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Nov 28, 2022, 3:30:44 AM11/28/22

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Little facts that make my head spin.

Shifting two numbers by 't' units.

Can you say *why* their sum is not

translation invariant while their

difference is? Namely,

a+b = a-(-b) = a-B, yet:

a+b = (a+t)+(b+t) - 2t

a-B = (a+t)-(B+t)

Julio

Shifting two numbers by 't' units.

Can you say *why* their sum is not

translation invariant while their

difference is? Namely,

a+b = a-(-b) = a-B, yet:

a+b = (a+t)+(b+t) - 2t

a-B = (a+t)-(B+t)

Julio

Nov 28, 2022, 9:29:30 AM11/28/22

to

On Mon, 28 Nov 2022 00:30:38 -0800 (PST), Julio Di Egidio

<ju...@diegidio.name> wrote:

>Little facts that make my head spin.

>

>Shifting two numbers by 't' units.

>

>Can you say *why* their sum is not

>translation invariant while their

>difference is? Namely,

If I have 3 apples and you have 2, we have 5 apples.
<ju...@diegidio.name> wrote:

>Little facts that make my head spin.

>

>Shifting two numbers by 't' units.

>

>Can you say *why* their sum is not

>translation invariant while their

>difference is? Namely,

But if I have 6 apples and you have 5, we have 11.

On the other hand, if I have 5 and you have 2, I have 3 more than you.

Then if I have 8 and you have 5, I still have 3 more than you.

Why do you think they should be invariant? Sum is both commutative

and associative; difference is neither.

--

Remove del for email

Nov 28, 2022, 12:02:07 PM11/28/22

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On Monday, 28 November 2022 at 15:29:30 UTC+1, Barry Schwarz wrote:

> On Mon, 28 Nov 2022 00:30:38 -0800 (PST), Julio Di Egidio

> <ju...@diegidio.name> wrote:

>

> > Little facts that make my head spin.

> >

> > Shifting two numbers by 't' units.

> >

> > Can you say *why* their sum is not

> > translation invariant while their

> > difference is? Namely,

>

> If I have 3 apples and you have 2, we have 5 apples.

> But if I have 6 apples and you have 5, we have 11.

>

> On the other hand, if I have 5 and you have 2, I have 3 more than you.

> Then if I have 8 and you have 5, I still have 3 more than you.

Yes. More generally:
> On Mon, 28 Nov 2022 00:30:38 -0800 (PST), Julio Di Egidio

> <ju...@diegidio.name> wrote:

>

> > Little facts that make my head spin.

> >

> > Shifting two numbers by 't' units.

> >

> > Can you say *why* their sum is not

> > translation invariant while their

> > difference is? Namely,

>

> If I have 3 apples and you have 2, we have 5 apples.

> But if I have 6 apples and you have 5, we have 11.

>

> On the other hand, if I have 5 and you have 2, I have 3 more than you.

> Then if I have 8 and you have 5, I still have 3 more than you.

a+b = (a+t)+(b+t) - 2t // not invariant

a-b = (a+t)-(b+t) // invariant

To "explain it", one could observe:

(a+t)+(b+t) = a+t+b+t

(a+t)-(b+t) = a+t-b-t

to point out that it is a change of sign in 't'

that is making all the difference, so to speak.

But that is just one way to see the why, maybe

not even that interesting, anyway I tend to

think just geometrically and then look for ways

to express my geometric intuition/reasoning in

maths. So, maybe the problem is really trivial,

or maybe a mathematician would have something

else to say.

> Why do you think they should be invariant?

> Sum is both commutative

> and associative; difference is neither.

dispense with subtraction, i.e. by noticing

that a-b = a+(-b)... except that, apparently,

and in a sense that I still have only partly

clear, we cannot.

(And, overall, because I have been spending

too much time lately tripping over fixed points,

symmetries, conservation laws...)

Julio

Nov 28, 2022, 5:17:42 PM11/28/22

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Now I take as the math equations that govern all of Physics as the calculus on New Ohm's Law Voltage = current*magnetic field*electric field, and that calculus delivers these EM laws that govern all of physics-- and governs all of mathematics also, for math is a subset of physics.

Here I use Angular Momentum L as electric field E, both are interchangeable.

0) domain law as Atomic Theory

1) Magnetic primal unit law Magnetic Field B = kg /A*s^2

2) V = C*B*L New Ohm's law, law of electricity

3) V' = (C*B*L)' Capacitor law

4) (V/C*L)' = B' Ampere-Maxwell law

5) (V/(B*L))' = C' Faraday law

6) (V/(C*B))' = L' the new law of Coulomb force with EM gravity force

All the calculus permutations possible are V', B', C', and E' (same as L').

Notice that in all the calculus, only one is Addition the Voltage derivative. While the other three are derivatives of division which ends up as Subtraction. For example, the quotient rule of calculus is (V/i*L)' = B' = (V'*i*L - V*i' *L - V*i*L') / (i*L)^2. Do you see that in every division derivative, we have two subtractions over one number.

