AP, King of Science, is asking a strange question if we can keep the 4 quadrants afterall and redefine function.

In the course of writing my 221st book of science to be published in the next few days, an idea emerged. It may pan out as an important idea or it may just fizzle away into nothing.

We all know that a circle graph is not a function for it violates the uniqueness of one y-value for every x-value. A semicircle placed properly can be a function.

We cannot have negative numbers and achieve a geometry proof of Fundamental Theorem of Calculus.

And we require 1st Quadrant Only for FTC.

So, well, what if we had 4 Quadrants and all 4 having only positive number values.
1
|
__1__0__1__2
|
1

ascii art diagram sketch of 4 quadrants with only positive number values

And if we tack on 3rd dimension with a z-axis all being positive numbers only.

What I am exploring is the idea-- we have no negative numbers. But if we have positive numbers only and 8 quadrants in 3D and 4 quadrants in 2D, can we extend and expand the concept of Function, so that a circle comes a function and keeps "uniqueness".

This would be of huge benefit to physics as EM is all about circuits, closed loops, and it would be benefical to circuit math.

I am the King of Science and must pursue every lead, no matter how strange.

AP

What drives me to this question is that all of Physics is Atomic physics and all of atomic physics is electricity and magnetism. Central to electricity is the electric circuit. You have no electric current without the closed loop circuit.

Yet math shuns the closed loop circuit as not even being a function.

So, looking at the 4 quadrant graph in 2D, I certainly know there are no negative numbers-- negative numbers are a pile of shit invented by kooks and retained by kooks.

But, what happens if we keep 4 quadrants in 2D and instead of negative numbers, fill the axes with just positive numbers.

Does this, then gain us the Closed Loop as a function???

So we know the formula of circle is (x-a)^2 + (y-b)^2 = radius^2. Where (a,b) is the center of circle. Of course when center is the origin (0,0) we have x^2+y^2 = r^2.

But now what happens with that equation if we had 4 quadrants all with axes of positive numbers.

It looks to me as if, we can treat the Circle as a function of mathematics.

And better yet treat the sphere in 3D as also a function, even though it is a closed loop.

AP, King of Science, especially Physics

On Wednesday, December 14, 2022 at 8:57:21 PM UTC-6, Archimedes Plutonium wrote:
> What drives me to this question is that all of Physics is Atomic physics and all of atomic physics is electricity and magnetism. Central to electricity is the electric circuit. You have no electric current without the closed loop circuit.
>
> Yet math shuns the closed loop circuit as not even being a function.
>
> So, looking at the 4 quadrant graph in 2D, I certainly know there are no negative numbers-- negative numbers are a pile of shit invented by kooks and retained by kooks.
>
> But, what happens if we keep 4 quadrants in 2D and instead of negative numbers, fill the axes with just positive numbers.
>
> Does this, then gain us the Closed Loop as a function???
>
> So we know the formula of circle is (x-a)^2 + (y-b)^2 = radius^2. Where (a,b) is the center of circle. Of course when center is the origin (0,0) we have x^2+y^2 = r^2.
>
> But now what happens with that equation if we had 4 quadrants all with axes of positive numbers.
>
> It looks to me as if, we can treat the Circle as a function of mathematics.
>
> And better yet treat the sphere in 3D as also a function, even though it is a closed loop.

Now every circle no matter where it is situated in the 4 quadrant plane graph, can be split in half into two semicircles. And every point on the circle, except the equator points have what can be seen as podal and antipodal points.

So a circle with center (0,0) and radius 3 will have its equator podal and antipodal be (3,0) and (3,0) where all axes are positive numbers only. Now the pole of this circle will be (0,3) for north and the same for south (0,3). In Old Math those would be (0,3) with (0,-3). For x= 1 we have 1+y^2 = 9 and y= sqrt8= 2.828. So we have the point of the circle in northern hemisphere as (1,2.828) and in Old Math we have that antipodal point be (1,-2.828). But in New Math we have no negative numbers so we have the southern hemisphere be (1,2.828) and in addition to the leftside we have this same two points above and below x-axis.

What I am trying to fathom here, is can I say the graph of a circle or any closed loop figure obeys the definition of Function once I discard negative numbers and replace them with all positive numbers and thus keeping the 4 quadrants in 2D?

We can easily call the semicircle as being the function and that we simply adjoin the mirror image of the semicircle.

