that has no solution...
It is why it is called the imaginary.
No negative roots...

Mitchell Raemsch

On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
> that has no solution...
> It is why it is called the imaginary.
> No negative roots...

https://thoughtfulsoftware.wordpress.com/2013/11/17/negative-caching/

https://en.wikipedia.org/wiki/Negative_cache

On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
> On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
> > that has no solution...
> > It is why it is called the imaginary.
> > No negative roots...
> https://thoughtfulsoftware.wordpress.com/2013/11/17/negative-caching/
>
> https://en.wikipedia.org/wiki/Negative_cache

The product is defined in the modern versions of complex analysis:
( a + bi )( c + di ) = ac - bd + ( ad + bc )i
and yes, this is a formula. Likewise the sum more trivially:
( a + bi ) + ( c + di ) = a + c + ( b + d )i
I believe it was Hamilton who mentions i as a sign in one of his many publications.
In this same publication he goes into the meaning of magnitude as well.
That there is a convergence of these concepts in a multi-type system: should we expect that nature is entirely typeless?
Spacetime itself seems to carry structure just as the very words 'space' and 'time' go in junction as one.
Electicity and Magnetism likewise.
Number and Sign have such a relation and so the form
s x
is a multitype system; sign being discrete in nature and obeying a specified modulo behavior; for instance modulo two in the real values. That the complex numbes arive without the need for additional formulas simply by engaging the modulo three form is a great simplification that alleviates the tension which causes the OP.

On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
> On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
> > On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
> > > that has no solution...
> > > It is why it is called the imaginary.
> > > No negative roots...
> > https://thoughtfulsoftware.wordpress.com/2013/11/17/negative-caching/
> >
> > https://en.wikipedia.org/wiki/Negative_cache
> The product is defined in the modern versions of complex analysis:
> ( a + bi )( c + di ) = ac - bd + ( ad + bc )i
> and yes, this is a formula. Likewise the sum more trivially:
> ( a + bi ) + ( c + di ) = a + c + ( b + d )i
> I believe it was Hamilton who mentions i as a sign in one of his many publications.
> In this same publication he goes into the meaning of magnitude as well.
> That there is a convergence of these concepts in a multi-type system: should we expect that nature is entirely typeless?
> Spacetime itself seems to carry structure just as the very words 'space' and 'time' go in junction as one.
> Electicity and Magnetism likewise.
> Number and Sign have such a relation and so the form
> s x
> is a multitype system; sign being discrete in nature and obeying a specified modulo behavior; for instance modulo two in the real values. That the complex numbes arive without the need for additional formulas simply by engaging the modulo three form is a great simplification that alleviates the tension which causes the OP.

Stop moving your i around; only its co effecient could change the complex plane
or the imaginary math... but not really...

On Sunday, January 2, 2022 at 3:16:46 PM UTC-5, mitchr...@gmail.com wrote:
> On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
> > On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
> > > On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
> > > > that has no solution...
> > > > It is why it is called the imaginary.
> > > > No negative roots...
> > > https://thoughtfulsoftware.wordpress.com/2013/11/17/negative-caching/
> > >
> > > https://en.wikipedia.org/wiki/Negative_cache
> > The product is defined in the modern versions of complex analysis:
> > ( a + bi )( c + di ) = ac - bd + ( ad + bc )i
> > and yes, this is a formula. Likewise the sum more trivially:
> > ( a + bi ) + ( c + di ) = a + c + ( b + d )i
> > I believe it was Hamilton who mentions i as a sign in one of his many publications.
> > In this same publication he goes into the meaning of magnitude as well.
> > That there is a convergence of these concepts in a multi-type system: should we expect that nature is entirely typeless?
> > Spacetime itself seems to carry structure just as the very words 'space' and 'time' go in junction as one.
> > Electicity and Magnetism likewise.
> > Number and Sign have such a relation and so the form
> > s x
> > is a multitype system; sign being discrete in nature and obeying a specified modulo behavior; for instance modulo two in the real values. That the complex numbes arive without the need for additional formulas simply by engaging the modulo three form is a great simplification that alleviates the tension which causes the OP.
> Stop moving your i around; only its co effecient could change the complex plane
> or the imaginary math... but not really...

Hey, I don't mean to support their clap-trap.
Rational values are done the same damn way.
Just toss dimensional analysis out the window in their case.

Polysign are the native form for the rotational number systems.
They are geometric from the get-go.
They are algebraic from the start.
They are general 'dimensional' though this language is passe.
They are the real deal.
Get over it.

On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com wrote:
> On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
> > On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
> > > On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
> > > > that has no solution...
> > > > It is why it is called the imaginary.
> > > > No negative roots...
> > > https://thoughtfulsoftware.wordpress.com/2013/11/17/negative-caching/
> > >
> > > https://en.wikipedia.org/wiki/Negative_cache
> > The product is defined in the modern versions of complex analysis:
> > ( a + bi )( c + di ) = ac - bd + ( ad + bc )i
> > and yes, this is a formula. Likewise the sum more trivially:
> > ( a + bi ) + ( c + di ) = a + c + ( b + d )i
> > I believe it was Hamilton who mentions i as a sign in one of his many publications.
> > In this same publication he goes into the meaning of magnitude as well.
> > That there is a convergence of these concepts in a multi-type system: should we expect that nature is entirely typeless?
> > Spacetime itself seems to carry structure just as the very words 'space' and 'time' go in junction as one.
> > Electicity and Magnetism likewise.
> > Number and Sign have such a relation and so the form
> > s x
> > is a multitype system; sign being discrete in nature and obeying a specified modulo behavior; for instance modulo two in the real values. That the complex numbes arive without the need for additional formulas simply by engaging the modulo three form is a great simplification that alleviates the tension which causes the OP.
> Stop moving your i around; only its co effecient could change the complex plane
> or the imaginary math... but not really...