Algebra of 3D Calculus, for remember we did the algebra of

V' = (iBL)'

i' = (V/BL)'

B' = (V/iL)'

L' = (V/iB)'

--- quoting 1st year calculus from Teaching True ---

Using the Product Rule which is (fgh)' = (f'gh + fg'h + fgh')

Capacitor Law (i*B*L)' = i'*B*L + i*B'L + i*B*L'

V' = (iBL)' = i'*B*L + i*B'*L + i*B*L' here we have three terms explaining capacitors

Ampere-Maxwell Law

Using the Quotient Rule, which is (f/gh)' = (f'gh - fg'h - fgh')/(gh)^2

(V/i*L)' = B' = (V'*i*L - V*i' *L - V*i*L') / (i*L)^2

Maxwell had two terms in the Ampere-Maxwell law-- the produced magnetic field and a displacement current, but above we see we have also a third new term.

Faraday Law

(V/B*L)' = i' = (V'*B*L - V*B' *L - V*B*L') / (B*L)^2

------------

V' = (iBL)' = i'*B*L + i*B'*L + i*B*L' reduces to

= iBL + iVL + iBL'

i' = V'*B*L/ (B*L)^2 - V*B' *L/ (B*L)^2 - V*B*L' / (B*L)^2 reduces to

i' = B^2*L/ (B*L)^2 - V^2 *L/ (B*L)^2 - V*B*L' / (B*L)^2 further reduces

= 1/L - V^2/B^2*L - VL'/BL^2

B' = V'*i*L/ (i*L)^2 - V*i' *L/ (i*L)^2 - V*i*L' / (i*L)^2 reduces to

B' = B*i*L/ (i*L)^2 - V*i *L/ (i*L)^2 - V*i*L' / (i*L)^2 further reduces to

= B/iL - V/iL - VL'/iL^2

L' = (V/i*B)' = (V'*i*B - V*i' *B - V*i*B') / (i*B)^2 reduces to

L' = i*B^2 / (i*B)^2 - V*i *B / (i*B)^2 - V^2*i / (i*B)^2 further reduces to

= 1/i - V/iB - V^2/iB^2

--------

(1) V' = iBL + iVL + iBL'

(2) i' = 1/L - V^2/B^2*L - VL'/BL^2

(3) B' = B/iL - V/iL - VL'/iL^2

(4) L' = 1/i - V/iB - V^2/iB^2

Alright, so I replace L' in (1) with 1/i - V/iB - V^2/iB^2

I get V' = iBL + iVL + iB*(1/i - V/iB - V^2/iB^2 )

= iBL + iVL + B - V - V^2/ B

Doing the replacement in (2)

i' = 1/L - V^2/B^2*L - VL'/BL^2

= 1/L - V^2/B^2*L - V*(1/i - V/iB - V^2/iB^2) /BL^2

= 1/L - V^2/B^2*L - (V/iBL^2) - (V^2/iB^2L^2) - (V^3/(iB^3L^2))

Doing the replacement in (3)

B' = B/iL - V/iL - VL'/iL^2

= B/iL - V/iL - V(1/i - V/iB - V^2/iB^2)/iL^2

= B/iL - V/iL - (V/i^2L^2) - (V^2/i^2*B*L^2) - (V^3/( i^2B^2L^2))

Julio likes geometry, and that is my preference also, I prefer to solve problems of both science, and math, best solved by geometry, for I believe in Darwin Evolution that what evolved us from apes was the throwing of rocks and stones to garner advantage, and throwing requires the brain to advance in geometry, more than advance in quantity = algebra. To hit a target needs geometry.

So in the above, I replace Voltage = current*magnetic field*electric field (note, the * symbol is generalized multiplication and can be dot vector product or cross vector product or even scalar product).

So, well in math we can convert Voltage = i*B*E, into volume and make it easy on ourselves, as Volume = length*width*height.

Now that Volume decomposes into calculus of four differential equations of Volume', length', width', height'.

The length, width, height differential equations will end up with two subtraction terms. The Volume differential equation will be pure addition of terms.

And now, this leads me to the clarity of Cubic versus General Set in Schrodinger Equation.

If we have volume of rectangular box such as Length = 10, Width = 4, Height = 3.

But what if we have the volume of Length = 10, Width = 10, Height = 10, then we are faced with two possibilities, for our volume can be the volume of a cube in 3D or the volume of a sphere with a radius of 5.

This is why Schrodinger Equation has a General Set along with a Cubic Set.

AP

Nov 28, 2022, 6:48:46 PM11/28/22

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Nov 28, 2022, 9:26:15 PM11/28/22

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On Monday, November 28, 2022 at 4:17:42 PM UTC-6, Archimedes Plutonium wrote:

> So, well in math we can convert Voltage = i*B*E, into volume and make it easy on ourselves, as Volume = length*width*height.

>

> Now that Volume decomposes into calculus of four differential equations of Volume', length', width', height'.

>

> The length, width, height differential equations will end up with two subtraction terms. The Volume differential equation will be pure addition of terms.

>

> And now, this leads me to the clarity of Cubic versus General Set in Schrodinger Equation.

>

> If we have volume of rectangular box such as Length = 10, Width = 4, Height = 3.