What I am attempting to do for physics and math, is preserve Function definition but also gain closed loop figures as being wholly functions also. And to obtain this desired gain, I throw out negative numbers and in place install only positive number axes.

Now a gain of this in function means nothing unless it is a gain in calculation. Can I obtain the area of a circle or of a sphere by having only positive number axes and then the integral gives me the volume of a sphere in 3D, and the integral gives me the area of a circle in 2D. Of course that is broadened to include all symmetrical figures-- ellipse, ellipsoid, oval, ovoid, etc.

A breakthrough in math is no breakthrough unless it makes some calculations much easier.

AP, King of Science, especially Physics

So, what I am getting at here, if true, is that 4 quadrants for 2D, and 8 quadrants for 3D was needed, not to carry negative numbers, for they are meaningless and delusional, but needed to make closed loop circuits be a Function.

Most of physics math involves electromagnetism of a closed loop circuit. And the way to get that math of a closed loop circuit to be a function is to have all 3 axes, x,y,z be positive numbers. This preserves "uniqueness" in the function definition.

Negatives numbers are fiction. But the need of 4 quadrants for a circle and a function to be a circle is a pressing need.

But if the 4 quadrants in 2D and 8 quadrants in 3D make no advances in integral for circle or sphere, then this adventure grinds to a halt.

AP

On Thursday, December 15, 2022 at 3:14:46 AM UTC-6, Archimedes Plutonium wrote:
> So, what I am getting at here, if true, is that 4 quadrants for 2D, and 8 quadrants for 3D was needed, not to carry negative numbers, for they are meaningless and delusional, but needed to make closed loop circuits be a Function.
>
> Most of physics math involves electromagnetism of a closed loop circuit. And the way to get that math of a closed loop circuit to be a function is to have all 3 axes, x,y,z be positive numbers. This preserves "uniqueness" in the function definition.
>
> Negatives numbers are fiction. But the need of 4 quadrants for a circle and a function to be a circle is a pressing need.
>
> But if the 4 quadrants in 2D and 8 quadrants in 3D make no advances in integral for circle or sphere, then this adventure grinds to a halt.
>
So now, to get any sort of traction or momentum in this new idea, I need to be able to get the volume of cube, volume of sphere via Calculus, upon graphs in 3D using only polynomials and using only positive numbers on all 3 axes.

In my bath last night-- I seem to do a lot of thinking in the bathtub. I can easily see the prefix number of volume of sphere, moving from x^2 to (1/3)x^3 as integral.

Recall the volume of sphere is (4/3)radius^3*(pi)

Now the volume of cube is side^3. And we need to position the cube as 1/2 cut-out positioned so the cube center is (0,0) in 2D and to where it is not
___
| |

but rather positioned as this

/\

with (0,0) as center in 2D. And so the angle at apex is 90degrees.

And so the area under /\ is 1/2 side^2 of square, full area of square is 2*(1/2)side^2.

Now for sphere in 3D, I am going to place that sphere with center on (0,0) in 2D and enclosed inside a square -- cube.

If this New Math works, everything that looks like a ellipse ellipsoid is enclosed inside a rectangle or rectangular box.

Likewise, everything that looks like a Oval is enclosed inside a isosceles trapezoid or its 3D analog.

So, well, I can get all square areas to be 2*(1/2)side^2 thus side^2 = x^2.

But can I get x^3 for the cube volume? Keeping in mind these are closed loop figures and thus not a function in Old Math. Here in New Math, we toss out negative numbers, keeping only positive numbers and the 4 quadrants in 2D and 8 quadrants in 3D.

So I need to get volume of cube as x^3.

Taking 1/2 of the cube centered at (0,0,0) and shaped like this /\ over (0,0,0).

The integral of x^2 is (1/3) x^3.

Can I get rid of the 1/3?? Well, there is just 1/2 of the cube volume, and (2/3)x^3 does not give the result.

So, what I need is to drop back down to 2D and envision the 1/2 cube as a triangle /\ and enclosed inside something that gets rid of the 1/3.

AP

Now I am making this more difficult than it actually is.

So the square is cut in half and placed center on (0,0) in 2D looking like this /\.

So I have a line Y=x and integral is 1/2x^2.

Since there are 2 of these /\ I end up with area of square as x^2.

Now let me see if I get volume of cube as x^3.

