> On Friday, 16 December 2022 at 01:33:15 UTC+1, FredJeffries wrote:
> > On Thursday, December 15, 2022 at 6:02:39 AM UTC-8, strue...@gmail.com wrote:
> > > Hello,
> > > I am new in this group. Is it possible to ask mathematical questions here?
> >
> > You can ask. But it's been years since anyone who's asked a serious question here has gotten a meaningful answer.

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 Fred, how about answering that question???

On Friday, December 16, 2022 at 2:49:02 PM UTC-8, Archimedes Plutonium wrote:
> 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 Fred, how about answering that question???

Lagrange polynomial for those four points yields

L(x) = (100/9)x^2 − (100/9)x + 1

https://planetcalc.com/8680/?xy=1%2F10%200%0A9%2F10%200%0A1%201%0A0%201&showSteps=1&interpolate=&showBasis=0

On Friday, December 16, 2022 at 6:10:43 PM UTC-6, FredJeffries wrote:
> On Friday, December 16, 2022 at 2:49:02 PM UTC-8, Archimedes Plutonium wrote:
>
> > 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 Fred, how about answering that question???
> Lagrange polynomial for those four points yields
>
> L(x) = (100/9)x^2 − (100/9)x + 1
>

Thanks, I am puzzled as to how that escaped being a 3rd degree polynomial. Checking on x= 0.5 turns out almost equaling -2. Trying to envision the geometry of that polynomial.

But to cut to the chase, I am looking for a polynomial that almost represents a circle or oval when graphed. Maybe I need 6 points, not 4. And that would be a 5th degree polynomial.

I am trying to avoid the polynomial of 4 points from going like this

/\____/\

When I want it to go more like a closed loop
___
|__|

Looking on the www for polynomial that looks like circle gives me these hits.

Now a quarter circle in polynomials of 5th degree is

Y = 1 - x - x^2/2 + x^3/6 + x^4/24 - x^5/120

But can we get a polynomial to sort of roll up like a spiral to approximate a circle. It not need to be fully closed but almost closed.

So how far can we get a polynomial to be in the shape of a circle, or oval or a spiral loop.

Of course this begs the question of a proof that no polynomial can be a closed figure in 2D.

I see it as a battle between addition of terms and subtraction of terms.

AP

On Thursday, December 15, 2022 at 6:33:15 PM UTC-6, FredJeffries wrote:
> On Thursday, December 15, 2022 at 6:02:39 AM UTC-8, strue...@gmail.com wrote:
> > Hello,
> > I am new in this group. Is it possible to ask mathematical questions here?
> You can ask. But it's been years since anyone who's asked a serious question here has gotten a meaningful answer.
> > If not, can me tell someone where I can ask the following question?
> Have you tried MathOverflow?
>
> https://mathoverflow.net/

On Friday, December 16, 2022 at 6:10:43 PM UTC-6, FredJeffries wrote:
> On Friday, December 16, 2022 at 2:49:02 PM UTC-8, Archimedes Plutonium wrote:
>
> > 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 Fred, how about answering that question???
> Lagrange polynomial for those four points yields
>
> L(x) = (100/9)x^2 − (100/9)x + 1
>

Thanks, I am puzzled as to how that escaped being a 3rd degree polynomial. Checking on x= 0.5 turns out almost equaling -2. Trying to envision the geometry of that polynomial.

But to cut to the chase, I am looking for a polynomial that almost represents a circle or oval when graphed. Maybe I need 6 points, not 4. And that would be a 5th degree polynomial.

I am trying to avoid the polynomial of 4 points from going like this

/\____/\

When I want it to go more like a closed loop
___
|__|

Looking on the www for polynomial that looks like circle gives me these hits.

Now a quarter circle in polynomials of 5th degree is

Y = 1 - x - x^2/2 + x^3/6 + x^4/24 - x^5/120

But can we get a polynomial to sort of roll up like a spiral to approximate a circle. It not need to be fully closed but almost closed.

So how far can we get a polynomial to be in the shape of a circle, or oval or a spiral loop.

Of course this begs the question of a proof that no polynomial can be a closed figure in 2D.

I see it as a battle between addition of terms and subtraction of terms.

AP

Asking Fred another question:

Fred, can you write for me the best polynomial that mathematicians have at this moment in time that looks like almost a Closed Loop in the graphing plane???

Is there a polynomial which sort of loops around like this ______O_____

What is the very best polynomial in math literature of a polynomial closed loop or almost closed loop???

AP

Alright, I am looking at a lot of Polynomial graphs and find not a single one of them with the ability to form a closed loop. However, if I take two polynomials and add them together over a interval that the one intersects with the other and forms a closed loop.