>

> But what if we have the volume of Length = 10, Width = 10, Height = 10, then we are faced with two possibilities, for our volume can be the volume of a cube in 3D or the volume of a sphere with a radius of 5.

>

> This is why Schrodinger Equation has a General Set along with a Cubic Set.

Now the above is not confined to just Cube and then a Sphere with radius 1/2 of cube side. But made more general with Rectangular Box volume replaced by ellipsoid volume. So if I have volume of rectangular box as Length= 10, Width = 4, Height= 3, I can also turn that into a volume of ellipsoid with semi-axes of 10/2, 4/2, 3/2. Keeping the curved-geometry volume less than the straightline geometry volume.
> So, well in math we can convert Voltage = i*B*E, into volume and make it easy on ourselves, as Volume = length*width*height.

>

> Now that Volume decomposes into calculus of four differential equations of Volume', length', width', height'.

>

> The length, width, height differential equations will end up with two subtraction terms. The Volume differential equation will be pure addition of terms.

>

> And now, this leads me to the clarity of Cubic versus General Set in Schrodinger Equation.

>

> If we have volume of rectangular box such as Length = 10, Width = 4, Height = 3.

>

> But what if we have the volume of Length = 10, Width = 10, Height = 10, then we are faced with two possibilities, for our volume can be the volume of a cube in 3D or the volume of a sphere with a radius of 5.

>

> This is why Schrodinger Equation has a General Set along with a Cubic Set.

Now, has anyone asked the question what is the side view of a torus, what does that figure become in 2D geometry??

So, you hold up a torus and look at it from a side view. You cannot see the donut hole. But what does the figure become in 2D??? It cannot be an ellipse? For the bottom and top are straightlines. It reminds me of these wood tabletops that you can pull apart and add more for a larger table.

(_______)

So the ends were part of a circle from a circle torus but the bottom and top are straight lines. And what would the area of such a figure be??? A formula for the area?

AP

Nov 28, 2022, 10:00:58 PM11/28/22

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On 11/28/2022 2:30 AM, Julio Di Egidio wrote:

> Little facts that make my head spin.

>

> Shifting two numbers by 't' units.

>

> Can you say *why* their sum is not

> translation invariant while their

> difference is? Namely,

>

>#1 a+b = a-(-b) = a-B, yet:
> Little facts that make my head spin.

>

> Shifting two numbers by 't' units.

>

> Can you say *why* their sum is not

> translation invariant while their

> difference is? Namely,

>

so B = -b

>#2 a+b = (a+t)+(b+t) - 2t

ok

> #3 a-B = (a+t)-(B+t)

However, a-(-b) = a+b = (a+t)+(b+t) - 2t

but is not equal to (a+t)-(B+t) which is (a+t) + b - t

>

> Julio

Equation #3 does not follow from #1 or #2

Message has been deleted

Nov 29, 2022, 4:43:03 AM11/29/22

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On Monday, November 28, 2022 at 8:26:15 PM UTC-6, Archimedes Plutonium wrote:

> (_______)

>

> So the ends were part of a circle from a circle torus but the bottom and top are straight lines. And what would the area of such a figure be??? A formula for the area?

Now I was looking in the math literature for a name of a 2D figure whose top and bottom were straight line segments but whose ends are parts of a circle.
> (_______)

>

> So the ends were part of a circle from a circle torus but the bottom and top are straight lines. And what would the area of such a figure be??? A formula for the area?

It does have 2 axes of symmetry, same as ellipse. So is it a special kind of ellipse?????

AP

Nov 29, 2022, 4:49:41 AM11/29/22

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Alright, it is called a stadium, and now I wonder if its area is less or greater than the area of a standard ellipse with the same axes of symmetry, I would guess the standard ellipse has more area, and whether the number quantity

of standard ellipse area subtract stadium is a significant number related to pi.

of standard ellipse area subtract stadium is a significant number related to pi.

Nov 29, 2022, 4:55:22 AM11/29/22

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In 3D, it is called a capsule, like a pill capsule. And I wonder if the volume of the capsule of axes A and B, is less than the volume of ellipsoid of axes A and B, I would guess so.

Nov 29, 2022, 5:12:58 AM11/29/22

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A couple of nice links I have found:

<https://www.researchgate.net/figure/A-capsule-geometry-consisting-of-a-cylinder-with-hemispherical-ends-The-capsule-geometry_fig9_316347573>

<http://balmoralsoftware.com/equability/hemicylinder/hemicylinder.htm>

Julio

Nov 29, 2022, 5:19:43 AM11/29/22

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> Thanks Julio and Barry, for this offers me more clarity on Schrodinger Equation of a cubic set solution versus general set. The Elements Beyond Uranium, Seaborg & Loveland, 1990, pages 72-73.

>

> Now I take as the math equations that govern all of Physics as the calculus on New Ohm's Law Voltage = current*magnetic field*electric field, and that calculus delivers these EM laws that govern all of physics-- and governs all of mathematics also, for math is a subset of physics.

>

> Here I use Angular Momentum L as electric field E, both are interchangeable.

