On Thursday, July 14, 2022 at 10:18:57 AM UTC-4, Timothy Golden wrote:
> On Tuesday, July 12, 2022 at 10:22:11 AM UTC-4, Timothy Golden wrote:
> > To challenge the real number as fundamental is enough to challenge the Cartesian representation of a 2D space since the 1D version would be suspect.
> > However in the realization that the real number is not fundamental but is instead a two-signed number the geometrical 1D form does in fact hold strong.
> > The question of whether two such 1D values suddenly spills into a plane is more the cup of tea being tasted, quaffed by some, but rejected by one here.
> >
> > Ahh... perhaps the vector that I am looking for lays in the deconstruction of higher spaces. If for instance we pop in on a 1D space, and by this I mean a populated version. Populated with what? Well, this is a physical problem isn't it? Did we merely have a single point in R? After all, this is the Cartesian representation. So to what degree then is the representation merely of a point versus a space? Is RxR then a space? Or is it merely a point? That point of course being a zero dimensional entity now requiring two dimensions as I've quipped on here in the past. As a geometer fills his plane with whatever suits his interest can we do the same on the real line? Even that will betray the level of possibilities, for a thickness of point distribution should entirely be possible, as when a and b have the same coordinate the density of the trace would be twice what either singleton would be. No wonder physics rests so far away from mathematics. But they take the same root. They as well decompose down to just a few interacting objects, reduce back down to one dimension; such simplistic instances are ideal to the textbook approach. As creatures of habit these methods are getting old now. Reused for generations. Then along comes some complicated text that takes it all to another level. Should we have had the complicated version first then all of those simplistic depictions will appear fraudulent won't they? In that reality is not RxRxR (the real number cubed somehow), ignoring it's two-signed nature, being unappreciated by apes who barely were even able to accept that version of the number whose purity in prior times was less questionable...
> >
> > Then to wind up with physical systems whose point particles engage spin tactics, as if to admit these point's solidity, does not receive discussion why exactly?
> > Is it because this realization will take the entire system apart? Is it because Cartesian thought was already known to fail? In that future constructions are best defended as variations on the failed past constructions; in avoiding such combative situations academia has speared itself through the foot into soft ground, the spear going down deeply into the soil so not to be pulled out easily. I suppose there is room to pivot around, but that is about it.
> >
> > I find those who speak so highly and deferentially about books to be frauds. They must be nearly the same ones who bow to academia. What of they who do not fit in to that structure? It's actually rather a lot of people. Do people of good common sense of physical reality not fit in? Of course the sheer quantity of genera of language that have devolved over time in overlapping identical words that mean completely different things such as the word 'group', or 'ring'; the escapement of such as such are such that their obscurity is welcome in the stifling towers. "We have AC.", said one academician to the other academician. "Yes, but for how long. You know the grid has issues..." said the other to the first. A third, overhearing the first two, interjects: "It's a perfectly good reason that we should have a student run nuclear power plant on campus." The other academician, being a bit earthy, says: "Well, that would close the loop, though Uranium doesn't just grow on trees." The first responded: "Well, maybe a student run chip burning power plant would do." The other says "You know, going undergound would solve all of this..." - https://www.rt.com/shows/going-underground/
> I love Afshin. I saw him once on the Boston common back when things were getting hot here. He is very short. I said hi but he ignored me. Understandable given his position and the skirmishes in the street. It was quiet where we were but he was on his way somewhere. The herd mentality was in full swing out on the streets. A flock of humans is a sure way to hide deceptive practices. One bottle thrown into the police and you are looking at a situation, let alone the signaling of a bottle with some fluid in it. How would you like to be the unlucky cop who has to open the cap on a plastic bottle only to smell urine? Talk about demoralizing.
>
> Back in the innocent land of mathematics the Cartesian product is suffering heavy losses here on this thread. It seems to require some sort of a superdimensional space to get going in. After all, upon instantiating your first line you have built one dimension. Put it down on a piece of paper that is planar already. Then stick down another, at any angle really other than colinear with the first. Plop down a third if you like: the paper doesn't care. What seems to really matter here is that the lines which we choose to represent a coordinate series by, upon establishing those references, hold their original direction even while they provide the means of translation. Yes, two is enough, but three does not fail, either. Would you call yourself a minimalist if you balk at three? Already you are up to six rays when three would have sufficed. Four rays for the minimalist, yet three would have sufficed, and just two of those three really necessary in the reduced form, you minimalist. Could it be that polysign could suffer the same fate as the Cartesian thought process? That it is merely one step better and that another step or so remain?
>
> The solid analysis does sort of point this way, but can't we arrive there by linear algebra and matrix theory? Does polysign somehow incur a point field as a solid better? It does not seem to. The constraint within a given Pn is a gain. That would have to be conceded by anyone capable of understanding the discussion. Then the natural point in T4, say, is more of interest in that it can contain some additional referential detail. It seems almost too much detail:
> a10
> a20 a21
> a30 a31 a32
> a40 a41 a42 a43
> these being neutral element first (NU); MU being next.
>
> Scrambling the threads again, the unity awareness of the NU and the MU and the mistake of the Cartesian to take the NU as more fundamental is a cross-fit betwixt the two.
> Clearly the MU can generate the NU, but not the other way around, so as to which is more fundamental... should it come first? Well then let's rewrite this trivium into existence:
> a11
> a21 a22
> a31 a32 a33
> a41 a42 a43 a44
> and now it as if we never even had the zero sign. Still, we know that the zero sign is correct. a10=a11, a20=a22, a30=a33, a40=a44, and these are true by definition really. These are the modulo statements; the line being a modulo two concept, the plane being a modulo three concept, volumetric space being a modulo four context is being supported here, and it is in a discrete context that it is so. This is completely foreign to the Cartesian way of thinking, as is the arithmetic product. Whether we even should include P4 into this deal is an open problem...
> P1 P2 P3 | P4 P5 P6 ...
> but the breakpoint remains so that support either way can be argued for. The kaleidoscopic P4 mimics P2P3, and some will insist I insert a Cartesian cross here.
> Yet shouldn't a decomposition by P1 be possible? Ah, or is it true that should one decompose by P1 you still get P4 back? The geometry would hold true. This is where the real number as two-signed somehow gains a foothold. Yes, P2 is here in polysign. Yes, all dot products yield a P2 value rather than a P1 value in ordinary vector mechanics. Yes, this is the source of projection as well. At least this is the source of ordinary vector projection. Possibly there is more. Certainly the mathematicains PP=P is pee-pee, but that is merely bad interpretation. Still, P2 is there and is still there in my code. Now dot your v's and cross your eyes, and oh, here, have a sip of apple juice...
>
> https://www.youtube.com/watch?v=EDZ_pNnA6qE
> https://www.youtube.com/results?search_query=bottle+of+pee+thrown+at+cop