I am in 3D now with the cube. Actually 1/2 of the cube positioned over (0,0,0) in 3D with axes x,y,z all possessing only positive numbers. For those that tuned in late, I am trying to preserve closed loops as being functions.

Now this cube solid looking like this |__| over (0,0,0) has a volume of 1/2 of full cube. This area is 1/2 side^2 and taking the integral is 1/6 side^3. Since there are 2 of these the volume is 1/3 side^3. No, I still need to get rid of the 1/3.

AP

If I am successful in this, I will have created the 3D calculus, computing volume from area in 2D.

But as it stands at the moment, that is a longshot.

AP

No, this is not working at all for integral of 2D to 3D volumes.

But there is some computation value in the periodic functions such as sine and cosine where they dip below the x axis or leftwards of 0.

This idea of preservation of closed loops as Functions is a worthy goal.

And the computation appears to be in periodic functions.

So I shift from volume to periodicity.

AP

No, this is not working at all for integral of 2D to 3D volumes.

But there is some computation value in the periodic functions such as sine and cosine where they dip below the x axis or leftwards of 0.

This idea of preservation of closed loops as Functions is a worthy goal.

And the computation appears to be in periodic functions.

So I shift from volume to periodicity.

I have the need of this new math-- where I make all the axes as positive numbers. There was no way in Old Math to graph the longitudinal wave, the sound wave, but my closed loop functions allows compression and rarefaction geometry graphs.

AP

Alright, I made tremendous huge progress here, and this definitely will be another book of mine. I cannot promise 3D calculus. But I do promise clarity for calculus.

I am struck awed by the infantile baby omission of a definition of Integral in Old Math. And one of the most important Theorems in all of math, geometry and algebra-- prove the volume of rectangular box is that of length x width x height. I do not mean being fed a observation in Grade School or High School but a formal proof that Volume = length x width x height. It is missing in Old Math.

I have resolved yesterday's hardships-- how to get rid of the 1/3 in the moving of x^2 area to an integral in 3D as (1/3)x^3. So that the volume of a cube is x^3. I have solved that. And it was a pitfall of Old Math. Going back to the fact that Old Math never proved volume of box. Old Math did have a definition of derivative as dy/dx, but left the definition of integral -- thoroughly in obfuscation as just saying "area under function graph".

Math is the science of precision, and something like "area under function graph" is not going to fly, or not going to cut the cake.

I am going to retain this idea that the function can include a closed loop figure such as the square or circle or any closed loop figure, provided we throw out Negative numbers in the 4 quadrants and have only positive numbers on all three axes of 3D. When we have only positive numbers, our function definition allows "uniqueness" to a closed loop.

But, to double check my claims of this uniqueness preservation, I need to double check if a Polynomial can be rendered of a closed loop. In this I earlier today asked this question in sci.math.

Thanks Julio, I was not aware that Fred Jeffries was a part of the stalkers-attack-insane crowd.

I was just about to ask Jeffries to run this through his computer for a Lagrange interpolation of a Polynomial

Polynomial Transformation Generator tool: The Polynomial Generator is this tool:

For 2 coordinate points, (x0, y0) (x1, y1), we produce the 1st degree polynomial, a straightline or line segment

P(x) = y0(x-x1) / (x0-x1)
+ y1(x-x0) / (x1-x0)

For 3 coordinate points, (x0, y0) (x1, y1) (x2, y2), we produce the 2nd degree polynomial, a compilation-curve

P(x) = y0(x-x1)(x-x2) / (x0-x1)(x0-x2)
+y1(x-x0)(x-x2) / (x1-x0)(x1-x2)
+y2(x-x0)(x-x1) / (x2-x0)(x2-x1)

For 4 coordinate points, (x0, y0) (x1, y1) (x2, y2) (x3, y3), we produce the 3rd degree polynomial, a compilation curve

P(x) = y0(x-x1)(x-x2)(x-x3) / (x0-x1)(x0-x2)(x0-x3)
+y1(x-x0)(x-x2)(x-x3) / (x1-x0)(x1-x2)(x1-x3)
+y2(x-x0)(x-x1)(x-x3) / (x2-x0)(x2-x1)(x2-x3)
+y3(x-x0)(x-x1)(x-x2) / (x3-x0)(x3-x1)(x3-x2)

4 points of (0.1,0) (0.9,0)(1,1)(0,1) That is almost a square, and I want to see if it is a Polynomial from those 4 points. To see if polynomials can achieve a circuit. I would be happy if it almost achieves a closed loop.