>

> 0) domain law as Atomic Theory

> 1) Magnetic primal unit law Magnetic Field B = kg /A*s^2

> 2) V = C*B*L New Ohm's law, law of electricity

> 3) V' = (C*B*L)' Capacitor law

> 4) (V/C*L)' = B' Ampere-Maxwell law

> 5) (V/(B*L))' = C' Faraday law

> 6) (V/(C*B))' = L' the new law of Coulomb force with EM gravity force

>

> All the calculus permutations possible are V', B', C', and E' (same as L').

>

> Notice that in all the calculus, only one is Addition the Voltage derivative. While the other three are derivatives of division which ends up as Subtraction. For example, the quotient rule of calculus is (V/i*L)' = B' = (V'*i*L - V*i' *L - V*i*L') / (i*L)^2. Do you see that in every division derivative, we have two subtractions over one number.

<snip>
>

> Now I take as the math equations that govern all of Physics as the calculus on New Ohm's Law Voltage = current*magnetic field*electric field, and that calculus delivers these EM laws that govern all of physics-- and governs all of mathematics also, for math is a subset of physics.

>

> Here I use Angular Momentum L as electric field E, both are interchangeable.

>

> 0) domain law as Atomic Theory

> 1) Magnetic primal unit law Magnetic Field B = kg /A*s^2

> 2) V = C*B*L New Ohm's law, law of electricity

> 3) V' = (C*B*L)' Capacitor law

> 4) (V/C*L)' = B' Ampere-Maxwell law

> 5) (V/(B*L))' = C' Faraday law

> 6) (V/(C*B))' = L' the new law of Coulomb force with EM gravity force

>

> All the calculus permutations possible are V', B', C', and E' (same as L').

>

> Notice that in all the calculus, only one is Addition the Voltage derivative. While the other three are derivatives of division which ends up as Subtraction. For example, the quotient rule of calculus is (V/i*L)' = B' = (V'*i*L - V*i' *L - V*i*L') / (i*L)^2. Do you see that in every division derivative, we have two subtractions over one number.

Thank you, that is quite along the lines of some of my ruminations. Indeed, volume is playing a crucial role in some recent theoretical physics (look up for "amplituhedron", though it gets immediately and utterly technical), where the volume of some combinatorial structures ends up being the amplitude of the corresponding physical configuration.

The amazing effectiveness of mathematics...

Julio

Nov 29, 2022, 6:40:39 AM11/29/22

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Nov 29, 2022, 2:21:20 PM11/29/22

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Julio, I want some advice from you on Wikipedia. I was stung by them in the 1990s when I did some editing of "their errors" spending quite a bit of my time in making the edit of Wikipedia errors when -- a few hours later, a Wikipedia editor reverted my input-- all a waste of time.

Wikipedia is wrong when they say a "ellipse is a oval", they are wrong in "Dandelin proof" for that is a fakery, and they are wrong when they say a "oval is ill defined or nebulous defined". And of course, their big mistake of geometry-- they claim a ellipse is a conic section yet it never was for a single right circular cone. However, if you take 2 right circular cones and join them together in <> that some cuts are indeed ellipses, but not a single right circular cone.

Julio-- I want to know what your attitude is on Wikipedia when you see they have mistakes in various entries. Has Wikipedia changed at all since 1990s when their entries are full of error, and is the best policy towards Wikipedia-- continue to ignore their many mistakes of math. For I have the feeling that many of those editors-- are just plain bullies that know little real mathematics.

What is your opinion????

Nov 29, 2022, 3:32:19 PM11/29/22

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Alright, well, the Stadium figure and the Capsule figure put new life into the proof that slant cut of single right circular cone is a Oval, never the Ellipse.

And calls upon me to make another proof argument for that idea.

D

B /slant cut plane

/ \/C

_A___________/ \

Now I hope that ascii art holds up its format once I post it, for often it falls apart in ineligible.

Now I label this line from the base of the right circular cone as A. I label B as the apex of cone. I label C as the intersection point of slant cut plane D.

Now, the purpose of this diagram is to focus in on where the D, slant cut plane intersects with A and the angle formed. Hard to see from that above graphic for I had only one angle choice in ascii art. I prefer a smaller angle for the cut so it reaches out to A.

And the proofs that slant cut is Oval, never ellipse has to do with the fact that the plane passes through the center perpendicular with apex and the side nearest the right wall is closer and thus smaller than the side nearest the left wall and thus not equal, making the figure impossible to be a ellipse. And rather, instead be a oval.

Now I am going to have to elaborate on this sketch by dropping perpendiculars down from C to base A, and down from B to base A, and down from the exit intersection to base A.

I am going to have to do that for proofs of theorems about slant cut not in a cylinder, but in a barrel or torus without donut hole. I have to do these perpendiculars because I need to find out if the Circle curve maximizes area and whether the slant cut intersects a Circle Arc.

Let me define Arc as a section of a Circle. Or a section of a ellipse, or a section of a oval.

So the silhouette of Sectioning.

Sectioning a cone is /\

Sectioning a cylinder is | |

Sectioning a barrel is (__)

Sectioning a barrel is the same as sectioning a torus without the donut hole involved.

Now all of this will sound remote to people not scientists, but all of this is vital to physics especially astronomy, for gravity uses ellipses, and atomic theory uses toruses. The atomic and hydrogen bombs are details of toruses and sectioning geometry.