Constraints, conventions, and assumptions; in these things a blurry place awaits where when one puts on a certain pair of glasses one can see it one way, then another pair and it looks a bit different. The lenses are but prefilters and it is in fact the human mind which is doing quite a lot of processing on a rough graphical form. Consider the series:
R, RxR, RxRxR, RxRxRxR
This is somewhat the Cartesian version of the polysign progression. There is a general dimensional awareness that is being bred, and in that mathematicians attempt to work in general this is a good thing. The early forms seem very simple. Especially the presentation of an actual value as merely a series of numbers. That these numbers translate over to geometry is why we even bother with such construction. Yet are these instances purely general mathematics? Where is their geometry? Is it hidden away in an unstated series of assumptions? Or doesn't it actually exist at all? And as we take the first say; R; and plot it down on a piece of paper to what degree is our sensibility of R actually formed in a plane from the outset? The same can be said of all of our work that is printed on the pages in the stacks and even those on the internet. That these pages afford to be flipped and stacked at all is a statement of a more outward space known as reality, which we are fortunate enough to live in a stable region of.

The last part my abovepost
"Much of mathematics is grounded in the study of equivalences, and order relations. Lattice
theory captures the mathematical structure of order relations. Even though equivalence relations
are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is
not as well known as that of orders. The former structure draws primarily on group theory and, to
a lesser extent, on the theory of lattices, categories, and groupoids."

https://en.wikipedia.org/wiki/Equivalence_relation#Algebraic_structure
In the concrete, and also, in the abstract, equivalences and orders. This is, reberverates in the differents aspects.

