Add onto my 150th book// Proof Research into Cylinder, Cone, Sphere, Ellipsoid, Ovoid, Torus; fixing the mistakes of Apollonius and Euclid // Math Research series, book 2 by Archimedes Plutonium
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Archimedes Plutonium
Dec 18, 2022, 9:05:07 PM (2 days ago)
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Add onto my 150th book// Proof Research into Cylinder, Cone, Sphere, Ellipsoid, Ovoid, Torus; fixing
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Archimedes Plutonium
Dec 18, 2022, 11:36:36 PM (2 days ago)
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So, all I need do is find out if the inscribed oval inside a isosceles trapezoid with only 4 points
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Archimedes Plutonium
Dec 19, 2022, 12:43:43 AM (2 days ago)
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Archimedes Plutonium Dec 19, 2022, 12:40 AM to sci.math Alright I did manage to find some pictures on
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Archimedes Plutonium
Dec 19, 2022, 2:38:56 AM (2 days ago)
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On Monday, December 19, 2022 at 12:40:27 AM UTC-6, Archimedes Plutonium wrote: > Alright I did
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Archimedes Plutonium
Dec 19, 2022, 12:39:46 PM (2 days ago)
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Alright, this is getting better all the time. It is the Kite that encloses the Oval, not the
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Archimedes Plutonium
Dec 19, 2022, 12:42:58 PM (2 days ago)
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Is the kite geometry, the most essential and primitive aerodynamic geometry? I ask this question
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Archimedes Plutonium
Dec 19, 2022, 11:36:18 PM (yesterday)
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Alright I can see from the Alamy egg enclosed inside of Kite, where we can take any and every ellipse
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Archimedes Plutonium
Dec 20, 2022, 12:03:51 AM (yesterday)
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And this leads to a mystery question, instead of attaching a smaller 1/2 ellipse or a larger 1/2
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Archimedes Plutonium
Dec 20, 2022, 10:32:10 AM (yesterday)
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Alright, I need to include the Stadium figure. And see if it has that magical number of 78.5% that
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Archimedes Plutonium<plutonium.archimedes@gmail.com>
Dec 20, 2022, 10:39:13 AM (yesterday)
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I am glad I wrote my 150th book for I find myself continually using it for reference. Proof Research into Cylinder, Cone, Sphere, Ellipsoid, Ovoid, Torus; fixing the mistakes of Apollonius and Euclid // Math Research series, book 2

by Archimedes Plutonium

150th published book

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Archimedes Plutonium<plutonium.archimedes@gmail.com>
Dec 20, 2022, 10:52:09 AM (yesterday)
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Now I cannot help but notice that phi the golden ratio is 1.618.... while 0..785^2 is 0.616 for a Sigma Error of just 618/616 = 0.3%, and in physics we say within 0.5% or less they are equal. The interpretation here is the quantum mechanics psi squared that pi digits and golden ratio are the same thing.

AP
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Archimedes Plutonium<plutonium.archimedes@gmail.com>
Dec 20, 2022, 12:28:45 PM (yesterday)
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Some philosophical geometry. I call it that because the answers are not clear enough.
1) Is pi involved in the parabola? I would say no because those are polynonomials.
2) Is pi involved in hyperbola? Not sure for it has the features of a circle equation. And instead of conics being >< apex to apex, if we had then <> base to base, then the hyperbola cut would be indead a ellipse, perhaps a circle.

3) Can we say the only smooth curves in geometry are circle, ellipse, oval? Anything else that is closed looped and a curve is not smooth but has a vertex or more.

4) The proof that the area of circle is pi*r^2 involves slicing the circle up into fine rectangles that form a rectangle of r height and base of pi*r. But can the proof of area begot from the fact of 1/4 of circle enclosed in square of radius^2? Can we argue that multiply this square of r^2 times pi is area circle?

AP
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Archimedes Plutonium<plutonium.archimedes@gmail.com>
Dec 20, 2022, 7:26:16 PM (23 hours ago)
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So to have a smooth curve, no vertices, requires 4 points as maximums where to the left and right of that point is a rise or fall but not both.