What I an striving for is to prove several theorems of geometry of cone, of capsule, of stadium, of a cylinder whose walls are curves --- a barrel if you please, or is a barrel a torus without the donut hole.

But first I want to review the history of the maximum rectangle or square that fits inside a circle, any given circle. Is the maximum in area going to be a rectangle or going to be a square. Let my check the literature at this moment.

Yes, apparently the Square is the largest rectangle inside a circle in terms of area. And we say a square is a subset of rectangle whose all four sides are equal.

Can we do the same with a ellipse and oval?

Now we start with a square and ask, what is the largest in terms of area of a ellipse or oval to fit inside the square? We do the same question with starting from a rectangle-- which is the larger area a ellipse or oval?? I believe the answer is a oval is the maximum area that fits inside either a rectangle or square. And if true, then I can say a Oval is a subset of a Ellipse whose 1/2 portions of the oval come from two different ellipses.

Quite the reverse of what Wikipedia editors say that a ellipse is a oval, when they should be saying that a oval is a ellipse.

AP

P.S. I need to turn this into my 220+K book of science.

And calls upon me to make another proof argument for that idea.

D

B /slant cut plane

/ \/C

_A___________/ \

Now I hope that ascii art holds up its format once I post it, for often it falls apart in ineligible.

Now I label this line from the base of the right circular cone as A. I label B as the apex of cone. I label C as the intersection point of slant cut plane D.

Now, the purpose of this diagram is to focus in on where the D, slant cut plane intersects with A and the angle formed. Hard to see from that above graphic for I had only one angle choice in ascii art. I prefer a smaller angle for the cut so it reaches out to A.

And the proofs that slant cut is Oval, never ellipse has to do with the fact that the plane passes through the center perpendicular with apex and the side nearest the right wall is closer and thus smaller than the side nearest the left wall and thus not equal, making the figure impossible to be a ellipse. And rather, instead be a oval.

Now I am going to have to elaborate on this sketch by dropping perpendiculars down from C to base A, and down from B to base A, and down from the exit intersection to base A.

I am going to have to do that for proofs of theorems about slant cut not in a cylinder, but in a barrel or torus without donut hole. I have to do these perpendiculars because I need to find out if the Circle curve maximizes area and whether the slant cut intersects a Circle Arc.

Let me define Arc as a section of a Circle. Or a section of a ellipse, or a section of a oval.

So the silhouette of Sectioning.

Sectioning a cone is /\

Sectioning a cylinder is | |

Sectioning a barrel is (__)

Sectioning a barrel is the same as sectioning a torus without the donut hole involved.

Now all of this will sound remote to people not scientists, but all of this is vital to physics especially astronomy, for gravity uses ellipses, and atomic theory uses toruses. The atomic and hydrogen bombs are details of toruses and sectioning geometry.

What I an striving for is to prove several theorems of geometry of cone, of capsule, of stadium, of a cylinder whose walls are curves --- a barrel if you please, or is a barrel a torus without the donut hole.

But first I want to review the history of the maximum rectangle or square that fits inside a circle, any given circle. Is the maximum in area going to be a rectangle or going to be a square. Let my check the literature at this moment.

Yes, apparently the Square is the largest rectangle inside a circle in terms of area. And we say a square is a subset of rectangle whose all four sides are equal.

Can we do the same with a ellipse and oval?

Now we start with a square and ask, what is the largest in terms of area of a ellipse or oval to fit inside the square? We do the same question with starting from a rectangle-- which is the larger area a ellipse or oval?? I believe the answer is a oval is the maximum area that fits inside either a rectangle or square. And if true, then I can say a Oval is a subset of a Ellipse whose 1/2 portions of the oval come from two different ellipses.

Quite the reverse of what Wikipedia editors say that a ellipse is a oval, when they should be saying that a oval is a ellipse.

AP

P.S. I need to turn this into my 220+K book of science.

Nov 29, 2022, 3:51:43 PM11/29/22

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I better capture Wikipedia's muddled and error filled math of ellipse and oval, before they dispose of it and catches me complaining of their error without evidence of their muddle headed error. This is a huge problem of Wikipedia science entrees-- written by and composed by people with few Logical marbles.

--- quoting Wikipedia ---

The shape is based on a stadium, a place used for athletics and horse racing tracks.

A stadium may be constructed as the Minkowski sum of a disk and a line segment.[6] Alternatively, it is the neighborhood of points within a given distance from a line segment. A stadium is a type of oval. However, unlike some other ovals such as the ellipses, it is not an algebraic curve because different parts of its boundary are defined by different equations.

--- end quoting ---

--- quoting Wikipedia ---

The shape is based on a stadium, a place used for athletics and horse racing tracks.

A stadium may be constructed as the Minkowski sum of a disk and a line segment.[6] Alternatively, it is the neighborhood of points within a given distance from a line segment. A stadium is a type of oval. However, unlike some other ovals such as the ellipses, it is not an algebraic curve because different parts of its boundary are defined by different equations.

--- end quoting ---

Nov 29, 2022, 9:00:16 PM11/29/22

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Alright, in this definition of Stadium figure it has to have No - Vertex.