Still I cant quite explain the cartesian product issue that you have.

By the way, if you want the R^3 instead of CxR in Pn, just change the p4 multiplication, use klein four-group instead of mod 4 ;)

"Much of mathematics is grounded in the study of equivalences, and order relations. Lattice
theory captures the mathematical structure of order relations. Even though equivalence relations
are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is
not as well known as that of orders. The former structure draws primarily on group theory and, to
a lesser extent, on the theory of lattices, categories, and groupoids."

https://en.wikipedia.org/wiki/Equivalence_relation#Algebraic_structure

In the concrete and in the abstract, order theoretic and equivalences, that reverberates.

By the way, if you want R^3 instead of RxC, change the p4 multiplication, swap the mod4 by klein four-groupe.

On Wednesday, July 20, 2022 at 3:38:36 PM UTC-4, michael Rodriguez wrote:
> "Much of mathematics is grounded in the study of equivalences, and order relations. Lattice
> theory captures the mathematical structure of order relations. Even though equivalence relations
> are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is
> not as well known as that of orders. The former structure draws primarily on group theory and, to
> a lesser extent, on the theory of lattices, categories, and groupoids."
>
> https://en.wikipedia.org/wiki/Equivalence_relation#Algebraic_structure
>
> In the concrete and in the abstract, order theoretic and equivalences, that reverberates.
>
> By the way, if you want R^3 instead of RxC, change the p4 multiplication, swap the mod4 by klein four-groupe.
[K4]
-------------------------------
- - = @
- + = *
- * = +
- @ = -
+ + = @
+ * = -
+ @ = +
* * = @
* @ = *
@ @ = @

Interesting. Unique in four. I'd have to try to do some graphics with it. I don't see it. There are claims that if a product is associative then it will still yield the RxC type of behavior. This looks pretty flat though, or symmetric, at least.
( - 1 + 1 * 1 )( - 1 + 1 * 1 ) = @a *1 +1 *1 @1 -1 +1 -1 @1 = - 2 + 2 * 2 @ 3 = @1
So there is one real axis. Excluding that line we have only the -,.+,and * components doing their completely symmetrical behavior, and bouncing us out of that plane onto the @ axis again. In that they are three signs we could sort of anticipate some P3 behavior, but
( - 1 + 1 )( * 1) = + 1 - 1
( - 1 + 1 )( - 1 + 1 ) = @ 1 * 1 * 1 @ 1 = * 2 @ 2
yet there are three of these aren't there? They are completely symmetrical. There must be some broken algebra within the distribution of terms maybe? Oh I think I see it now:
( - 1 + 1 * 1 @ 1 )( - 1 ) = @ 1 * 1 + 1 - 1 = 0
No that is OK. I wonder to what degree is it necessary to mangle the space in order to force the RxC claim? There were some transformations that Hagen had in his work.
Could this break that theorem? This Klein four group is a neat find. You must have something up your sleeve to go here...

Possibly you have a claim here on emergent 3D space. That would be awesome! The K4 product has a kind of balance that seems to disrupt the RxC claim, but it has to be said that we are substituting over native product and keeping the geometry of polysign. I think this may get into improper transforms https://en.wikipedia.org/wiki/Improper_rotation
As you try to turn a shell inside out there is an intermediate where the shell is flat as a pancake. And another extreme where it is an axial form. These P4 gives with the native product survey. https://drive.google.com/file/d/1m-CmjBZKOtBMHDJQmRLaUQACT0e9GUfz/view?usp=sharing

That is basically a product survey. Same needs to be done for K4 I think. No promises but I'd like to get to it.