The i's usually attached to what is mostly a polar term. It's funny that polar means angular.
In the orthogonal it means asymptotic.

The imaginary unit with the property of dropping out in that it annihilates itself
to -1 as its square, and all its thusly modular even powers, there is that the powers
and half powers are worked out what results why i the value, in the orthogonal terms,
really is a placeholder for cancellation after arithmetic of polar moments.

(That's its modular clock.)

I.e., complex analysis, with respect to how the complex is related to the real
besides its branches and analytic continuations for example, i.e. usually
for the real part of z, here is for lattice terms and moments with the usual
idea that moments are constructive.

When it results in terms the imaginary reduces out and results a blind side,
it's so - it always results there's a pillar in the way.
..
Basically for the two moments as opposite corners of a 2x2 square, each
moment of radius 1, is for an atomized moment of a grid.

Whether arc is radians or pi radians is a thing.

(Here the radius is 1 but the distance two root two - 2, or 2(root2-1).)

That the distance to the next is four....

Then, "i is a formula", ..., is that the properties of the elements,
are about Argand plane, with Wessel plane, is that in the
"i^2 = -1 its only root" there are multiple ways that in the
interpretation for formula, that complex analysis is applied,
what results for example usual real or discrete analysis.

Because it simply defines what product spaces exist....

On Sunday, January 2, 2022 at 12:43:14 PM UTC-8, Ross A. Finlayson wrote:
> On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com wrote:
> > On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
> > > On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
> > > > On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
> > > > > that has no solution...
> > > > > It is why it is called the imaginary.
> > > > > No negative roots...
> > > > https://thoughtfulsoftware.wordpress.com/2013/11/17/negative-caching/
> > > >
> > > > https://en.wikipedia.org/wiki/Negative_cache
> > > The product is defined in the modern versions of complex analysis:
> > > ( a + bi )( c + di ) = ac - bd + ( ad + bc )i
> > > and yes, this is a formula. Likewise the sum more trivially:
> > > ( a + bi ) + ( c + di ) = a + c + ( b + d )i
> > > I believe it was Hamilton who mentions i as a sign in one of his many publications.
> > > In this same publication he goes into the meaning of magnitude as well.
> > > That there is a convergence of these concepts in a multi-type system: should we expect that nature is entirely typeless?
> > > Spacetime itself seems to carry structure just as the very words 'space' and 'time' go in junction as one.
> > > Electicity and Magnetism likewise.
> > > Number and Sign have such a relation and so the form
> > > s x
> > > is a multitype system; sign being discrete in nature and obeying a specified modulo behavior; for instance modulo two in the real values. That the complex numbes arive without the need for additional formulas simply by engaging the modulo three form is a great simplification that alleviates the tension which causes the OP.
> > Stop moving your i around; only its co effecient could change the complex plane
> > or the imaginary math... but not really...
> The i's usually attached to what is mostly a polar term. It's funny that polar means angular.
> In the orthogonal it means asymptotic.
>
>
> The imaginary unit with the property of dropping out in that it annihilates itself
> to -1 as its square, and all its thusly modular even powers, there is that the powers
> and half powers are worked out what results why i the value, in the orthogonal terms,
> really is a placeholder for cancellation after arithmetic of polar moments.
>
> (That's its modular clock.)
>
>
> I.e., complex analysis, with respect to how the complex is related to the real
> besides its branches and analytic continuations for example, i.e. usually
> for the real part of z, here is for lattice terms and moments with the usual
> idea that moments are constructive.
>
> When it results in terms the imaginary reduces out and results a blind side,
> it's so - it always results there's a pillar in the way.
> .
> Basically for the two moments as opposite corners of a 2x2 square, each
> moment of radius 1, is for an atomized moment of a grid.
>
> Whether arc is radians or pi radians is a thing.
>
> (Here the radius is 1 but the distance two root two - 2, or 2(root2-1).)
>
> That the distance to the next is four....
>
> Then, "i is a formula", ..., is that the properties of the elements,
> are about Argand plane, with Wessel plane, is that in the
> "i^2 = -1 its only root" there are multiple ways that in the
> interpretation for formula, that complex analysis is applied,
> what results for example usual real or discrete analysis.
>
> Because it simply defines what product spaces exist....

The i formula is said not to have a solution.
And math named it accordingly...