So does that eliminate all curves except for semicircle arc, and for semiellipse arc?

I think it even allows for a what can be described as the semi-oval arc.

But it restricts the formation of a Stadium figure to no other arcs, not the semi-arc.

You cannot take a sliver of a circle curve or ellipse curve or oval curve and avoid a Vertex.

Only on a point of the circle, ellipse, oval where the curve arc is rising, then reaches its peek, then declines can I avoid forming a vertex to two parallel lines for Stadium figure.

Now, I see no way of proving that. One of those things that you can see in the mind, but cannot prove.

AP

So does that eliminate all curves except for semicircle arc, and for semiellipse arc?

I think it even allows for a what can be described as the semi-oval arc.

But it restricts the formation of a Stadium figure to no other arcs, not the semi-arc.

You cannot take a sliver of a circle curve or ellipse curve or oval curve and avoid a Vertex.

Only on a point of the circle, ellipse, oval where the curve arc is rising, then reaches its peek, then declines can I avoid forming a vertex to two parallel lines for Stadium figure.

Now, I see no way of proving that. One of those things that you can see in the mind, but cannot prove.

AP

Nov 30, 2022, 12:25:16 AM11/30/22

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I am asking how one goes about proving no vertices and some-- with vertices if you do not have the semi-curve.

My best guess is that to avoid a vertex, you have to use a curve in which a peak point simultaneous with a lowest point is at a "moment of the end of a rise and the beginning of a fall". Perhaps calculus of 0 slope at a point and the two parallel lines are 0 slope.

Yes, so it is not that difficult.

Now a new question arises, a very important one, which will give the maximum area if we enclose the stadium in a rectangle? The semicircle stadium, the semiellipse stadium, the semioval X 2 if we enclose the figure in a rectangle. I will guess the semicircle stadium is the most dense area. What this helps to answer is in physics, are the toruses of protons are they circular or elliptical or oval toruses??

AP

Nov 30, 2022, 1:51:45 AM11/30/22

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Nov 30, 2022, 2:49:00 AM11/30/22

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So the only legitimate numbers in AP's vision of mathematics are Grid Systems-- all created from Counting numbers with Mathematical Induction. And Earle was appreciative of this as well as Stanford Univ faculty allowing them more coffee and bisquit breaks.

The 10 Decimal Grid is this:

9.0, 9.1, 9.2, 9.3, 9.4, 9.5 9.6, 9.7, 9.8, 9.9, 10.0

8.0, 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9,

7.0, 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8, 7.9,

6.0, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9,

5.0, 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9,

4.0, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9,

3.0, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9,

2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9,

1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9,

0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9,

The 100 Grid is a mathematical induction starting with 0 then 0.01, the 1000 Grid starts with 0 then 0.001.

And Earle was immensely pleased with that Simplification, a true simplification.

But as for amplituhedron, is a mockery, and only cloaked as a simplification as if the authors are trying to hide more than reveal.

The beauty of the Decimal Grid Numbers is that they are essential and required in order to do a geometry proof of the Fundamental Theorem of Calculus. You cannot do a geometry proof of FTC with the Reals because you need empty space from one number to the next. You cannot do a geometry FTC if the numbers are a continuum. You need discrete numbers to do a FTC.

So that is true, real Simplification, for I threw out every number in Old Math except for the Counting numbers. And above, if you multiply every number by 10, you get the Counting numbers 0 to 1 to 100, same goes for all the other grids. That Numbers are formed from one generator only--- Math Induction. Not were a string of kooks in math with their sack full of new types of numbers clogs up the system. Just Counting Numbers and math induction.

AP

Nov 30, 2022, 5:14:36 AM11/30/22

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On Tuesday, 29 November 2022 at 20:21:20 UTC+1, Archimedes Plutonium wrote:

> On Tuesday, November 29, 2022 at 4:12:58 AM UTC-6, ju...@diegidio.name wrote:

<snip>
> On Tuesday, November 29, 2022 at 4:12:58 AM UTC-6, ju...@diegidio.name wrote:

> Hi, Julio, I thought I was going to have to move my talk of capsule and stadium

> to another thread, so as not to interrupt your thread here, but seeing that you

> joined in on capsule, I think I will stay here in this thread.

nothing terribly interesting...

> Julio, I want some advice from you on Wikipedia. I was stung by them in the

the worst brainwashing and global fraud. But Wikipedia is rather the example

of blind auto-organization and what we, the stupid fucking white men, do with

it. So, don't blame it, WP is in fact the thermometer of our globalized infamy

and abysmal stupidity.

> Wikipedia is wrong when they say a "ellipse is a oval",

in the slant cut of a cone, the central axis passes by one of the foci of the

ellipse, not its center. Moreover, if you don't trust the algebra, look at the

pictures or do an experiment: go to the beach, construct a cone with sand,

then cut it with some foil or something (pardon my English), and see what

you get: it is not an oval, it is an ellipse... And here are a couple of nice

pictures, from WP's articles:

<https://en.wikipedia.org/wiki/Conic_section#/media/File:Eccentricity.svg>

<https://en.wikipedia.org/wiki/Apollonius_of_Perga#/media/File:Conic_Sections.svg>

The end of the world begins with Plato-Aristotle, but it took centuries to go

from a deranged philosophy to the total inculture and insanity that we are

down to even the most mundane aspects our lives. indeed you can trust

the Greeks for their geometry/logic, and only the Greeks...