On Friday, July 22, 2022 at 11:22:18 AM UTC-4, Timothy Golden wrote:
> On Wednesday, July 20, 2022 at 3:38:36 PM UTC-4, michael Rodriguez wrote:
> > "Much of mathematics is grounded in the study of equivalences, and order relations. Lattice
> > theory captures the mathematical structure of order relations. Even though equivalence relations
> > are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is
> > not as well known as that of orders. The former structure draws primarily on group theory and, to
> > a lesser extent, on the theory of lattices, categories, and groupoids."
> >
> > https://en.wikipedia.org/wiki/Equivalence_relation#Algebraic_structure
> >
> > In the concrete and in the abstract, order theoretic and equivalences, that reverberates.
> >
> > By the way, if you want R^3 instead of RxC, change the p4 multiplication, swap the mod4 by klein four-groupe.
> [K4]
> -------------------------------
> - - = @
> - + = *
> - * = +
> - @ = -
> + + = @
> + * = -
> + @ = +
> * * = @
> * @ = *
> @ @ = @
>
> Interesting. Unique in four. I'd have to try to do some graphics with it. I don't see it. There are claims that if a product is associative then it will still yield the RxC type of behavior. This looks pretty flat though, or symmetric, at least.
> ( - 1 + 1 * 1 )( - 1 + 1 * 1 ) = @a *1 +1 *1 @1 -1 +1 -1 @1 = - 2 + 2 * 2 @ 3 = @1
> So there is one real axis. Excluding that line we have only the -,.+,and * components doing their completely symmetrical behavior, and bouncing us out of that plane onto the @ axis again. In that they are three signs we could sort of anticipate some P3 behavior, but
> ( - 1 + 1 )( * 1) = + 1 - 1
> ( - 1 + 1 )( - 1 + 1 ) = @ 1 * 1 * 1 @ 1 = * 2 @ 2
> yet there are three of these aren't there? They are completely symmetrical. There must be some broken algebra within the distribution of terms maybe? Oh I think I see it now:
> ( - 1 + 1 * 1 @ 1 )( - 1 ) = @ 1 * 1 + 1 - 1 = 0
> No that is OK. I wonder to what degree is it necessary to mangle the space in order to force the RxC claim? There were some transformations that Hagen had in his work.
> Could this break that theorem? This Klein four group is a neat find. You must have something up your sleeve to go here...
>
> Possibly you have a claim here on emergent 3D space. That would be awesome! The K4 product has a kind of balance that seems to disrupt the RxC claim, but it has to be said that we are substituting over native product and keeping the geometry of polysign. I think this may get into improper transforms https://en.wikipedia.org/wiki/Improper_rotation
> As you try to turn a shell inside out there is an intermediate where the shell is flat as a pancake. And another extreme where it is an axial form. These P4 gives with the native product survey. https://drive.google.com/file/d/1m-CmjBZKOtBMHDJQmRLaUQACT0e9GUfz/view?usp=sharing
>
> That is basically a product survey. Same needs to be done for K4 I think. No promises but I'd like to get to it.

Wow: https://drive.google.com/file/d/1B2D5KTg4L89fuFYqTQMZqRDHgsorJFYR/view?usp=sharing
This does deserve further work. Got to put keys on the sphere and get color gradation too. That would bring it up to standard product study graphic.
nSigned K4( const nSigned & z1, const nSigned & z2 )
{ // Klein 4 group style product
const unsigned int K4M[4][4] { {0,1,2,3},
{1,0,3,2},
{2,3,0,1},
{3,2,1,0} };
nSigned zk(4);

if( z1.n != 4 || z2.n != 4 )
{ cerr << "\n Klein 4 group product error: must be P4 values.\n";
throw zk;
}
for( int i = 0; i < 4; i++ )
for( int j = 0; j < 4; j++ )
zk[K4M[i][j]] += z1[i] * z2[j];
return zk;
}
void K4SpiralStudy( )
{ // orient the sphere to the identity axis
// Normalize the magnitude
int n = 4;
DualAnimation anim(200, 70.0, "K4SpiralStudy.gif");
RandomProjection p( n-1 );
anim.SetProjection( p );
SpiralShell u1( n-1, 0.05); // the red dot
SpiralShell u2( n-1, 0.05); // the shell
Cart n1, n2;
while( u1.Next())
{
anim.i1.GraphSignVectors();
anim.i2.GraphSignVectors();
while( u2.Next() )
{
n1 = K4( u1.pos , u2.pos);
n1.Unitize();
n2 = K4( u1.pos , u2.last);
n2.Unitize();
anim.i1.CartLine( n1, n2, u2.color );
anim.i2.CartLine( u2.cpos, u2.clast, u2.color );