So Julio, my asking of Jeffries is probably going to be met with the hate-stalkers-guild in sci.math and of no answer at all, but more emoji hate stickers.

Thanks I thought Jeffries was above the fray, but apparently not.

AP

Alright, AP does not have many geometry textbooks at home, and especially none which are Grade School level. I have a High School level geometry textbook-- Harold Jacobs, 2nd edition, Geometry, 1987, page 552.

I am surprised of how stupid Old Math was with a proof of volume of rectangular box is length times width times height.

Usually Jacobs is excellent on math clarity. But in this instance, he is obfuscation.

He starts with Postulate 15 -- The volume of any prism is Bh, in which B is the area of one of its bases and h is the length of its altitude.

With that, Jacobs proceeds to make volume of rectangular box a corollary.

Corollary 1 : The volume of a rectangular solid is lwh, in which l, w, and h are its length, width, and height.

Ugly ugly ugly Old Math, and they failed on volume of rectangular box.

It would have been much better for Jacobs to have completely omitted that proof altogether, and stuck with classroom Observation.

Define Volume as a 1 by 1 by 1 cube. Simply define it as such.

Now note, we have a rectangular box, of 10 by 3 by 5 say. When we ask for volume, we are asking how many of these 1by1by1 cubes fit inside.

So in Grade School what we do is divide the 10 into 10 segments. We divide the 3 and 5 into 1 unit segments. Then we ask the students, how many of those 1 by 1 by 1 cubes are inside that box of 10 by 3 by 5.

Later we show the students that instead of counting we just multiply 10*3*5 = 150. Maybe I should have started with a smaller box say 2 by 3 by 4.

Anyway this teaches not only definition-- Volume is how many 1by1by1 cubes, but that V= L*W*H is obtained by a proof of mathematical induction. Not from Jacobs obfuscation.

AP

So I resolved this problem of getting rid of the 1/3 in the going from 2D of x^2 to going to 3D for x^3 as volume of cube.

You see, I go from a Y = x to a integral of 1/2x^2 and since there are 2 of those triangles, I have 2*(1/2)( x^2) for area in 2D.

But I was stuck with going from x^2 to volume as x^3 in 3D. Stuck because I saw no way of getting rid of the 1/3. Until today.

To do the area, I needed Y = x.

To do the volume in 3D, do I need Y =1, then Z = 1 then X=1 ????

Each of those integrated is x and x and x, giving me x*x*x = x^3 and the 1/3 never arises to give me problems.

AP, King of Science, especially Physics

Alright this came out beautifully for me.

There are no negative numbers, and there is no function in Old Math that is a closed loop. The best we can do here is a semicircle.

The very best that a Polynomial function graph can do is a semicircle.

So well, Physics needs functions that are Closed Looped, a full circle or full ellipse or full oval. How does physics get that????

The answer lies in that we do nothing to change the definition of function and keep its uniqueness. However we change our Graph Coordinate System. We have 4 quadrants in 2D and 8 in 3D. All three axes in a Coordinate System have only Positive number values.

So, what we end up doing is finding a Polynomial to be 1/2 a circle and another independent polynomial by the other 1/2 circle and join them together. If the first semicircle is on the x-axis in 1st quadrant then the 4th quadrant semicircle polynomial joins up and completes the circle.

This involves some theorems-- the maximum curve of a polynomial is 1/2 circle, 1/2 ellipse, 1/2 oval.

Another theorem is -- every 1/2 circle is joined by a independent polynomial to make a full circle from 2 polynomials.

Theorem-- we need all the axes of graphing to be positive numbers only and so when you need to graph a (1,-2) that becomes a (1,2) in 4th quadrant.

What this does is preserve function definition and it allows expansion from 1 quadrant only. And most important, it allows a closed loop be a function in math and physics. Closed loops are the most important function in all of physics for it is all Light waves.

AP, King of Science, especially Physics

Alright, there is going to be very many beautiful theorems out of this work.. Perhaps the most beautiful is the idea that the Polynomial function has the capacity to become a semicircle but no more. To become a semiellipse, but no more, and the capacity to become a semioval, but no more.