Gottfried Wilhelm Leibniz stated "He who understands Archimedes and

Apollonius will admire less the achievements of the foremost men of

later times."

Julio

Nov 30, 2022, 5:30:53 AM11/30/22

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On Wednesday, 30 November 2022 at 11:14:36 UTC+1, Julio Di Egidio wrote:

> On Tuesday, 29 November 2022 at 20:21:20 UTC+1, Archimedes Plutonium wrote:

> > On Tuesday, November 29, 2022 at 4:12:58 AM UTC-6, ju...@diegidio.name wrote:

> <snip>

> On Tuesday, 29 November 2022 at 20:21:20 UTC+1, Archimedes Plutonium wrote:

> > On Tuesday, November 29, 2022 at 4:12:58 AM UTC-6, ju...@diegidio.name wrote:

> <snip>

> > Wikipedia is wrong when they say a "ellipse is a oval",

>

> No, they aren't

Eh, I give too many things for granted. I was thinking
>

> No, they aren't

and I meant they are not wrong that the slant cut of a

cone is an ellipse. Of course an oval is not an ellipse,

unless one asks the ignorant, the former having only

one axis of symmetry.

Julio

Nov 30, 2022, 9:28:21 AM11/30/22

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Nov 30, 2022, 9:58:59 AM11/30/22

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If you shift a by t and b by -t then the sum is invariant,

but the difference is not anymore invariant:

a+b = a+t + (b-t)

a-b =\= a+t - (b-t)

This clearly shows that the infinity borderline maybe

1.4...*10^-604 and not just 1.000... *10^-604.

but the difference is not anymore invariant:

a+b = a+t + (b-t)

a-b =\= a+t - (b-t)

This clearly shows that the infinity borderline maybe

1.4...*10^-604 and not just 1.000... *10^-604.

Nov 30, 2022, 10:01:57 AM11/30/22

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Julio Di Egidio has brought this to us :

Of course it is easy to find contradictions.

https://www.maa.org/external_archive/joma/Volume8/Kalman/Ellipse1.html

https://www.maa.org/external_archive/joma/Volume8/Kalman/Ellipse1.html

Nov 30, 2022, 10:10:44 AM11/30/22

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Typical application of the (t,-t) invariant. Stretching

a circle into an ellipse. On both sides of a diameter,

the circle satisfies, taking x1 negative on the left side,

and x2 positive on the right side, diameter goes through zero:

x2 + x1 = 0

If you stretch it into an ellipse, you need t stretch,

depending on y, x1 by -t and x2 by t:

(x2+t) + (x1-t) = 0

Now multiplication has an interesting property, if

the stretching factor is r, then since multiplication

preserves sign, we get for t=r*x2-x2, and since

x2=-x1, we also have -t=r*x1+x1, and therefore:

r*x2 + r*x1 = 0

This clearly shows again that Archimedes Plutions

a circle into an ellipse. On both sides of a diameter,

the circle satisfies, taking x1 negative on the left side,

and x2 positive on the right side, diameter goes through zero:

x2 + x1 = 0

If you stretch it into an ellipse, you need t stretch,

depending on y, x1 by -t and x2 by t:

(x2+t) + (x1-t) = 0

Now multiplication has an interesting property, if

the stretching factor is r, then since multiplication

preserves sign, we get for t=r*x2-x2, and since

x2=-x1, we also have -t=r*x1+x1, and therefore:

r*x2 + r*x1 = 0

This clearly shows again that Archimedes Plutions

infinity borderline maybe 1.4...*10^-604 and not just

1.000... *10^-604.

https://www.maa.org/external_archive/joma/Volume8/Kalman/Ellipse2.html
1.000... *10^-604.

Nov 30, 2022, 10:30:27 AM11/30/22

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in which a circle is an ellipse, but an oval is not an ellipse any more

than an ellipse is a circle... Words mean what they mean.

Julio

Nov 30, 2022, 1:10:34 PM11/30/22

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Thats a quite different statement "an ellipse is an oval",

and this statement here:

AP's Proof-Ellipse was never a Conic Section

and this statement here:

AP's Proof-Ellipse was never a Conic Section

Dec 1, 2022, 2:42:12 AM12/1/22

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Hi, Julio, rarely are you wrong, but in this instance you are. The axis of a cone does not shift to a focus in a slant cut.

And the Ancient Greeks were **exceptional geniuses in science** we both agree on that, however they did make some mistakes and Apollonius slant cut in cone is one such mistake.

A proof is simple-- cone and oval have 1 axis of symmetry, a ellipse requires 2, and so a slant cut in cone is oval.

But another argument is likely even easier to make, for a cone is vastly different in geometry to a cylinder. Even High School kids can see they are vastly different. So we cannot expect a slant cut in cone to deliver a ellipse when a slant cut in cylinder assuredly delivers a ellipse.