}
anim.i2.CartPoint( u1.cpos, 3, 0xf00000 ); // graph the red dot
anim.Add( 10, true );
cerr << "+" << flush;
}
//cerr << "3.\n" << flush;

On Tuesday, July 26, 2022 at 8:46:38 AM UTC-4, Timothy Golden wrote:
> On Friday, July 22, 2022 at 11:22:18 AM UTC-4, Timothy Golden wrote:
> > On Wednesday, July 20, 2022 at 3:38:36 PM UTC-4, michael Rodriguez wrote:
> > > "Much of mathematics is grounded in the study of equivalences, and order relations. Lattice
> > > theory captures the mathematical structure of order relations. Even though equivalence relations
> > > are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is
> > > not as well known as that of orders. The former structure draws primarily on group theory and, to
> > > a lesser extent, on the theory of lattices, categories, and groupoids.."
> > >
> > > https://en.wikipedia.org/wiki/Equivalence_relation#Algebraic_structure
> > >
> > > In the concrete and in the abstract, order theoretic and equivalences, that reverberates.
> > >
> > > By the way, if you want R^3 instead of RxC, change the p4 multiplication, swap the mod4 by klein four-groupe.
> > [K4]
> > -------------------------------
> > - - = @
> > - + = *
> > - * = +
> > - @ = -
> > + + = @
> > + * = -
> > + @ = +
> > * * = @
> > * @ = *
> > @ @ = @
> >
> > Interesting. Unique in four. I'd have to try to do some graphics with it. I don't see it. There are claims that if a product is associative then it will still yield the RxC type of behavior. This looks pretty flat though, or symmetric, at least.
> > ( - 1 + 1 * 1 )( - 1 + 1 * 1 ) = @a *1 +1 *1 @1 -1 +1 -1 @1 = - 2 + 2 * 2 @ 3 = @1
> > So there is one real axis. Excluding that line we have only the -,.+,and * components doing their completely symmetrical behavior, and bouncing us out of that plane onto the @ axis again. In that they are three signs we could sort of anticipate some P3 behavior, but
> > ( - 1 + 1 )( * 1) = + 1 - 1
> > ( - 1 + 1 )( - 1 + 1 ) = @ 1 * 1 * 1 @ 1 = * 2 @ 2
> > yet there are three of these aren't there? They are completely symmetrical. There must be some broken algebra within the distribution of terms maybe? Oh I think I see it now:
> > ( - 1 + 1 * 1 @ 1 )( - 1 ) = @ 1 * 1 + 1 - 1 = 0
> > No that is OK. I wonder to what degree is it necessary to mangle the space in order to force the RxC claim? There were some transformations that Hagen had in his work.
> > Could this break that theorem? This Klein four group is a neat find. You must have something up your sleeve to go here...
> >
> > Possibly you have a claim here on emergent 3D space. That would be awesome! The K4 product has a kind of balance that seems to disrupt the RxC claim, but it has to be said that we are substituting over native product and keeping the geometry of polysign. I think this may get into improper transforms https://en.wikipedia.org/wiki/Improper_rotation
> > As you try to turn a shell inside out there is an intermediate where the shell is flat as a pancake. And another extreme where it is an axial form.. These P4 gives with the native product survey. https://drive.google.com/file/d/1m-CmjBZKOtBMHDJQmRLaUQACT0e9GUfz/view?usp=sharing
> >
> > That is basically a product survey. Same needs to be done for K4 I think. No promises but I'd like to get to it.
> Wow: https://drive.google.com/file/d/1B2D5KTg4L89fuFYqTQMZqRDHgsorJFYR/view?usp=sharing
> This does deserve further work. Got to put keys on the sphere and get color gradation too. That would bring it up to standard product study graphic.
> nSigned K4( const nSigned & z1, const nSigned & z2 )
> { // Klein 4 group style product
> const unsigned int K4M[4][4] =
> { {0,1,2,3},
> {1,0,3,2},
> {2,3,0,1},
> {3,2,1,0} };
>
> nSigned zk(4);
>
> if( z1.n != 4 || z2.n != 4 )
> { cerr << "\n Klein 4 group product error: must be P4 values.\n";
> throw zk;
> }
>
> for( int i = 0; i < 4; i++ )
> for( int j = 0; j < 4; j++ )
> zk[K4M[i][j]] += z1[i] * z2[j];
> return zk;
> }
> void K4SpiralStudy( )
> { // orient the sphere to the identity axis
> // Normalize the magnitude
> int n = 4;
> DualAnimation anim(200, 70.0, "K4SpiralStudy.gif");
> RandomProjection p( n-1 );
> anim.SetProjection( p );
> SpiralShell u1( n-1, 0.05); // the red dot
> SpiralShell u2( n-1, 0.05); // the shell
> Cart n1, n2;
> while( u1.Next())
> {
> anim.i1.GraphSignVectors();
> anim.i2.GraphSignVectors();
> while( u2.Next() )
> {
> n1 = K4( u1.pos , u2.pos);
> n1.Unitize();
> n2 = K4( u1.pos , u2.last);
> n2.Unitize();