And so, to become a full circle or full ellipse or full oval, we need 4 quadrants in which all 2 axes are positive numbers only. That way, we find the opposite semicircle, semiellipse, semioval, and make these 2 independent polynomials be the full circle, full ellipse, full oval.

Negative numbers never existed in math or physical world, but we still need 4 quadrants in 2D in order to keep the definition of Function yet allowing us to have a full circle as a function. Because we borrow the 4th quadrant to complete the semicircle in 1st quadrant.

But the beauty is, that Polynomials have this innate capacity to create a semicircle, and no more of the circle. So why should polynomials be semi-circle seeking entities , but no more? And we see this in physics that the Light wave is a Double Transverse wave of two E fields and two B fields. So you need 4 semicircle waves. You need 8 quadrants in mathematics 3D to handle a Double Transverse wave. You need 4 Polynomials , 4 independent Polynomials coming together to make a Light Wave of Physics.

Physics of the future no longer needs to write for a circle the equation (x-a)^2 + (y-b)^2 = radius^2. Instead we write 2 polynomials over a specific interval.

And for the first time in physics history we write polynomials for Light Waves.

AP

Alright, as reference I am viewing graphed polynomials in Stewart, Redlin, Watson College Algebra 4th edition, 2004.

Page 314 is two polynomials graphed P(x) = 3x^5 -5x^3 +2x and Q(x) = 3x^5 from interval -1 to 1 but in New Math all the axes are positive numbers. It shows 2 closed loops of irregular shapes. Now I need two polynomials forming a closed loop that is near to being a circle, ellipse, oval.

Now on page 331 is graphed the polynomial (x+3)x(x-1)(x-5). So if AP is correct there is another polynomial when graphed would make a closed loop near ellipse from x = 1 to x=5. This is done because we can call the 4th quadrant as being all positive number axis.

Now on page 319 is the graphed polynomial x^3 -2x^2 -4x +8 and if we consider all axes as only positive numbers we can swing that portion of the graph from 0 to -2 back around and be a closed loop from 0 to +2. So in this case we begin to see how a single Polynomial can form a closed loop. And where we do not need two independent polynomials. We could say that when all axes are positive number axes that we pick and chose what parts of the function graph go in which quadrant. In this way we can form a circle by one single polynomial function.

AP

On Saturday, December 17, 2022 at 8:20:41 PM UTC-6, Archimedes Plutonium wrote:
> Alright, as reference I am viewing graphed polynomials in Stewart, Redlin, Watson College Algebra 4th edition, 2004.
On page 332 asks for a formula of graph. The formula is (x+2)^2(x-1)^2 a polynomial of course. The graph looks like a W.

And if we graph the portion of the W from x=0 to about x=-1.5 into the first quadrant we end up with almost a closed loop ellipse or oval. There probably are two vertices in the attachment.

Or, instead look for a polynomial that scoots that portion over and have 2 independent polynomials forming a closed loop figure.

All of this is possible if we discard the negative numbers from the axes of 2D and 3D. We keep the definition of function the same but modify the axes to be positive numbers only.

And this is so much needed in physics, formulas of polynomials for closed loop EM waves.

Now, it maybe possible that by having all axes as positive numbers and all functions as polynomials, that we dispense and throw out the Parametric Equations of Old Math. Maybe the parametric equations were a "groping feeling passion of Old Math" groping for polynomials to take their place, only not knowing where to find polynomials.

I remember in Europe, that woman were groped on the trains. Well, it maybe the case that parametric equations of Old Math was a mere groping exercise for which one day, polynomials would take them over.

AP

Alright, a nice proof of the fact that polynomials at most can be a semi-closed-loop. And a proof that does not invoke the definition of function-- for every x value there is a unique y value.

So I look in the College Algebra book of polynomial graphs and see x^3 and then -x^3, both graphed together and see a U in 1st and 2nd quadrant and upside down U in 3rd and 4th quadrant.

I look at (x-2)^4 and see a U shaped graph in 1st quadrant.

So a proof that any given polynomial at most can be only 1/2 of a closed-loop, involves the idea that a closed loop circle is of a power of 2 for x and y, and the polynomial of power 2 is a parabola, a open figure. Since the smallest power of 2 in polynomial is a parabola, no polynomial will be closed loop figure. And powers higher than 2 only make the openness more apparent.

AP