On Sunday, January 2, 2022 at 3:43:14 PM UTC-5, Ross A. Finlayson wrote:
> On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com wrote:
> > On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
> > > On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
> > > > On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
> > > > > that has no solution...
> > > > > It is why it is called the imaginary.
> > > > > No negative roots...
> > > > https://thoughtfulsoftware.wordpress.com/2013/11/17/negative-caching/
> > > >
> > > > https://en.wikipedia.org/wiki/Negative_cache
> > > The product is defined in the modern versions of complex analysis:
> > > ( a + bi )( c + di ) = ac - bd + ( ad + bc )i
> > > and yes, this is a formula. Likewise the sum more trivially:
> > > ( a + bi ) + ( c + di ) = a + c + ( b + d )i
> > > I believe it was Hamilton who mentions i as a sign in one of his many publications.
> > > In this same publication he goes into the meaning of magnitude as well.
> > > That there is a convergence of these concepts in a multi-type system: should we expect that nature is entirely typeless?
> > > Spacetime itself seems to carry structure just as the very words 'space' and 'time' go in junction as one.
> > > Electicity and Magnetism likewise.
> > > Number and Sign have such a relation and so the form
> > > s x
> > > is a multitype system; sign being discrete in nature and obeying a specified modulo behavior; for instance modulo two in the real values. That the complex numbes arive without the need for additional formulas simply by engaging the modulo three form is a great simplification that alleviates the tension which causes the OP.
> > Stop moving your i around; only its co effecient could change the complex plane
> > or the imaginary math... but not really...
> The i's usually attached to what is mostly a polar term. It's funny that polar means angular.
> In the orthogonal it means asymptotic.
>
>
> The imaginary unit with the property of dropping out in that it annihilates itself
> to -1 as its square, and all its thusly modular even powers, there is that the powers
> and half powers are worked out what results why i the value, in the orthogonal terms,
> really is a placeholder for cancellation after arithmetic of polar moments.
>
> (That's its modular clock.)
>
>
> I.e., complex analysis, with respect to how the complex is related to the real
> besides its branches and analytic continuations for example, i.e. usually
> for the real part of z, here is for lattice terms and moments with the usual
> idea that moments are constructive.
>
> When it results in terms the imaginary reduces out and results a blind side,
> it's so - it always results there's a pillar in the way.
> .
> Basically for the two moments as opposite corners of a 2x2 square, each
> moment of radius 1, is for an atomized moment of a grid.
>
> Whether arc is radians or pi radians is a thing.
>
> (Here the radius is 1 but the distance two root two - 2, or 2(root2-1).)
>
> That the distance to the next is four....
>
> Then, "i is a formula", ..., is that the properties of the elements,
> are about Argand plane, with Wessel plane, is that in the
> "i^2 = -1 its only root" there are multiple ways that in the
> interpretation for formula, that complex analysis is applied,
> what results for example usual real or discrete analysis.
>
> Because it simply defines what product spaces exist....

The construction fails to be general dimensional.
The question of what the 3D complex numbers are turns out to be a disappointment.
Polysign however manage the boring P4 as a mere step in a progression. P5 and P6 for instance are far more interesting.

On 1/2/2022 1:16 PM, Timothy Golden wrote:
> On Sunday, January 2, 2022 at 3:43:14 PM UTC-5, Ross A. Finlayson wrote:
>> On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com wrote:
>>> On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
>>>> On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
>>>>> On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
>>>>>> that has no solution...
>>>>>> It is why it is called the imaginary.
>>>>>> No negative roots...
>>>>> https://thoughtfulsoftware.wordpress.com/2013/11/17/negative-caching/
>>>>>
>>>>> https://en.wikipedia.org/wiki/Negative_cache
>>>> The product is defined in the modern versions of complex analysis:
>>>> ( a + bi )( c + di ) = ac - bd + ( ad + bc )i
>>>> and yes, this is a formula. Likewise the sum more trivially:
>>>> ( a + bi ) + ( c + di ) = a + c + ( b + d )i
>>>> I believe it was Hamilton who mentions i as a sign in one of his many publications.
>>>> In this same publication he goes into the meaning of magnitude as well.
>>>> That there is a convergence of these concepts in a multi-type system: should we expect that nature is entirely typeless?
>>>> Spacetime itself seems to carry structure just as the very words 'space' and 'time' go in junction as one.
>>>> Electicity and Magnetism likewise.
>>>> Number and Sign have such a relation and so the form
>>>> s x
>>>> is a multitype system; sign being discrete in nature and obeying a specified modulo behavior; for instance modulo two in the real values. That the complex numbes arive without the need for additional formulas simply by engaging the modulo three form is a great simplification that alleviates the tension which causes the OP.
>>> Stop moving your i around; only its co effecient could change the complex plane
>>> or the imaginary math... but not really...
>> The i's usually attached to what is mostly a polar term. It's funny that polar means angular.
>> In the orthogonal it means asymptotic.
>>
>>
>> The imaginary unit with the property of dropping out in that it annihilates itself
>> to -1 as its square, and all its thusly modular even powers, there is that the powers
>> and half powers are worked out what results why i the value, in the orthogonal terms,
>> really is a placeholder for cancellation after arithmetic of polar moments.
>>
>> (That's its modular clock.)
>>
>>
>> I.e., complex analysis, with respect to how the complex is related to the real
>> besides its branches and analytic continuations for example, i.e. usually
>> for the real part of z, here is for lattice terms and moments with the usual
>> idea that moments are constructive.
>>
>> When it results in terms the imaginary reduces out and results a blind side,
>> it's so - it always results there's a pillar in the way.
>> .
>> Basically for the two moments as opposite corners of a 2x2 square, each
>> moment of radius 1, is for an atomized moment of a grid.
>>
>> Whether arc is radians or pi radians is a thing.
>>
>> (Here the radius is 1 but the distance two root two - 2, or 2(root2-1).)
>>
>> That the distance to the next is four....
>>
>> Then, "i is a formula", ..., is that the properties of the elements,
>> are about Argand plane, with Wessel plane, is that in the
>> "i^2 = -1 its only root" there are multiple ways that in the
>> interpretation for formula, that complex analysis is applied,
>> what results for example usual real or discrete analysis.
>>
>> Because it simply defines what product spaces exist....
>
> The construction fails to be general dimensional.
> The question of what the 3D complex numbers are turns out to be a disappointment.