Oops. I'm reusing code here. We are looking at unitized values in that graphic.
I was just about to investigate magnitude conservation but that lead is shut now.
This is going to get confusing because as google drive fills in with images I'm going to lose organization.
This last link will go stale I suppose. Get it while it's hot!
But it's sometimes not so hot.

> anim.i1.CartLine( n1, n2, u2.color );
> anim.i2.CartLine( u2.cpos, u2.clast, u2.color );
>
> }
> anim.i2.CartPoint( u1.cpos, 3, 0xf00000 ); // graph the red dot
> anim.Add( 10, true );
> cerr << "+" << flush;
> }
> //cerr << "3.\n" << flush;

On Tuesday, July 26, 2022 at 9:05:36 AM UTC-4, Timothy Golden wrote:
> On Tuesday, July 26, 2022 at 8:46:38 AM UTC-4, Timothy Golden wrote:
> > On Friday, July 22, 2022 at 11:22:18 AM UTC-4, Timothy Golden wrote:
> > > On Wednesday, July 20, 2022 at 3:38:36 PM UTC-4, michael Rodriguez wrote:
> > > > "Much of mathematics is grounded in the study of equivalences, and order relations. Lattice
> > > > theory captures the mathematical structure of order relations. Even though equivalence relations
> > > > are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is
> > > > not as well known as that of orders. The former structure draws primarily on group theory and, to
> > > > a lesser extent, on the theory of lattices, categories, and groupoids."
> > > >
> > > > https://en.wikipedia.org/wiki/Equivalence_relation#Algebraic_structure
> > > >
> > > > In the concrete and in the abstract, order theoretic and equivalences, that reverberates.
> > > >
> > > > By the way, if you want R^3 instead of RxC, change the p4 multiplication, swap the mod4 by klein four-groupe.

I corrected the unitization out and got a lucky projection too. The red axis is the identity axis and happens to be pointing down. The three real line effects of the nonidentity signs are showing nicely as the red dot progresses through those signs.
https://drive.google.com/file/d/1s3TERz-ZKZYMKXGe208Y2hd5cqhU8OP7/view?usp=sharing
Although it is a low res image it is high res stepping. Zooming in helps expose the red dot a bit.
Pretty sure this is a good K4P survey. Altering the acronym a bit to account for the polysign interpretation. I guess others would try this as orthogonal values, eh? Or not at all maybe. It's a bit reminiscent of XOR logic, but claiming an n-form seems impossible. Still, if you could grow it and it took on various forms then all would be well. I guess that puts us right onto the matrix form; commutativity as an assumption requiring symmetric values; the tatrix as an embedded form.

