On Wednesday, September 1, 2021 at 6:57:11 AM UTC-7, timba...@gmail.com wrote:
> On Tuesday, August 31, 2021 at 10:14:24 AM UTC-4, Ross A. Finlayson wrote:
> > On Thursday, August 26, 2021 at 7:57:47 AM UTC-7, timba...@gmail.com wrote:
> > > On Monday, August 23, 2021 at 5:52:02 PM UTC-4, Timothy Golden wrote:
> > > > On Monday, August 23, 2021 at 9:54:26 AM UTC-4, Timothy Golden wrote:
> > > > > On Monday, August 23, 2021 at 9:44:25 AM UTC-4, Timothy Golden wrote:
> > > > > > On Thursday, August 19, 2021 at 4:29:00 PM UTC-4, Timothy Golden wrote:
> > > > > > > Rotation As Fundamental
> > > > > > > ----------------------------------------
> > > > > > > 1 rotation.
> > > > > > > 2 rotations.
> > > > > > > 1/2 rotation.
> > > > > > > 1/3 rotation.
> > > > > > > 3 rotations.
> > > > > > >
> > > > > > > The interpretation is not clear as to which is the partial rotation and which is the multiple of a singular full rotation. This is to say that one third of three rotations is unity as well as that three times one third of a rotation is as well one rotation. That these in multiplicity congrue is not a surprise. A complete and congruent rotational spectrum is permitted. Interpreting
> > > > > > > 1.5 rotations
> > > > > > > 1 rotations
> > > > > > > 1.9 rotations
> > > > > > > 0.9 rotations
> > > > > > >
> > > > > > > Yet should the reciprocal rotation be entered into the mix? In effect three or 3 codes for the number of turns it takes to get one rotation. This natural unital form does pre-exist the real number. It's own modulo format is physically corresponding. That the winding number gets thrown away and that it remains available as well if you wish; this is not unlike polysign where for instance the real numbers have two components yet if you like we can always zero one of them. This then exposes how Pn are n-1 dimensional. In effect the bidirectional line is the first geometry to break naught. As if it unfolds is how the generations nay siblings of polysign unfold. So it is possible to implement an n-ary gradation down to ultrafine resolution should a large n be selected; winding be damned. These are the strength of the reciprocal rotations ( rr ) and strangely at zero they command an infinity. This zero happens to relate to P2 where a zero angle rotation implies an out and back sort of structure be maintained. This is a biray concept; holding them to naught. -1+1=0 in an out and back sort of way; just as the geometry commands, the signon implies, and the real number overlooks. Here is the duplicity again. Seemingly optional yet possibly profound.
> > > > > > So polysign numbers are naturally rotational. Discrete sign n under product do perform discrete rotations. Powers of minus unity (MU) cover the space. In this way polysign form a more fundamental system than the Cartesian system allows for. The rules of complex analysis are superfluous under polysign, since P3 recovers C as P2 recovers R. Pn are all rule behaved with no exceptions. This includes the underling P1 which goes unaddressed in modern mathematics... presuming polysign is still not admitted into modern mathematics, which may not be the case much longer.
> > > > > > I speak here as an outsider to the existent system, but one who has some level of training in it. Regardless, the rotations of polysign are not so straightforward as the OP, which has gone strictly to rotations as a sole means of a metric. As we consider n-signed numbers (Pn) their dimensional form due to the (metric?) balanced geometry sum over s of sx (where x is being held constant) equals zero:
> > > > > > - 1 = 0 (P1)
> > > > > > - 1 + 1 = 0 (P2)
> > > > > > - 2.3 + 2.3 * 2.3 = 0 (P3)
> > > > > > - z + z * z # z = 0 (P4)
> > > > > > ...