I do not know where they screwed up on the Dandelin ordeal and mega-mistake. And that Dandelin stuff shows one that if you assume a falsehood-- the cut is a ellipse-- that it can deliver any fake result you so desire.

I do not know how Apollonius never reviewed the cylinder cuts. I am told that in Ancient Greek times, the academicians rarely soiled their hands in hands on experiment. Maybe that was the case in conics, they imagined it all, and never gave hands on demonstration. No, I do not use a sand pit but better a paper cone of a magazine cover and a Kerr or Ball lid and shows me that the lower portion of the circle is augmented with a vastly larger crescent area than near the apex entry. This is a proof that the Oval is the slant cut.

And the Ancient Greeks were **exceptional geniuses in science** we both agree on that, however they did make some mistakes and Apollonius slant cut in cone is one such mistake.

A proof is simple-- cone and oval have 1 axis of symmetry, a ellipse requires 2, and so a slant cut in cone is oval.

But another argument is likely even easier to make, for a cone is vastly different in geometry to a cylinder. Even High School kids can see they are vastly different. So we cannot expect a slant cut in cone to deliver a ellipse when a slant cut in cylinder assuredly delivers a ellipse.

I do not know where they screwed up on the Dandelin ordeal and mega-mistake. And that Dandelin stuff shows one that if you assume a falsehood-- the cut is a ellipse-- that it can deliver any fake result you so desire.

I do not know how Apollonius never reviewed the cylinder cuts. I am told that in Ancient Greek times, the academicians rarely soiled their hands in hands on experiment. Maybe that was the case in conics, they imagined it all, and never gave hands on demonstration. No, I do not use a sand pit but better a paper cone of a magazine cover and a Kerr or Ball lid and shows me that the lower portion of the circle is augmented with a vastly larger crescent area than near the apex entry. This is a proof that the Oval is the slant cut.

Dec 1, 2022, 3:58:31 AM12/1/22

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You forgot to insert the word "wrong":

> A simply wrong proof is -- cone and oval have 1 axis of

> A simply wrong proof is -- cone and oval have 1 axis of

> symmetry, a ellipse requires 2, and so a slant cut in cone is oval.

https://en.wikipedia.org/wiki/Ellipse#/media/File:Ellipse-conic.svg
Dec 1, 2022, 4:41:31 AM12/1/22

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On Thursday, 1 December 2022 at 09:58:31 UTC+1, Mostowski Collapse wrote:

> You forgot to insert the word "wrong":

While you are a retarded piece of shit converted just polluter of ponds.
> You forgot to insert the word "wrong":

Congratulations.

*Troll alert*

Julio

Dec 1, 2022, 5:03:00 AM12/1/22

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On Thursday, 1 December 2022 at 08:42:12 UTC+1, Archimedes Plutonium wrote:

> Hi, Julio, rarely are you wrong,

I have never shied way from mistakes, my own
> Hi, Julio, rarely are you wrong,

and not only, since that is where we actually

learn something new. A long story in itself...

> but in this instance you are. The axis of a cone does not shift to a focus in a slant cut.

cone that shits, it is the second focus of the

ellipse that, the more we slant the cut, shifts

away from the focus that stays coincident to

the cone's central axis.

> And the Ancient Greeks were **exceptional geniuses in science** we both agree on that,

the western civilization overall remains the

beginning of the end (of a huge cycle, if you

believe in that kind of stuff: nothing ever ends

really, not even this ridiculous tragedy).

> Apollonius slant cut in cone is one such mistake.

>

> A proof is simple-- cone and oval have 1 axis of

> symmetry, a ellipse requires 2, and so a slant cut in cone is oval.

look at the cone right from above: then e.g. a

right cone has exactly the same symmetries

as a circle.

I remember it was counter-intuitive the first

time I have seen it, but you are simply refusing

to look at all the evidence and demonstrations,

nor will you do a sand experiment or similar:

and there I see you stuck into never admitting

even a minor mistake on your part, which

might even be understandable given the

systematic personal attacks you have endure

around here, but it is also the end of science.

My 2c. Take care.

Julio

Dec 1, 2022, 6:43:56 AM12/1/22

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To some, an ellipse is a special case of oval just like a square is a

special case of rectangle and a circle is a special case of ellipse.

Mostowski Collapse pretended :

special case of rectangle and a circle is a special case of ellipse.

Mostowski Collapse pretended :

Dec 1, 2022, 8:47:03 AM12/1/22

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https://www.geogebra.org/m/sQu4Zfsd

That might clarify things a bit, because it's hard to describe

such complicated and dynamic geometric 3D constructions

in words.

(It might also help to modify such visualizations so you

can see them both in 3D and in 2D side by side to verify

claims about the shape of the intersection).

Dec 1, 2022, 10:03:56 AM12/1/22

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I didn't object on that, the difference between the two statements is:

"an ellipse is an oval" ---> True

"Ellipse was never a Conic Section" --> False

"an ellipse is an oval" ---> True

"Ellipse was never a Conic Section" --> False

Dec 1, 2022, 10:23:46 AM12/1/22

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Dec 1, 2022, 11:03:43 AM12/1/22