Triplex numbers?

https://www.soler7.com/Fractals/Matrices%20to%20Triplex.pdf

> Polysign however manage the boring P4 as a mere step in a progression. P5 and P6 for instance are far more interesting.

On 1/2/2022 1:16 PM, Timothy Golden wrote:
> On Sunday, January 2, 2022 at 3:43:14 PM UTC-5, Ross A. Finlayson wrote:
>> On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com wrote:
>>> On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
>>>> On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
>>>>> On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
[...]
> The construction fails to be general dimensional.
> The question of what the 3D complex numbers are turns out to be a disappointment.
> Polysign however manage the boring P4 as a mere step in a progression. P5 and P6 for instance are far more interesting.

Can you generate a Mandelbulb using Polysign?

After serious thinking Chris M. Thomasson wrote :
> On 1/2/2022 1:16 PM, Timothy Golden wrote:
>> On Sunday, January 2, 2022 at 3:43:14 PM UTC-5, Ross A. Finlayson wrote:
>>> On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com
>>> wrote:
>>>> On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
>>>>> On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
>>>>>> On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com
>>>>>> wrote:
> [...]
>> The construction fails to be general dimensional.
>> The question of what the 3D complex numbers are turns out to be a
>> disappointment.
>> Polysign however manage the boring P4 as a mere step in a progression. P5
>> and P6 for instance are far more interesting.
>
> Can you generate a Mandelbulb using Polysign?

How many mathematicians does it take to change a mandelbulb?

On 1/3/2022 8:16 AM, FromTheRafters wrote:
> After serious thinking Chris M. Thomasson wrote :
>> On 1/2/2022 1:16 PM, Timothy Golden wrote:

>>> The construction fails to be general dimensional.
>>> The question of what the 3D complex numbers are
>>> turns out to be a disappointment.
>>> Polysign however manage the boring P4 as a mere step
>>> in a progression. P5 and P6 for instance are far more
>>> interesting.
>>
>> Can you generate a Mandelbulb using Polysign?
>
> How many mathematicians does it take to change
> a mandelbulb?

sqrt(-1)

On Sunday, January 2, 2022 at 9:47:51 PM UTC-5, Chris M. Thomasson wrote:
> On 1/2/2022 1:16 PM, Timothy Golden wrote:
> > On Sunday, January 2, 2022 at 3:43:14 PM UTC-5, Ross A. Finlayson wrote:
> >> On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com wrote:
> >>> On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
> >>>> On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
> >>>>> On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
> [...]
> > The construction fails to be general dimensional.
> > The question of what the 3D complex numbers are turns out to be a disappointment.
> > Polysign however manage the boring P4 as a mere step in a progression. P5 and P6 for instance are far more interesting.
> Can you generate a Mandelbulb using Polysign?

Well, you prescribe the Mandelbulb but sadly the Mandelbrot set in the proper 3D form is merely an extruded Mandelbrot set.
If, Chris, you wish to maintain an algebraically well behaved product; preserving the good behaviors of those complex numbers that came before and did yield the original, then you see that P4, or the 3D version has similarity to RxC, whose product is assumed, though arbitrarily, to be for instance:
( 1R + 2C + 3iC )( 4R + 5C + 6iC) = (1)(4)R + ((2)(5) - (3)(6))C + ((2)(6) + (3)(5))iC
and here in this complete format does lay the possibility of introducing quite some distortion whereas in sum the straightforward
( 1R + 2C + 3iC ) + ( 4R + 5C + 6iC) = ( 1 + 4 ) R + ( 2 + 5 ) C + ( 3 + 6 ) i C

is clearly correct.

The magic lays elsewhere.

On Monday, January 3, 2022 at 9:36:54 AM UTC-5, Timothy Golden wrote:
> On Sunday, January 2, 2022 at 9:47:51 PM UTC-5, Chris M. Thomasson wrote:
> > On 1/2/2022 1:16 PM, Timothy Golden wrote:
> > > On Sunday, January 2, 2022 at 3:43:14 PM UTC-5, Ross A. Finlayson wrote:
> > >> On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com wrote:
> > >>> On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
> > >>>> On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
> > >>>>> On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
> > [...]
> > > The construction fails to be general dimensional.
> > > The question of what the 3D complex numbers are turns out to be a disappointment.
> > > Polysign however manage the boring P4 as a mere step in a progression.. P5 and P6 for instance are far more interesting.
> > Can you generate a Mandelbulb using Polysign?
> Well, you prescribe the Mandelbulb but sadly the Mandelbrot set in the proper 3D form is merely an extruded Mandelbrot set.
> If, Chris, you wish to maintain an algebraically well behaved product; preserving the good behaviors of those complex numbers that came before and did yield the original, then you see that P4, or the 3D version has similarity to RxC, whose product is assumed, though arbitrarily, to be for instance:
> ( 1R + 2C + 3iC )( 4R + 5C + 6iC) = (1)(4)R + ((2)(5) - (3)(6))C + ((2)(6) + (3)(5))iC
> and here in this complete format does lay the possibility of introducing quite some distortion whereas in sum the straightforward
> ( 1R + 2C + 3iC ) + ( 4R + 5C + 6iC) = ( 1 + 4 ) R + ( 2 + 5 ) C + ( 3 + 6 ) i C
>
> is clearly correct.
>
>
> The magic lays elsewhere.

Nextly, let's ask ourselves, if R is a subset of C, and we issue a command like:
Multiply( a, R, b, c, C )
is it true that the effect would be:
return( ab, c, C) ?
I do see that my strange formatting is leading, but I do believe that it will translate cleanly to whatever style you like.