What about a stadium? we see the ends as 2 maximum but not the 4, instead we see a constant, no rise, no fall as the parallel line segments take off.

Quadrilaterals have 4 vertices as they close loop around.

Triangles have 3 vertices and closed looped.

A parabola has 1 maximum. A hyperbola has 2 maximums.

Curiously and caught my attention fast, a polynomial of 5th degree has 4 maximums.

AP
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So, the mystery question for today is whether a hyperbola with equation x^2/a^2 - y^2/b^2 = 1 while circle is the same formula only a addition signage rather than subtraction, whether the hyperbola contains pi or whether the hyperbola follows in the footsteps of the parabola y= x^2 and is just a pure polynomial.

I am going to side with the idea that hyperbola Y = 1/x is not a polynomial and that the Conic sections when two cones are placed <> rather than >< that the hyperbola is a formula of a closed loop circuit.

AP

On Wednesday, December 21, 2022 at 6:50:33 PM UTC-6, Archimedes Plutonium wrote:
> So, the mystery question for today is whether a hyperbola with equation x^2/a^2 - y^2/b^2 = 1 while circle is the same formula only a addition signage rather than subtraction, whether the hyperbola contains pi or whether the hyperbola follows in the footsteps of the parabola y= x^2 and is just a pure polynomial.
>
> I am going to side with the idea that hyperbola Y = 1/x is not a polynomial and that the Conic sections when two cones are placed <> rather than >< that the hyperbola is a formula of a closed loop circuit.
>

So, the mystery of the hyperbola, and one can see that in the formula x^2/a^2 - y^2/b^2 = 1 if we let a,b = 1 we have x^2 -y^2 = 1 whereas circle is x^2 + y^2 = 1. Now, whose idea in math history was it, to think that Y = 1/x is a hyperbola ???? Whose idea???? Was it Newton?

So, I suspect what happened here in math history was another huge blunder. They looked upon Y = 1/x and said, low and behold, "What curve is this?" Probably at time of Christmass they said this. And it was a curve they could not relate to, so they said, the conic sections have a parabola and hyperbola, so let us call it a hyperbola for it is more open than a parabola. Even though you cannot get 1/x from x^2 - y^2 = 1.

So the Hyperbola has some circle and ellipse features-- its formula is the same except subtraction instead of addition. While it has some polynomial parabola features-- it is not a closed curve but open and parabolas are polynomials. So is the Hyperbola a polynomial or is it half of a closed curve with its 2 branches separated? If we do conic sections as >< with apex of cone on apex (really silly way if you ask me) for the logical conic sections should be <> base to base, then the parabola remains but the hyperbola disappears altogether as a closed curve of ellipse, or circle, no ovals though.

So, what I am going to do to make more commonsense logic is define a new type class of Polynomials. The Inverse Polynomial. I see no-one in math literature to define such a thing. The Inverse Polynomial are all normal polynomials but entirely divided into 1.

Some examples of Inverse Polynomials.

1/x
1/x^2
1/(x^3 +2x+1)
1/(x^5 -3x^4+2x^2 +x - 2)

Alright, you get the idea, every normal polynomial has a inverse polynomial and the Y= 1/x is a inverse polynomial. Is it a hyperbola??????

AP

Alright, several posts back I claimed the Circle, Ellipse, Oval each had 4 poles in their closed loop, matching the 4 vertices of Square, Rectangle, Kite that encloses them respectively.

Now some would argue that the circle has an infinite number of poles, just as a circle is infinitely symmetrical with infinite lines of symmetry.

But in my definition of pole I need to clarify the circle. I forgot "change of direction". My definition of a maximum point for a pole is that a point to the left is rising while point to the right is falling. I cannot say that the two equator points are falling as they go to the opposite pole. But I can say the equator points are changing to the opposite direction.

So, given any arbitrary circle. Look at a point and the point to the left and right, one is rising the other is falling. Call this point of the circle the North Pole. Go along that maximum point until you reach the Equator 1 pole where the point to the left is one direction, the point to the right is the opposite direction. Keep going until you reach another maximum point, the South Pole where the left point of maximum is rising while the point right of South Pole was falling. Keep going until another maximum is reached and this is Equator2, the point to the right changes direction from point to the left, not a change of rise and fall, but change of direction.