This is likely a proof by the fact that polynomials are functions and cannot have two y values. Simple as that.

But here in New Math we make the axes all be positive numbers, and that does not violate closed loops as functions. Where I take the semicircle of one polynomial, combine with a second polynomial and form a full circle.

Now I cannot add the one to the other and form 1 polynomial and keep the closed loop. I have to treat each polynomial separately.

So, well, I have my answer. Polynomials can be closed loops if two are taken independently together.

AP

On Friday, December 16, 2022 at 9:09:02 PM UTC-5, Archimedes Plutonium wrote:
> On Friday, December 16, 2022 at 6:10:43 PM UTC-6, FredJeffries wrote:
> > On Friday, December 16, 2022 at 2:49:02 PM UTC-8, Archimedes Plutonium wrote:
> >
> > > 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 Fred, how about answering that question???
> > Lagrange polynomial for those four points yields
> >
> > L(x) = (100/9)x^2 − (100/9)x + 1
> >
> Thanks, I am puzzled as to how that escaped being a 3rd degree polynomial.. Checking on x= 0.5 turns out almost equaling -2. Trying to envision the geometry of that polynomial.
>

[snip]

Hmmm... Earlier this month, the real AP wrote here:

"Negative numbers are the witches and hobgoblins of insane kook mathematicians. "
--Dec. 7, 2022
https://groups.google.com/g/sci.math/c/dd4_2ryBLcQ/m/L0_NprJoCgAJ

He has been consistent over the years, too. He wrote here:

“No negative numbers exist.”
--December 22, 2018

Dan

On Saturday, December 17, 2022 at 12:09:30 PM UTC-5, Dan Christensen wrote:
> On Friday, December 16, 2022 at 9:09:02 PM UTC-5, Archimedes Plutonium wrote:
> > On Friday, December 16, 2022 at 6:10:43 PM UTC-6, FredJeffries wrote:
> > > On Friday, December 16, 2022 at 2:49:02 PM UTC-8, Archimedes Plutonium wrote:
> > >
> > > > 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 Fred, how about answering that question???
> > > Lagrange polynomial for those four points yields
> > >
> > > L(x) = (100/9)x^2 − (100/9)x + 1
> > >
> > Thanks, I am puzzled as to how that escaped being a 3rd degree polynomial. Checking on x= 0.5 turns out almost equaling -2. Trying to envision the geometry of that polynomial.
> >
> [snip]
>
> Hmmm... Earlier this month, the real AP wrote here:
>
> "Negative numbers are the witches and hobgoblins of insane kook mathematicians. "
> --Dec. 7, 2022
> https://groups.google.com/g/sci.math/c/dd4_2ryBLcQ/m/L0_NprJoCgAJ
>
> He has been consistent over the years, too. He wrote here:
>
> “No negative numbers exist.”
> --December 22, 2018
>

https://groups.google.com/g/sci.math/c/4WNA17GSTNI/m/iRTKuy30CAAJ

> Dan

On Friday, December 16, 2022 at 6:09:02 PM UTC-8, Archimedes Plutonium wrote:
> On Friday, December 16, 2022 at 6:10:43 PM UTC-6, FredJeffries wrote:
> > On Friday, December 16, 2022 at 2:49:02 PM UTC-8, Archimedes Plutonium wrote:
> >
> > > 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 Fred, how about answering that question???
> > Lagrange polynomial for those four points yields
> >
> > L(x) = (100/9)x^2 − (100/9)x + 1
> >
> Thanks, I am puzzled as to how that escaped being a 3rd degree polynomial..

Your choice of points was too symmetric -- symmetric about the line x = 1/2

Try fudging them away from the symmetry just a little and the x^3 term will show up

On Saturday, December 17, 2022 at 3:33:27 PM UTC-6, FredJeffries wrote:
> On Friday, December 16, 2022 at 6:09:02 PM UTC-8, Archimedes Plutonium wrote:
> > On Friday, December 16, 2022 at 6:10:43 PM UTC-6, FredJeffries wrote:
> > > On Friday, December 16, 2022 at 2:49:02 PM UTC-8, Archimedes Plutonium wrote:
> > >
> > > > 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 Fred, how about answering that question???
> > > Lagrange polynomial for those four points yields
> > >
> > > L(x) = (100/9)x^2 − (100/9)x + 1
> > >
> > Thanks, I am puzzled as to how that escaped being a 3rd degree polynomial.
> Your choice of points was too symmetric -- symmetric about the line x = 1/2
>
> Try fudging them away from the symmetry just a little and the x^3 term will show up

Has anyone in mathematics proven that the maximum curvature of a polynomial is a semicircle, semiellipse, semioval, and a proof that does not utilize function definition-- every x has a unique y value. I do not call that a proof.

AP