But what I mean to challenge here is that if I can multiply a real by a complex and I can multiply a complex by a real then where were these operations when I multiplied a real and a complex by a real and a complex?
Should we fully engage our operators to their distributive brilliance then we could have such a product as:
( R1 + C1 )( R2 + C2 ) = R1R2 + R1C2 + C1R2 + C1C2
and now the arbitrary nature of the presumed product outlined above takes place.

As to how one manages such ambiguity: one choice is to stop regarding the real value as a subset of the complex number. This complaint then bounces back down the chain and we no longer develop the real value as a subset concept. We now develop the free-standing real value. It is a type on its own that means something rather different than the discrete form which in the current epoch is regarded as a subset. No: the continuous nature of the number is not alleviated by any selection of value. These qualities are flat across the spectrum of its possibilities.

On 1/3/2022 6:36 AM, Timothy Golden wrote:
> On Sunday, January 2, 2022 at 9:47:51 PM UTC-5, Chris M. Thomasson wrote:
>> On 1/2/2022 1:16 PM, Timothy Golden wrote:
>>> On Sunday, January 2, 2022 at 3:43:14 PM UTC-5, Ross A. Finlayson wrote:
>>>> On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com wrote:
>>>>> On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
>>>>>> On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
>>>>>>> On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
>> [...]
>>> The construction fails to be general dimensional.
>>> The question of what the 3D complex numbers are turns out to be a disappointment.
>>> Polysign however manage the boring P4 as a mere step in a progression. P5 and P6 for instance are far more interesting.
>> Can you generate a Mandelbulb using Polysign?
>
> Well, you prescribe the Mandelbulb but sadly the Mandelbrot set in the proper 3D form is merely an extruded Mandelbrot set.

The Mandelbulb is not an extruded Mandelbrot set. However, check this
out, read my pseudo-code:

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=17187

It has to use triplex numbers.

> If, Chris, you wish to maintain an algebraically well behaved product; preserving the good behaviors of those complex numbers that came before and did yield the original, then you see that P4, or the 3D version has similarity to RxC, whose product is assumed, though arbitrarily, to be for instance:
> ( 1R + 2C + 3iC )( 4R + 5C + 6iC) = (1)(4)R + ((2)(5) - (3)(6))C + ((2)(6) + (3)(5))iC
> and here in this complete format does lay the possibility of introducing quite some distortion whereas in sum the straightforward
> ( 1R + 2C + 3iC ) + ( 4R + 5C + 6iC) = ( 1 + 4 ) R + ( 2 + 5 ) C + ( 3 + 6 ) i C
>
> is clearly correct.
>
>
> The magic lays elsewhere.

On 1/3/2022 6:36 AM, Timothy Golden wrote:
> On Sunday, January 2, 2022 at 9:47:51 PM UTC-5, Chris M. Thomasson wrote:
>> On 1/2/2022 1:16 PM, Timothy Golden wrote:
>>> On Sunday, January 2, 2022 at 3:43:14 PM UTC-5, Ross A. Finlayson wrote:
>>>> On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com wrote:
>>>>> On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
>>>>>> On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
>>>>>>> On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
>> [...]
>>> The construction fails to be general dimensional.
>>> The question of what the 3D complex numbers are turns out to be a disappointment.
>>> Polysign however manage the boring P4 as a mere step in a progression. P5 and P6 for instance are far more interesting.
>> Can you generate a Mandelbulb using Polysign?
>
> Well, you prescribe the Mandelbulb but sadly the Mandelbrot set in the proper 3D form is merely an extruded Mandelbrot set.
> If, Chris, you wish to maintain an algebraically well behaved product; preserving the good behaviors of those complex numbers that came before and did yield the original, then you see that P4, or the 3D version has similarity to RxC, whose product is assumed, though arbitrarily, to be for instance:
> ( 1R + 2C + 3iC )( 4R + 5C + 6iC) = (1)(4)R + ((2)(5) - (3)(6))C + ((2)(6) + (3)(5))iC
> and here in this complete format does lay the possibility of introducing quite some distortion whereas in sum the straightforward
> ( 1R + 2C + 3iC ) + ( 4R + 5C + 6iC) = ( 1 + 4 ) R + ( 2 + 5 ) C + ( 3 + 6 ) i C
>
> is clearly correct.
>
>
> The magic lays elsewhere.

An extruded 2d Mandelbort can look like:

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=17147

On 1/3/2022 6:21 AM, Jim Burns wrote:
> On 1/3/2022 8:16 AM, FromTheRafters wrote:
>> After serious thinking Chris M. Thomasson wrote :
>>> On 1/2/2022 1:16 PM, Timothy Golden wrote:
>
>>>> The construction fails to be general dimensional.
>>>> The question of what the 3D complex numbers are
>>>> turns out to be a  disappointment.
>>>> Polysign however manage the boring P4 as a mere step
>>>> in a  progression. P5 and P6 for instance are far more
>>>> interesting.
>>>
>>> Can you generate a Mandelbulb using Polysign?
>>
>> How many mathematicians does it take to change
>> a mandelbulb?
>
> sqrt(-1)
>

LOL! :^)

On 1/3/2022 6:16 PM, Chris M. Thomasson wrote:
> On 1/3/2022 6:21 AM, Jim Burns wrote:
>> On 1/3/2022 8:16 AM, FromTheRafters wrote:
>>> After serious thinking Chris M. Thomasson wrote :
>>>> On 1/2/2022 1:16 PM, Timothy Golden wrote:

>>>>> The construction fails to be general dimensional.
>>>>> The question of what the 3D complex numbers are
>>>>> turns out to be a  disappointment.
>>>>> Polysign however manage the boring P4 as a mere step
>>>>> in a  progression. P5 and P6 for instance are far more
>>>>> interesting.
>>>>
>>>> Can you generate a Mandelbulb using Polysign?
>>>
>>> How many mathematicians does it take to change
>>> a mandelbulb?
>>
>> sqrt(-1)
>
> LOL! :^)

sqrt(-LOL)

(FYI, I don't know what that means.)