Now I want to say something about the Relativity of a circle. So we pick any point we want and it must be a North Pole, but this arbitrary selection nails down where the other 3 poles the other 3 maximum points are. And we thought only Physics had relativity, and here in mathematics, with circle we have also a Relativity with a reference frame that marks out four poles of the circle.

Now can we say this Circle Relativity relates to Physics Relativity where the Physics relativity is all reduced down to the simple idea-- we must have a stationary coil with moving bar magnet in Faraday Law be exactly the same as a moving coil with stationary bar magnet. All of Special Relativity is reduced to that statement. But here I am asking if this Relativity of math circle with its 4 poles is equivalent to physics Relativity???

AP, King of Science, especially Physics

I need to start over to explain maximum and 4 poles better.

Starting with triangles to explain maximum.

I need Function Graph and 1st quadrant only.

Take any triangle, and all have one leg that you place on the x-axis so that the other 2 legs from a function graph with a maximum point. Obviously a right triangle must have its 90 degree angle so no perpendicular is on the x-axis.

So I define pole as the maximum. I define the point of intersection with the x-axis as two more poles.

Now I go to closed curve figures, circle, ellipse, oval.

Obviously I can only do a semi of these figures to obey function graph.

So the circle is semicircle with equator on x-axis. I have a north pole maximum. I have two points of equator as pole.

Same goes for ellipse.

Same goes for oval, only the oval is 2 different ellipses.

In this manner, I well define the 4 poles of the circle, ellipse, oval. Now the poles are maximums in different arrangements. So if a ellipse is turned so the poles are equator points then the equator points become north and south pole.

And if I expand the coordinate system to 4 quadrants all positive numbers, the south pole is then seen as a maximum also.

This is how I should start the definitions, keeping them in line with function graph definition.

Now I can better tackle the question of whether a hyperbola is a polynomial like the parabola is a polynomial. Or whether the hyperbola is a cut away circle, ellipse.

AP

On Thursday, December 22, 2022 at 3:24:27 PM UTC-6, Archimedes Plutonium wrote:
> I need to start over to explain maximum and 4 poles better.
>
> Starting with triangles to explain maximum.
>
> I need Function Graph and 1st quadrant only.
>
> Take any triangle, and all have one leg that you place on the x-axis so that the other 2 legs from a function graph with a maximum point. Obviously a right triangle must have its 90 degree angle so no perpendicular is on the x-axis.
>
> So I define pole as the maximum. I define the point of intersection with the x-axis as two more poles.
>
> Now I go to closed curve figures, circle, ellipse, oval.
>
> Obviously I can only do a semi of these figures to obey function graph.
>
> So the circle is semicircle with equator on x-axis. I have a north pole maximum. I have two points of equator as pole.
>
> Same goes for ellipse.
>
> Same goes for oval, only the oval is 2 different ellipses.
>
> In this manner, I well define the 4 poles of the circle, ellipse, oval. Now the poles are maximums in different arrangements. So if a ellipse is turned so the poles are equator points then the equator points become north and south pole.
>
> And if I expand the coordinate system to 4 quadrants all positive numbers, the south pole is then seen as a maximum also.
>
> This is how I should start the definitions, keeping them in line with function graph definition.
>
> Now I can better tackle the question of whether a hyperbola is a polynomial like the parabola is a polynomial. Or whether the hyperbola is a cut away circle, ellipse.
>

Now in this definition of the maximum for triangles whose 1 leg is placed on the x-axis, all triangles except the acute triangle-- all angles less than 90 degrees -- have at least one leg that cannot be on the x-axis for it defies the definition of graph function. All right triangles must have the hypotenuse on the x-axis to obey graph function definition.

So what I am doing here is letting graph function definition define what is a maximum and what the 4 poles are in circle, ellipse, oval.

This is good because it draws in calculus for maximum.

