalforum, Alef has done a great job in computing a 3D
>>>> Mandelbrot fractal with my system. See here
>>>> https://fractalforums.org/index.php?topic=4675.msg33509#msg33509
>>>> Beautiful image.
>>>
>>> Great! I need to look at the thread in more detail. It would be nice to
>>> implement it over in ShaderToy. Here is a Mandelbulb:
>>>
>>> https://www.shadertoy.com/view/MdXSWn
>>>
>>>
>> Can you view the shadertoy link?
> Yes, I see it. This is a power 8 Mandelbrot I think.

Power of 8 Mandelbulb. You can use the program as a base, and just put
your iteration into the ray marcher. The fractal would just, show up. :^)

Le lundi 28 mars 2022 à 23:46:28 UTC+2, Chris M. Thomasson a écrit :
> On 3/27/2022 9:30 AM, PengKuan Em wrote:
> Power of 8 Mandelbulb. You can use the program as a base, and just put
> your iteration into the ray marcher. The fractal would just, show up. :^)
Yes. In this forum Alef has done z^2+c and z^3+c and the result is amazing. Below are the links
https://fractalforums.org/gallery/201-230322083340.jpeg
https://fractalforums.org/index.php?action=dlattach;topic=4675.0;attach=26488;image
https://fractalforums.org/index.php?action=dlattach;topic=4675.0;attach=26491;image
https://fractalforums.org/gallery/201-250322092456.jpeg

On 3/28/2022 3:03 PM, PengKuan Em wrote:
> Le lundi 28 mars 2022 à 23:46:28 UTC+2, Chris M. Thomasson a écrit :
>> On 3/27/2022 9:30 AM, PengKuan Em wrote:
>> Power of 8 Mandelbulb. You can use the program as a base, and just put
>> your iteration into the ray marcher. The fractal would just, show up. :^)
> Yes. In this forum Alef has done z^2+c and z^3+c and the result is amazing. Below are the links
> https://fractalforums.org/gallery/201-230322083340.jpeg
> https://fractalforums.org/index.php?action=dlattach;topic=4675.0;attach=26488;image
> https://fractalforums.org/index.php?action=dlattach;topic=4675.0;attach=26491;image
> https://fractalforums.org/gallery/201-250322092456.jpeg

I had a feeling that your work would create interesting fractals: I was
right! Now, I want to use your algebra to create a new version of my
stacked Mandelbulb. Should have some more time tomorrow. For what its
worth, read this:

https://www.fractalforums.com/index.php?action=gallery;sa=view;id=17187
(has pseudo code in the comments...)

My stacking volumetric should work fine with your algebra. It's going to
be fun to compare and contrast my original work with your technique.

I create a stack of images and visualize them as a whole using a
volumetric renderer.

Chris M. Thomasson wrote:

> I had a feeling that your work would create interesting fractals: I was
> right! Now, I want to use your algebra to create a new version of my
> stacked Mandelbulb. Should have some more time tomorrow. For what its
> worth, read this:

BRITISH EMPIRE CENSORS SPUTNIKNEWS - Censorship Update 20
https://www.bitchute.com/video/CX8mU38ZZJ3w/
BRITISH EMPIRE [UK, USA, NZ, OZ] WILL NOT TOLLERATE ANY NARRATIVE OTHER
THAN THE BRITISH EMPIRE NARRATIVE. This collective NARCISSISTIC
PERSONALITY DISORDER mobilises in an orchestrated attack on INDEPENDENT
JOURNALISM, a second opinion that exposes 700 years of The Empire...

Le dimanche 17 avril 2022 à 22:07:40 UTC+2, Chris M. Thomasson a écrit :
> I had a feeling that your work would create interesting fractals: I was
> right! Now, I want to use your algebra to create a new version of my
> stacked Mandelbulb. Should have some more time tomorrow. For what its
> worth, read this:
>
> https://www.fractalforums.com/index.php?action=gallery;sa=view;id=17187
> (has pseudo code in the comments...)
>
> My stacking volumetric should work fine with your algebra. It's going to
> be fun to compare and contrast my original work with your technique.
>
> I create a stack of images and visualize them as a whole using a
> volumetric renderer.

Hi, Chris M. Thomasson,

I can never thank you enough for showing me the direction to post in this fractalforum. It is such a success that I’m astonished myself.

I have already seen your post of stacking volumetric there.
https://fractalforums.org/index.php?topic=4675.msg33629#msg33629
But I do not know what the images were about and did not know how to answer you.

If you have any question about the algebra, please ask me, I will give you all I have to help.