On 1/3/2022 4:20 PM, Jim Burns wrote:
> On 1/3/2022 6:16 PM, Chris M. Thomasson wrote:
>> On 1/3/2022 6:21 AM, Jim Burns wrote:
>>> On 1/3/2022 8:16 AM, FromTheRafters wrote:
>>>> After serious thinking Chris M. Thomasson wrote :
>>>>> On 1/2/2022 1:16 PM, Timothy Golden wrote:
>
>>>>>> The construction fails to be general dimensional.
>>>>>> The question of what the 3D complex numbers are
>>>>>> turns out to be a  disappointment.
>>>>>> Polysign however manage the boring P4 as a mere step
>>>>>> in a  progression. P5 and P6 for instance are far more
>>>>>> interesting.
>>>>>
>>>>> Can you generate a Mandelbulb using Polysign?
>>>>
>>>> How many mathematicians does it take to change
>>>> a mandelbulb?
>>>
>>> sqrt(-1)
>>
>> LOL! :^)
>
> sqrt(-LOL)
>
> (FYI, I don't know what that means.)
>

LOL means Lots of Laughs.

On 1/3/2022 9:48 PM, Chris M. Thomasson wrote:
> On 1/3/2022 4:20 PM, Jim Burns wrote:
>> On 1/3/2022 6:16 PM, Chris M. Thomasson wrote:
>>> On 1/3/2022 6:21 AM, Jim Burns wrote:
>>>> On 1/3/2022 8:16 AM, FromTheRafters wrote:
>>>>> After serious thinking Chris M. Thomasson wrote :
>>>>>> On 1/2/2022 1:16 PM, Timothy Golden wrote:

>>>>>>> The construction fails to be general dimensional.
>>>>>>> The question of what the 3D complex numbers are
>>>>>>> turns out to be a  disappointment.
>>>>>>> Polysign however manage the boring P4 as a mere step
>>>>>>> in a  progression. P5 and P6 for instance are far more
>>>>>>> interesting.
>>>>>>
>>>>>> Can you generate a Mandelbulb using Polysign?
>>>>>
>>>>> How many mathematicians does it take to change
>>>>> a mandelbulb?
>>>>
>>>> sqrt(-1)
>>>
>>> LOL! :^)
>>
>> sqrt(-LOL)
>>
>> (FYI, I don't know what that means.)
>>
>
> LOL means Lots of Laughs.

Just so. Or perhaps "Laughing Out Loud".

However, I don't know what the square root of
minus laughing out loud means.

Not that any of this matters. It's a giggle.

On Monday, January 3, 2022 at 5:57:35 PM UTC-5, Chris M. Thomasson wrote:
> On 1/3/2022 6:36 AM, Timothy Golden wrote:
> > On Sunday, January 2, 2022 at 9:47:51 PM UTC-5, Chris M. Thomasson wrote:
> >> On 1/2/2022 1:16 PM, Timothy Golden wrote:
> >>> On Sunday, January 2, 2022 at 3:43:14 PM UTC-5, Ross A. Finlayson wrote:
> >>>> On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com wrote:
> >>>>> On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
> >>>>>> On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
> >>>>>>> On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
> >> [...]
> >>> The construction fails to be general dimensional.
> >>> The question of what the 3D complex numbers are turns out to be a disappointment.
> >>> Polysign however manage the boring P4 as a mere step in a progression.. P5 and P6 for instance are far more interesting.
> >> Can you generate a Mandelbulb using Polysign?
> >
> > Well, you prescribe the Mandelbulb but sadly the Mandelbrot set in the proper 3D form is merely an extruded Mandelbrot set.
> > If, Chris, you wish to maintain an algebraically well behaved product; preserving the good behaviors of those complex numbers that came before and did yield the original, then you see that P4, or the 3D version has similarity to RxC, whose product is assumed, though arbitrarily, to be for instance:
> > ( 1R + 2C + 3iC )( 4R + 5C + 6iC) = (1)(4)R + ((2)(5) - (3)(6))C + ((2)(6) + (3)(5))iC
> > and here in this complete format does lay the possibility of introducing quite some distortion whereas in sum the straightforward
> > ( 1R + 2C + 3iC ) + ( 4R + 5C + 6iC) = ( 1 + 4 ) R + ( 2 + 5 ) C + ( 3 + 6 ) i C
> >
> > is clearly correct.
> >
> >
> > The magic lays elsewhere.
> An extruded 2d Mandelbort can look like:
>
> http://www.fractalforums.com/index.php?action=gallery;sa=view;id=17147

http://bandtechnology.com/PolySigned/Mandelbrot/MandelbrotStudy.html

These are slices; projecting high dimension (and computing) would require considerably more work.
The emergence of orthogonal (square) signals is not so mysterious; in high Pn the angles between the rays also approaches ninety degrees, though technically polysign are based on nonorthogonal coordinate systems.