And it tells me that the hyperbola is a circle or ellipse but a special type of closed figure when cones are oriented <> base to base rather than apex to apex ><

It tells me that the 1/x is not a hyperbola at all but a polynomial like a parabola and belongs to the parabola class geometry described by a inverse polynomial. Graph functions such as 1/x or 1/(x^2+3) etc etc.

Now why should all smooth curved figures that are closed looped have 4 poles?? If we put two parabolas together, there be 2 vertex, but if we put two ellipses or a ellipse + circle joined them at their equator poles we end up with a Smooth closed loop.

Why should parabolas joined at equator points end up as vertices??

So somewhere in this research I have to connect pi, that pi is contained in ellipse and oval and also the hyperbola, the hyperbola from cones oriented base to base <>.

AP

A NEW CONIC SECTION NEVER SEEN BEFORE IN MATH HISTORY

So now I do not know why the Ancient Greeks were not smart enough to put cones base to base <> and study them rather than put cones apex to apex ><. I see that as really quite stupid, even though the Ancient Greeks were geniuses.

Perhaps, the Ancient Greeks such as Apollonius never put them apex to apex and that is only a recent math error and not the fault of the Ancient Greeks at all.

For what benefit accrues from putting two cones apex to apex, rather than studying the single cone??? I suspect the only benefit is that you have the other half of the hyperbola. And this is the reason I think the Ancient Greeks never had the apex to apex for they did not care that the hyperbola was 1 branch. So I think someone in modern times put the apex to apex.

But AP is the very first to put 2 cones together base to base and those are the only worthwhile study.

For a magnificent conic section of the Hyperbola as a closed loop figure was missed. It is totally absent in math history. The perpendicular to bases cut in <> is a rhombus starting at apex and thus is a closed straight line figure of 4 vertices. However the perpendicular to base cut on any point not the apex itself of the base-points ends up being a new, totally new figure in geometry history for it is a closed loop figure with 4 rounded pencil loop vertices in the shape of a rhombus but with rounded vertices.

So, in my list of smooth curves that are closed loop I need to add this 4th one-- circle, ellipse, oval, rounded rhombus.

AP, King of Science, especially Physics

On Friday, December 23, 2022 at 7:15:08 PM UTC-6, Archimedes Plutonium wrote:
> A NEW CONIC SECTION NEVER SEEN BEFORE IN MATH HISTORY
>
> So now I do not know why the Ancient Greeks were not smart enough to put cones base to base <> and study them rather than put cones apex to apex ><. I see that as really quite stupid, even though the Ancient Greeks were geniuses.
>
> Perhaps, the Ancient Greeks such as Apollonius never put them apex to apex and that is only a recent math error and not the fault of the Ancient Greeks at all.
>
> For what benefit accrues from putting two cones apex to apex, rather than studying the single cone??? I suspect the only benefit is that you have the other half of the hyperbola. And this is the reason I think the Ancient Greeks never had the apex to apex for they did not care that the hyperbola was 1 branch. So I think someone in modern times put the apex to apex.
>
> But AP is the very first to put 2 cones together base to base and those are the only worthwhile study.
>
> For a magnificent conic section of the Hyperbola as a closed loop figure was missed. It is totally absent in math history. The perpendicular to bases cut in <> is a rhombus starting at apex and thus is a closed straight line figure of 4 vertices. However the perpendicular to base cut on any point not the apex itself of the base-points ends up being a new, totally new figure in geometry history for it is a closed loop figure with 4 rounded pencil loop vertices in the shape of a rhombus but with rounded vertices.
>
> So, in my list of smooth curves that are closed loop I need to add this 4th one-- circle, ellipse, oval, rounded rhombus.

At least I think the 4 vertices are all rounded, not sure of the two bases vertices as rounded. But the vertices off the shoulders of the cones are rounded-vertices and hence no vertices at all.

So how do I determine if the hyperbola cut when it reaches the joined together bases is a rounded-vertice or is a regular vertex??

We often think that geometry is easy and seldom fools us. But here is another, one in many expamples of where geometry fools us often and that need to demonstrate that this is a rounded vertex or a regular vertex.