PengKuan

On 4/17/2022 4:12 PM, PengKuan Em wrote:
> Le dimanche 17 avril 2022 à 22:07:40 UTC+2, Chris M. Thomasson a écrit :
>> I had a feeling that your work would create interesting fractals: I was
>> right! Now, I want to use your algebra to create a new version of my
>> stacked Mandelbulb. Should have some more time tomorrow. For what its
>> worth, read this:
>>
>> https://www.fractalforums.com/index.php?action=gallery;sa=view;id=17187
>> (has pseudo code in the comments...)
>>
>> My stacking volumetric should work fine with your algebra. It's going to
>> be fun to compare and contrast my original work with your technique.
>>
>> I create a stack of images and visualize them as a whole using a
>> volumetric renderer.
>
> Hi, Chris M. Thomasson,
>
> I can never thank you enough for showing me the direction to post in this fractalforum. It is such a success that I’m astonished myself.
>
> I have already seen your post of stacking volumetric there.
> https://fractalforums.org/index.php?topic=4675.msg33629#msg33629
> But I do not know what the images were about and did not know how to answer you.
>
> If you have any question about the algebra, please ask me, I will give you all I have to help.
>

Humm... You might want to reach out to a person by the name of David
Makin. He is working on something called the MultiPlex. He is a
programmer, and basically created an old school game, Crystal Dragon:

https://www.dungeoncrawlers.org/game/crystal-dragon

should I try to get you two in touch?

On 4/17/2022 4:12 PM, PengKuan Em wrote:
> Le dimanche 17 avril 2022 à 22:07:40 UTC+2, Chris M. Thomasson a écrit :
>> I had a feeling that your work would create interesting fractals: I was
>> right! Now, I want to use your algebra to create a new version of my
>> stacked Mandelbulb. Should have some more time tomorrow. For what its
>> worth, read this:
>>
>> https://www.fractalforums.com/index.php?action=gallery;sa=view;id=17187
>> (has pseudo code in the comments...)
>>
>> My stacking volumetric should work fine with your algebra. It's going to
>> be fun to compare and contrast my original work with your technique.
>>
>> I create a stack of images and visualize them as a whole using a
>> volumetric renderer.
>
> Hi, Chris M. Thomasson,
>
> I can never thank you enough for showing me the direction to post in this fractalforum. It is such a success that I’m astonished myself.
>
> I have already seen your post of stacking volumetric there.
> https://fractalforums.org/index.php?topic=4675.msg33629#msg33629
> But I do not know what the images were about and did not know how to answer you.
>
> If you have any question about the algebra, please ask me, I will give you all I have to help.
>

I basically need a power, multiplication and addition function to get it
right. Then I set the power to 2 to get a Mandelbrot. And increase per
slice in order to generate the volume.

On 4/17/2022 4:12 PM, PengKuan Em wrote:
> Le dimanche 17 avril 2022 à 22:07:40 UTC+2, Chris M. Thomasson a écrit :
>> I had a feeling that your work would create interesting fractals: I was
>> right! Now, I want to use your algebra to create a new version of my
>> stacked Mandelbulb. Should have some more time tomorrow. For what its
>> worth, read this:
>>
>> https://www.fractalforums.com/index.php?action=gallery;sa=view;id=17187
>> (has pseudo code in the comments...)
>>
>> My stacking volumetric should work fine with your algebra. It's going to
>> be fun to compare and contrast my original work with your technique.
>>
>> I create a stack of images and visualize them as a whole using a
>> volumetric renderer.
>
> Hi, Chris M. Thomasson,
>
> I can never thank you enough for showing me the direction to post in this fractalforum. It is such a success that I’m astonished myself.
>
> I have already seen your post of stacking volumetric there.
> https://fractalforums.org/index.php?topic=4675.msg33629#msg33629
> But I do not know what the images were about and did not know how to answer you.
>
> If you have any question about the algebra, please ask me, I will give you all I have to help.

I need to get back into these 3d stacks. They are quite interesting:

https://www.facebook.com/chris.thomasson.31/posts/612895216536147

Le lundi 18 avril 2022 à 06:42:34 UTC+2, Chris M. Thomasson a écrit :
> Humm... You might want to reach out to a person by the name of David
> Makin. He is working on something called the MultiPlex. He is a
> programmer, and basically created an old school game, Crystal Dragon:
>
> https://www.dungeoncrawlers.org/game/crystal-dragon
>
> should I try to get you two in touch?
Yes, with pleasure. I have even a function that make 3D object rotate in apace using the 3D complex. I think he might be interested.

For your need of algebraic function, I have put all the useful functions in 3D complex system in this document, you can download it and see if they satisfy your need. For download, you have to click on the button on the top-right corner of the view window, the button is for reading in a new tab. Then, you have the download button
https://pengkuanonmaths.blogspot.com/2022/04/procedure-to-convert-2d-formula-into-3d.html

PengKuan