On 1/3/2022 8:55 PM, Jim Burns wrote:
> On 1/3/2022 9:48 PM, Chris M. Thomasson wrote:
>> On 1/3/2022 4:20 PM, Jim Burns wrote:
>>> On 1/3/2022 6:16 PM, Chris M. Thomasson wrote:
>>>> On 1/3/2022 6:21 AM, Jim Burns wrote:
>>>>> On 1/3/2022 8:16 AM, FromTheRafters wrote:
>>>>>> After serious thinking Chris M. Thomasson wrote :
>>>>>>> On 1/2/2022 1:16 PM, Timothy Golden wrote:
>
>>>>>>>> The construction fails to be general dimensional.
>>>>>>>> The question of what the 3D complex numbers are
>>>>>>>> turns out to be a  disappointment.
>>>>>>>> Polysign however manage the boring P4 as a mere step
>>>>>>>> in a  progression. P5 and P6 for instance are far more
>>>>>>>> interesting.
>>>>>>>
>>>>>>> Can you generate a Mandelbulb using Polysign?
>>>>>>
>>>>>> How many mathematicians does it take to change
>>>>>> a mandelbulb?
>>>>>
>>>>> sqrt(-1)
>>>>
>>>> LOL! :^)
>>>
>>> sqrt(-LOL)
>>>
>>> (FYI, I don't know what that means.)
>>>
>>
>> LOL means Lots of Laughs.
>
> Just so. Or perhaps "Laughing Out Loud".
>
> However, I don't know what the square root of
> minus laughing out loud means.
>
> Not that any of this matters. It's a giggle.
>

Touche! :^)

On 1/4/2022 6:38 AM, Timothy Golden wrote:
> On Monday, January 3, 2022 at 5:57:35 PM UTC-5, Chris M. Thomasson wrote:
>> On 1/3/2022 6:36 AM, Timothy Golden wrote:
>>> On Sunday, January 2, 2022 at 9:47:51 PM UTC-5, Chris M. Thomasson wrote:
>>>> On 1/2/2022 1:16 PM, Timothy Golden wrote:
>>>>> On Sunday, January 2, 2022 at 3:43:14 PM UTC-5, Ross A. Finlayson wrote:
>>>>>> On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com wrote:
>>>>>>> On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
>>>>>>>> On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
>>>>>>>>> On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
>>>> [...]
>>>>> The construction fails to be general dimensional.
>>>>> The question of what the 3D complex numbers are turns out to be a disappointment.
>>>>> Polysign however manage the boring P4 as a mere step in a progression. P5 and P6 for instance are far more interesting.
>>>> Can you generate a Mandelbulb using Polysign?
>>>
>>> Well, you prescribe the Mandelbulb but sadly the Mandelbrot set in the proper 3D form is merely an extruded Mandelbrot set.
>>> If, Chris, you wish to maintain an algebraically well behaved product; preserving the good behaviors of those complex numbers that came before and did yield the original, then you see that P4, or the 3D version has similarity to RxC, whose product is assumed, though arbitrarily, to be for instance:
>>> ( 1R + 2C + 3iC )( 4R + 5C + 6iC) = (1)(4)R + ((2)(5) - (3)(6))C + ((2)(6) + (3)(5))iC
>>> and here in this complete format does lay the possibility of introducing quite some distortion whereas in sum the straightforward
>>> ( 1R + 2C + 3iC ) + ( 4R + 5C + 6iC) = ( 1 + 4 ) R + ( 2 + 5 ) C + ( 3 + 6 ) i C
>>>
>>> is clearly correct.
>>>
>>>
>>> The magic lays elsewhere.
>> An extruded 2d Mandelbort can look like:
>>
>> http://www.fractalforums.com/index.php?action=gallery;sa=view;id=17147
>
> http://bandtechnology.com/PolySigned/Mandelbrot/MandelbrotStudy.html
>
> These are slices; projecting high dimension (and computing) would require considerably more work.
> The emergence of orthogonal (square) signals is not so mysterious; in high Pn the angles between the rays also approaches ninety degrees, though technically polysign are based on nonorthogonal coordinate systems.
>

I am just wondering if polysign can compute a Mandelbulb. I know the
triplex numbers can because they extend complex numbers into three
dimensions. The escape time aspect for the 3d mandelbulb works on a
sphere, instead of a circle for 2d mandelbrot.