> > > > > > these discrete rotations are geometrically bound to these simplex coordinate systems. Under this thinking the real line as rotational is exposed. The real line as capable of carrying bidirectional segments to form a one dimensional simplex is necessary. In other words a structure on the real line whose evaluation is naught has a double segment which you will observe by taking your n-simplex down to n=2. Of course we go further in polysign to n=1 as well, and possibly from this perspective it can be adopted by some who deny the possibility of generalized sign. That these subtle differences exist in polysign from the ordinary geometry should be stressed for fear that they go overlooked as if they are merely a recover of the Cartesian train of thought. They are not. They breed the concept of dimension independently of the Cartesian thinking. They demand their geometry from the balance. That simple statement of balance secures their geometry and their algebra. That simple statement is not an ordinary part of the real number construction as real analysis (RA) teaches it. RA posits that an inverse exists and inherently defines that inverse as
> > > > > > - x
> > > > > > which clearly is the usage of the sign to imply the inverse, and as well as it works in RA, it has bound and gagged their progression. This sad fact will have to be addressed in the future, and for now the readers who bother to understand this discussion will have to sit on their hands a bit... decide whether or not to participate in the new math, while holding onto their position in the old structure. There is a direct conflict of interest here which endangers both the status quo mathematician and the early adopter of polysign. This said, there is tremendous room for development of polysign numbers into calculus; into physics; on and on even into discrete mathematics. Clearly the effect of discrete sign upon the continuum of magnitude is profound. The inability of people trained upon the 'real' number to adopt it is suggestive. The simplicity of polysign and its generality are fully established.
> > > > > >
> > > > > > I do not claim a sealed argument in terms of rotations as units within polysign. In effect a value z in Pn has these arms or rays
> > > > > > MU^n z
> > > > > > whose sum will return the user to their starting position. All such combinations form a signon: a structure that packs the space. The signon is not the simplex, though it is closely related, and further that signon has a significant detail in that the structure has unidirectional segments within its structure that include interior segments. These features are not an ordinary part of geometry. This harkens back to the consideration in P2 of the simplex geometry and that double coverage of what was the real line. It alone is the one system which possesses these exact returns in one step. Sometimes P2 appears as degenerate as P1, and yet as they are the low down forms of a general system we must not declare such exceptions as laws, but rather as constructive opportunities. Polysign promise some interdimensional thought system remains to be declared and leveraged formally. To what degree am I maintaining a photon as a unidirectional zero dimensional entity and betraying any P2 activity? If the photon is indeed a particle, and its transmission is at the speed of light, and relativity holds, then its collapse of general space is guaranteed on its own terms and its instantaneous action within its own reference frame whereby it ages exactly zero seconds from transmission to reception; this ladder of detail suggests that the return trip of an antiphoton is trivial. This is a brand new thought, but the idea that such thinking is coming into existence via a pure math form is promising. I do not however claim it to be complete. The point to me is that interpretation is largely what we are doing, yet the falsification of RA and the usage of it in the basis of physics will have to be regarded as a sore point to future generations. I am merely an early follower of polysign.. My own abilities are badly limited. Yet that such thinking can come from taking these fundamental ideas seriously ought to provide enough evidence to an onlooker that there is something here; much as the habituated human mind would like to deny the possibility. Sign can and has been generalized. This concept is called polysign numbers. They have rotational qualities. They have geometry. They have multidimensional qualities, though this term is couched in Cartesian logic. They make no use of that Cartesian logic. This is how profound they be.
> > > > > Not sure if this could augment an interpretation, and the general dimensional form cannot be denied:
> > > > > http://bandtechnology.com/ConicalStudy/conic.html
> > > > The idea that we discuss a supercone and a subcone essentially imply that the plane does take the interpretation that the plane is a cone. That the sub-cone folds down to a ray: This becomes a subdimensional interpretation. The realists as fundamentalists might claim that the double-cone is natural, yet here is it a degenerate real interpretation? To claim that a P1-P3 structure exists without P2 in the middle is an interdimensional concept that lacks general dimension. The ray and the cone.... peas in a pod... the square of the sphere is a cone... On this subject it may be a good time to drone... the square of the line is a ray... and now this does lay prone.... The center of the cone is a ray. Hur-ray Hurrah, Here ray, Here rise, Here Lays Her Lies. The ray is more fundamental than the line will ever be.
> > > >
> > > > The line as bidirectional is under-appreciated. I don't mind having some corrections and stipulations to fill in, and to what degree will every law be supported by theory versus theory be supported by law? It seems foolhardy to claim photon pairing, though I do think this has been tried. To claim the possibility of an inverse photon... should we be discussing BTU's of photons? The ray pair can exist in this way out and back.
> > > Having stretched and squeezed a continuum albeit with a discontinuity should we try it the other way around? All that quantum physics business and the idea that some other mapping could exist with a
> > > continuous/discrete
> > > awareness... Then too what if there is a third and our old FOXP2 has us struck on a bifold? Skewered on a bipole? Solidified in a bivolumetric way?