On Tuesday, July 26, 2022 at 10:02:43 AM UTC-4, Timothy Golden wrote:
> On Tuesday, July 26, 2022 at 9:05:36 AM UTC-4, Timothy Golden wrote:
> > On Tuesday, July 26, 2022 at 8:46:38 AM UTC-4, Timothy Golden wrote:
> > > On Friday, July 22, 2022 at 11:22:18 AM UTC-4, Timothy Golden wrote:
> > > > On Wednesday, July 20, 2022 at 3:38:36 PM UTC-4, michael Rodriguez wrote:
> > > > > "Much of mathematics is grounded in the study of equivalences, and order relations. Lattice
> > > > > theory captures the mathematical structure of order relations. Even though equivalence relations
> > > > > are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is
> > > > > not as well known as that of orders. The former structure draws primarily on group theory and, to
> > > > > a lesser extent, on the theory of lattices, categories, and groupoids."
> > > > >
> > > > > https://en.wikipedia.org/wiki/Equivalence_relation#Algebraic_structure
> > > > >
> > > > > In the concrete and in the abstract, order theoretic and equivalences, that reverberates.
> > > > >
> > > > > By the way, if you want R^3 instead of RxC, change the p4 multiplication, swap the mod4 by klein four-groupe.
> I corrected the unitization out and got a lucky projection too. The red axis is the identity axis and happens to be pointing down. The three real line effects of the nonidentity signs are showing nicely as the red dot progresses through those signs.
> https://drive.google.com/file/d/1s3TERz-ZKZYMKXGe208Y2hd5cqhU8OP7/view?usp=sharing
> Although it is a low res image it is high res stepping. Zooming in helps expose the red dot a bit.
> Pretty sure this is a good K4P survey. Altering the acronym a bit to account for the polysign interpretation. I guess others would try this as orthogonal values, eh? Or not at all maybe. It's a bit reminiscent of XOR logic, but claiming an n-form seems impossible. Still, if you could grow it and it took on various forms then all would be well. I guess that puts us right onto the matrix form; commutativity as an assumption requiring symmetric values; the tatrix as an embedded form.

Sorry this last gif was so large. You have to download it apparently. 12.7 MB is not thoughtful at all.
It is thorough though. No idea why they won't just pass it through to your browser.

At some level we should just be happy to have this 3D math, though that is enabling a misnomer. Every time I explain that P4 is 3D I am dissing this thread's spirit. How to proceed? Explain that some angular progression ultimately leading to ninety degrees is to blame? That the square is not actually natural until you've made it there? And that even then you'll be off by one? That you already obeyed the simplex rules but stopped at two?
When you are off a little you can typically correct by offsetting by some small amount. When you are off by a lot nothing will come clean. To say that polysign is only marginally better is like saying that you can do what polysign can do, and yet it has not been done. Admittedly my projections are Cartesian. My code has only gotten part way to where my mind has gone.

On 6/23/2022 7:04 AM, Timothy Golden wrote:
>
> On a sheet of clean paper draw two horizontal lines,
> labeling one X, and the other Y.
X is usually horizontal and y is vertical. Their intersection is the origin.
> Now choose a position along X and label it 'a', and choose a position along Y and label it 'a'. This is consistent with the Cartesian representation of (X,Y). Indeed if you were to shift the depiction of X or Y to anywhere on the paper the representation will remain the same, and this then embodies the concept of independence: that X and Y are completely unrelated to each other, other than the fact that they are exact copies of R. In effect this is a sufficient representation of (X,Y) and any further work in geometrical rendering will require additional constraints which will reduce the independence of X and Y. By reducing the independence then the Cartesian product must be weakened. So how is it that the curriculum has come to blur this detail? Shall we blur it back the other way and ask where does a ninety degree angle come from? It comes from thin air, and any angle other than zero degrees will suffice to preserve graphically the rendering. Even the units of the two 'axes' should not be assumed to be equivalent, yet if you choose to preserve that then clearly the criteria of independence of X and Y has been further blurred. Possibly the angular argument should be pushed to zero degrees, and the origins spread sufficiently so that both images fit cleanly upon a single line. Thence even the representation of finite systems will be achieved on a single line via some remapping method. Of course the same label will show up twice on that line, but with the bounds being sufficiently offset an unambiguous interpretation can be made of the two dimensions out of a one dimensional system, and clearly R was that.
>
> The graphical interpretation of the line as geometrical is unambiguous. What is ambiguous is the notion that granting two of these suddenly engages the plane. Was this plane constructed via the Cartesian product? Where in the Cartesian product are these demands? If anything the independence of the Cartesian product components ensures that such clear definition is lacking.
>
> What we have then is a system that has been engaged in by convention. Even the name "Cartesian" is a misnomer. He never used this and he never confused these two things. He did bring algebra into geometry, but his geometry was very Euclidean in nature. His usage of negative values as 'false' values is well established.
>
> That sign develops the real line via the balance of two rays such that
> - 1 + 1 = 0
> is where the geometrical integrity of the line begins. That the plane's geometrical integrity begins with:
> - 1 + 1 * 1 = 0
> has been overlooked by everyone until now afaict. This is the proper constraint that yields the plane and two dimensional thinking. Yet this term 'dimensional' has been tied to the real line as if that real line was fundamental, but it is not. So we have cracked open a rather large can with a rock here. Dipping a twig in we can get a bit of the juice out to taste it and it tastes pretty good. This can has been sitting around for four hundred years.
> Unopened, untampered with, and yet perfectly preserved.
>
>