AP

On Friday, December 23, 2022 at 7:37:31 PM UTC-6, Archimedes Plutonium wrote:
> On Friday, December 23, 2022 at 7:15:08 PM UTC-6, Archimedes Plutonium wrote:
> > A NEW CONIC SECTION NEVER SEEN BEFORE IN MATH HISTORY
> >
> > So now I do not know why the Ancient Greeks were not smart enough to put cones base to base <> and study them rather than put cones apex to apex ><. I see that as really quite stupid, even though the Ancient Greeks were geniuses.
> >
> > Perhaps, the Ancient Greeks such as Apollonius never put them apex to apex and that is only a recent math error and not the fault of the Ancient Greeks at all.
> >
> > For what benefit accrues from putting two cones apex to apex, rather than studying the single cone??? I suspect the only benefit is that you have the other half of the hyperbola. And this is the reason I think the Ancient Greeks never had the apex to apex for they did not care that the hyperbola was 1 branch. So I think someone in modern times put the apex to apex.
> >
> > But AP is the very first to put 2 cones together base to base and those are the only worthwhile study.
> >
> > For a magnificent conic section of the Hyperbola as a closed loop figure was missed. It is totally absent in math history. The perpendicular to bases cut in <> is a rhombus starting at apex and thus is a closed straight line figure of 4 vertices. However the perpendicular to base cut on any point not the apex itself of the base-points ends up being a new, totally new figure in geometry history for it is a closed loop figure with 4 rounded pencil loop vertices in the shape of a rhombus but with rounded vertices.
> >
> > So, in my list of smooth curves that are closed loop I need to add this 4th one-- circle, ellipse, oval, rounded rhombus.
> At least I think the 4 vertices are all rounded, not sure of the two bases vertices as rounded. But the vertices off the shoulders of the cones are rounded-vertices and hence no vertices at all.
>
> So how do I determine if the hyperbola cut when it reaches the joined together bases is a rounded-vertice or is a regular vertex??
>
> We often think that geometry is easy and seldom fools us. But here is another, one in many expamples of where geometry fools us often and that need to demonstrate that this is a rounded vertex or a regular vertex.
>

Alright, I am resolving this question in this manner. I am going to define a base to base joining as a planar circle a point thick joined to another planar circle a point think so the equator as the two bases have two points.

Now the apex is a single one point so a hyperbola in apex cut will have to push the apex aside to one of the two sections, for you cannot cut a point in half. (And this points out a deep logical flaw of Old Math in their continuum with an infinity of points between any two given points, for that is in essence translated into geometry as saying you can cut a geometry point into half or even an infinity of new points-- which goes to show how bereft and derelict Old Math professors were of a logical brain to even do mathematics.)

So, what I am going to end up doing in the Hyperbola sections is disqualify the apex to a perpendicular cut because you simply cannot cut a single point in half, leaving that cut as undefined.

Then any other hyperbola cuts along the shoulders of the two cones base to base <> I am going to say the cut through the bases involves two points back to back to form a rounded corner rather than a regular rhombus vertex, instead this fully 4 rounded vertices rhombus.

So, what I have done in effect is eliminate the hyperbola from math and from physics. But in physics, there never was a hyperbola trajectory in astronomy nor in particle physics. All open trajectories were some form of parabola.

AP, King of Science, especially Physics

So let me summarize at this point, so as not to get lost in thought.

The Parabola is indeed a open curve and follows a polynomial equation.

The Hyperbola comes not from apex to apex cones >< but from base to base cones <> and is a closed figure.

The formula of function Y = 1/x is not a hyperbola but instead a new class of functions called Inverse Polynomial.

The formula of Hyperbola as x^2/a^2 - y^2/b^2 = 1 is similar to formula of circle x^2 + y^2 = r^2 and circle more general (x-a)^2 + (y-b)^2 = r^2.

I have to find something in physics to force all conic sections to be base to base <>.

Once I find this "forcing and compliance" then the parabola and hyperbola become misshapened ellipses and rectangles.

AP

Alright progress is well under way on these topics. I have found the Physics forcing-agent, that all conic sections are closed curves; we just did not use a Logical Mind in Conic Science.