> > > No, let's have none of that...
> > > But for our n-ary awareness the m-ary of the n-ary can feel quite scary yet next comes lary. I was once hazed by a herd of baby sat kids. They stuck me in the bathroom after lary did a number two. They even closed the window. What a gag. No, really, it was pretty bad.
> > >
> > > We do see in the polysign progression a natural form
> > > a10
> > > a20 a21
> > > a30 a31 a32
> > > -------------------------
> > > a40 a41 a42 a43
> > > a50 a51 a52 a53 a54
> > > ...
> > > The bar is meaningful (though it is arguably optional) it is worth discussing and has been discussed plenty. Emergent spacetime and all that. Unidirectional time in support. Complex values to boot. General dimensional algebra not too shabby. Whatever, the point is to go on from here rather than end here. Progressive mindset and so forth.
> > >
> > > Some of the first rotational values that we are trained upon are our digits
> > > 1234567890 (Hurray for qwerty getting this right)
> > > Monday Tuesday Wednesday Thursday Friday Saturday Sunday
> > > which seems quite contrite but for the fantastic eight and its infinite potential seemingly double zero 8,
> > > as if to say,
> > > "return here for more of the same"
> > > and no doubt we will find the eight that way.
> > > Four sevens are not going to save the day, however.
> > > The point is more the ring and its encryption as a series. That we ought not to repeat ourselves lest we seem to be stumbling or bumbling idiotically through it all as blathering turds attempting to outdo each other with dirty vagrancies about others smelly parts, chuckling all the way. It's sad how humans can devolve this way. So composure to the rescue, https://en.wikipedia.org/wiki/Nicolae_Ceau%C8%99escu https://upload.wikimedia.org/wikipedia/commons/b/bc/Ro-Nicolae_Ceau%C8%99escu.ogg ,
> > > for shoe makers and bus drivers should look the other way.
> > >
> > > Anyway the early rotational form is way back there; not any further along... a discrete form that we are forced to express ourselves in here. That this discrete nature is a lie: readily proven by the number of days in a year and worse in a month, especially where the moon is concerned.
> > > That we arrived in minutes and seconds...
> > > Can we possibly get thirds?
> > > It strikes me that in all of this we have an option to go back to one rotation as fundamental. Relative reference begins here.
> > >
> > > All in all, when it comes time to assigning smelly digits on a global basis, my own cuntry and it's shenanigans have gone farther I believe than anyone else, and so we do in fact deserve credit as the dirtiest douchebags; chuckling our way along as we stick our fingers in other people's faces. Possibly the French have their own odour about them, and the English too.... Like good mathematicians we ought only to care about the digits and not who they belong to. Peace On.
> > All about the continuous and discrete the lines and points,
> > there's for that the continuous is infinitely divisible,
> > then that the integral defines whole measures,
> > about that re-Vitali-izing for measure theory has the
> > discrete basically double the points on the line,
> > thast thre are so many one-sided points in the line,
> > and at least twice more two-sided on the line,
> > those polydimensional to all the lines on and in.
> Seems pretty cryptic. Where you say: "discrete basically double the points on the line" could you expound? I have bumped into a double coverage on the real line, which is found as the P2 signon: http://bandtechnology.com/PolySigned/Lattice/Lattice.html which is arguably such an instance as you are discussing between continuous and discrete. Can you imagine if we've swapped a few things around twice over and come out the other side thinking we are coherent? This is just the sort of remap we ought to hope for.
>
> From the lattice side of things: our numerics develop such a lattice with natural values. These arguably in polysign are Qn rather than Pn. We can then go on to looking at our new fangled graph paper in Pn though it was developed via Qn principles. Whether the cells are rectilinear or simplex based should not matter should it? Clearly simplex based are more natural and more extensive in that they develop algebra in any discrete dimension. They all exist all at once:
> Q1 Q2 Q3 Q4 Q5 ...
> and this is the nary experience. The geometry of these are arguably like graph paper. Ohhh.... does this mean that graph paper should only be used pixelated? I never got too far into that. This is roughtly the problem though with lines having no thickness. Performed on our displays here they had better have at least one pixel of thickness. So the pencil is to blame...

More later though I've written some many posts here about
one-sided points in a line
two-sided points on a line
3/4/5 sided points according to lines defining points in a plane
.....

The continuous and discrete of course is one of the central understandings
of the objects of mathematics.