Chris M. Thomasson wrote:

> On 6/23/2022 7:04 AM, Timothy Golden wrote:
>>
>> On a sheet of clean paper draw two horizontal lines,
>> labeling one X, and the other Y.
>
> X is usually horizontal and y is vertical. Their intersection is the
> origin.

you american must be thinking we are stupid. The nonexistent "uKraine" is
the *milky_cow* of the bidans. They invest money, military and personnel
in their *_milky_cow_*. How such crap is legal in america, no idea.

here you have it, the gay actor on cocaine, no party, no politics, blessed
"president" and military admiral expert by the bidens. And the capitalist
europe are buying it and ask for more.

What’s Wrong with the Actor 😳
https://www.bitchute.com/video/vDn6I5JdfmtP/

kind of exotic, the p4 study gif

On Saturday, September 3, 2022 at 4:16:24 PM UTC-4, michael Rodriguez wrote:
> kind of exotic, the p4 study gif
Yes. Assuming you're talking about your K4 product. How it embeds by the rules of abstract algebra or associative algebra is of interest. Claims that this will map to RxC are I suppose contingent on stating a transform. The symmetry of three of the signs expose that reasoning as limited, I think.

On Monday, September 5, 2022 at 11:24:15 AM UTC-4, Timothy Golden wrote:
> On Saturday, September 3, 2022 at 4:16:24 PM UTC-4, michael Rodriguez wrote:
> > kind of exotic, the p4 study gif
> Yes. Assuming you're talking about your K4 product. How it embeds by the rules of abstract algebra or associative algebra is of interest. Claims that this will map to RxC are I suppose contingent on stating a transform. The symmetry of three of the signs expose that reasoning as limited, I think.

That all rests on some Cartesian product is exactly what I am convinced is not true. Still, the kaleidoscopic effect may be accurate. It is possible that I myself have not fully challenged the Cartesian paradigm as I was brought up in its hold. I am trying and I am prying, and the lid has budged.

To what degree the tatrix is receiving this treatment from us now is trouble I suspect. I can't help but do it:
P1 P2 P3 P4 ...
is the tatrix. Another form:
a10
a20 a21
a30 a31 a32
...
exists. This is the tatrix too. Now a third form:
p1, p2, p3, p4, p5, ...
of deltas will develop a tatrix as well when we let those p be coded correctly, or rather structured correctly. While this becomes a mixed modulo concept it is done with an eye for a segregated modulo structure. It is the last code of the number here that I am punting on. It is the one which we operate upon as assumed as modulo-ten. Yes, we already broke it, but is it there in the number? Is there a decimal point in the number? A sign? Is the number itself a sequence of digits? Certainly the meaning of the number is contingent upon its interpretation, and no confusion should be possible, and in this way to claim that a number does contain its modulo footprint is entirely reasonable. Of course then to respect this both in sign and in magnitude; well is it an option or should we have to code twice? That's confusing. Even the lowly decimal point that wants to point out unity; what of its number and its modulo flavor? Is this a pure number system?

We are approaching it, and yet here the initial problem was to challenge the Cartesian assumption as if P2 and P3 have some magical orthogonality between them. They are clearly independent. So independent that they do not deserve this treatment, especially given the lengths to which we have gone against the grain of the Cartesian mindset. In this way my own claims of emergent spacetime as
P1 P2 P3
really are withering on the vine. We have to step it up:
P1 P2 P3 | P4
This holds the spirit of challenging the Cartesian assumption. That we land with ten components seems a bit of a slip. That they can be taken down to six is quite a gain. Ah, the six again...

Room for electromagnetics seems extravagant, but that's all been done. Just not on the progressive form.