On Tuesday, January 4, 2022 at 3:12:17 PM UTC-8, Chris M. Thomasson wrote:
> On 1/4/2022 6:38 AM, Timothy Golden wrote:
> > On Monday, January 3, 2022 at 5:57:35 PM UTC-5, Chris M. Thomasson wrote:
> >> On 1/3/2022 6:36 AM, Timothy Golden wrote:
> >>> On Sunday, January 2, 2022 at 9:47:51 PM UTC-5, Chris M. Thomasson wrote:
> >>>> On 1/2/2022 1:16 PM, Timothy Golden wrote:
> >>>>> On Sunday, January 2, 2022 at 3:43:14 PM UTC-5, Ross A. Finlayson wrote:
> >>>>>> On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com wrote:
> >>>>>>> On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
> >>>>>>>> On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
> >>>>>>>>> On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
> >>>> [...]
> >>>>> The construction fails to be general dimensional.
> >>>>> The question of what the 3D complex numbers are turns out to be a disappointment.
> >>>>> Polysign however manage the boring P4 as a mere step in a progression. P5 and P6 for instance are far more interesting.
> >>>> Can you generate a Mandelbulb using Polysign?
> >>>
> >>> Well, you prescribe the Mandelbulb but sadly the Mandelbrot set in the proper 3D form is merely an extruded Mandelbrot set.
> >>> If, Chris, you wish to maintain an algebraically well behaved product; preserving the good behaviors of those complex numbers that came before and did yield the original, then you see that P4, or the 3D version has similarity to RxC, whose product is assumed, though arbitrarily, to be for instance:
> >>> ( 1R + 2C + 3iC )( 4R + 5C + 6iC) = (1)(4)R + ((2)(5) - (3)(6))C + ((2)(6) + (3)(5))iC
> >>> and here in this complete format does lay the possibility of introducing quite some distortion whereas in sum the straightforward
> >>> ( 1R + 2C + 3iC ) + ( 4R + 5C + 6iC) = ( 1 + 4 ) R + ( 2 + 5 ) C + ( 3 + 6 ) i C
> >>>
> >>> is clearly correct.
> >>>
> >>>
> >>> The magic lays elsewhere.
> >> An extruded 2d Mandelbort can look like:
> >>
> >> http://www.fractalforums.com/index.php?action=gallery;sa=view;id=17147
> >
> > http://bandtechnology.com/PolySigned/Mandelbrot/MandelbrotStudy.html
> >
> > These are slices; projecting high dimension (and computing) would require considerably more work.
> > The emergence of orthogonal (square) signals is not so mysterious; in high Pn the angles between the rays also approaches ninety degrees, though technically polysign are based on nonorthogonal coordinate systems.
> >
> I am just wondering if polysign can compute a Mandelbulb. I know the
> triplex numbers can because they extend complex numbers into three
> dimensions. The escape time aspect for the 3d mandelbulb works on a
> sphere, instead of a circle for 2d mandelbrot.

i is a formula math says has only an imaginary solution.
And they are right...

On 1/4/2022 7:24 PM, mitchr...@gmail.com wrote:
> On Tuesday, January 4, 2022 at 3:12:17 PM UTC-8, Chris M. Thomasson wrote:
>> On 1/4/2022 6:38 AM, Timothy Golden wrote:
>>> On Monday, January 3, 2022 at 5:57:35 PM UTC-5, Chris M. Thomasson wrote:
>>>> On 1/3/2022 6:36 AM, Timothy Golden wrote:
>>>>> On Sunday, January 2, 2022 at 9:47:51 PM UTC-5, Chris M. Thomasson wrote:
>>>>>> On 1/2/2022 1:16 PM, Timothy Golden wrote:
>>>>>>> On Sunday, January 2, 2022 at 3:43:14 PM UTC-5, Ross A. Finlayson wrote:
>>>>>>>> On Sunday, January 2, 2022 at 12:16:46 PM UTC-8, mitchr...@gmail.com wrote:
>>>>>>>>> On Sunday, January 2, 2022 at 9:32:41 AM UTC-8, timba...@gmail.com wrote:
>>>>>>>>>> On Sunday, January 2, 2022 at 11:00:28 AM UTC-5, FredJeffries wrote:
>>>>>>>>>>> On Saturday, January 1, 2022 at 10:27:44 AM UTC-8, mitchr...@gmail.com wrote:
>>>>>> [...]
>>>>>>> The construction fails to be general dimensional.
>>>>>>> The question of what the 3D complex numbers are turns out to be a disappointment.
>>>>>>> Polysign however manage the boring P4 as a mere step in a progression. P5 and P6 for instance are far more interesting.
>>>>>> Can you generate a Mandelbulb using Polysign?
>>>>>
>>>>> Well, you prescribe the Mandelbulb but sadly the Mandelbrot set in the proper 3D form is merely an extruded Mandelbrot set.
>>>>> If, Chris, you wish to maintain an algebraically well behaved product; preserving the good behaviors of those complex numbers that came before and did yield the original, then you see that P4, or the 3D version has similarity to RxC, whose product is assumed, though arbitrarily, to be for instance:
>>>>> ( 1R + 2C + 3iC )( 4R + 5C + 6iC) = (1)(4)R + ((2)(5) - (3)(6))C + ((2)(6) + (3)(5))iC
>>>>> and here in this complete format does lay the possibility of introducing quite some distortion whereas in sum the straightforward
>>>>> ( 1R + 2C + 3iC ) + ( 4R + 5C + 6iC) = ( 1 + 4 ) R + ( 2 + 5 ) C + ( 3 + 6 ) i C
>>>>>
>>>>> is clearly correct.
>>>>>
>>>>>
>>>>> The magic lays elsewhere.
>>>> An extruded 2d Mandelbort can look like:
>>>>
>>>> http://www.fractalforums.com/index.php?action=gallery;sa=view;id=17147
>>>
>>> http://bandtechnology.com/PolySigned/Mandelbrot/MandelbrotStudy.html
>>>
>>> These are slices; projecting high dimension (and computing) would require considerably more work.
>>> The emergence of orthogonal (square) signals is not so mysterious; in high Pn the angles between the rays also approaches ninety degrees, though technically polysign are based on nonorthogonal coordinate systems.
>>>
>> I am just wondering if polysign can compute a Mandelbulb. I know the
>> triplex numbers can because they extend complex numbers into three
>> dimensions. The escape time aspect for the 3d mandelbulb works on a
>> sphere, instead of a circle for 2d mandelbrot.
>
> i is a formula math says has only an imaginary solution.
> And they are right...

Is the y-axis imaginary? ;^)