Why the Cone in the first place?? It is the 3rd dimension representative of the Right Triangle, and we all know in Electromagnetic theory the primal importance that electricity is at right angles to magnetism, and the triangle is the most primitive 3D closed figure.

But then, what is the Forcing-Agent for all curves to be closed loop curves and that a open ended curve is imagination gone amok with missing parts and pieces?? And here the forcing-agent is the EM Spectrum of Light. No Light wave is open ended such as straight line ray, parabola ray, hyperbola ray. All EM Waves are closed loop waves. Now a Longitudinal closed loop wave is a series of circles or ellipses or ovals as compression and rarefaction, while Visible Light waves are Double Transverse Waves in a closed loop.

All energy in physics is a closed loop wave and all matter in physics is a closed loop wave. And so if nothing Physical in the Universe is a open curve, then a open curve in mathematics is mere illusion, delusion or imagination run amok.

The only reason in Old Math that they had parabola and hyperbola as open curves is they mismanaged definitions and concepts. For when you start with a open figure, many of your cuts are open. So a cone with its interior given a cut, to be honest, the solid cone has a closed figure as the cut. To get a parabola or hyperbola, you wishful thinking that the base and interior are nonexistent. And thus the parabola has a straight line segment at end of cut, same as hyperbola, but in Old Math they wished this straight line to go away. Many in Old Math just extending the cone, or in some cases the cylinder as to ellipse cuts. But extending the cone or cylinder does not remedy the error of logic. What solves the error is to put two cones base to base <> and thus we see the parabola is no longer a parabola but often a oval a and the hyperbola is no longer a open ended curve but a closed curve and many times is a ellipse.

The equation of ellipse is x^2/a^2 + y^2/b^2 = 1. The equation of hyperbola is the exact same as ellipse but for a change in signage to subtraction rather than addition. I made a mistake earlier in continually comparing hyperbola to circle, when the comparison was with ellipse.

AP

What started out as nice clean cut research has turned into torment.

Now the idea is that the focus in ellipse is the focal point in parabola and even in hyperbola. Using the directrix argument.

I am getting lost in a quagmire.

And perhaps the best I can do is turn to straightline figures-- square, rectangle, kite and come back to curved -straightline figures. Maybe in January 2023 I can unclog myself.

AP

Alright, I need to start this entire adventure into planar cuts into 3D objects all over again.

Some years back I was puzzled with the idea that the planar cut into a cube can yield a rectangle, but that if you start the planar cut from a 90degree angle edge, the 90 degrees is not preserved. So why does the 90degrees be preserved if I start the cut using a side, not edge, and producing a rectangle. I remember at the time worried as to whether the rectangle was really a rectangle and whether the 90 degrees had been corrupted.

So, let me make a Conjecture: a very general conjecture. All figures in 2D geometry can be obtained from a planar cut in a 3D object in 3D geometry. Sounds plausible, but can there be a proof?

But today's action is taking boxes, and we have many from the food packaging and take a sharp box cutter-- not for any kids to do--- for they surely will get cut and trim from a 90 degree corner and what I end up with are various size angles of triangles. So the question still remains-- if you take a planar cut into the side of a cube you can get a rectangle. But if you take a cut into a 90degree corner that is not parallel to base, you lose that 90degrees to a smaller angle. Now, is there any cut into a cube that delivers a greater than 90 degree angle? Not to my knowledge as of yet. And this poses the next big question, is any 3D object able to produce a angle larger than the largest angle contained???

AP

So reflecting back to that time in which I was stumped on whether a slant cut in cube to make a rectangle, that the angle of 90 degrees was not corrupted. For a slant cut in a corner edge does not preserve the 90 degrees unless the cut is parallel to the base of cube.

I overcame the corruption of the angle to form the rectangle by thinking of the side walls keeping the angle in place. The sidewalls of cube not allowing the corruption of the 90 degree angle and acting as so to speak "handrails" for the cut.

AP

Alright, back to this Conjecture on angles. Given the 3D cube or rectangular box. I see no way in which I can extract a planar figure by a cut that has a angle larger than 90 degrees.

So the Conjecture would look like this: Statement: no planar cut into a 3D object can yield a angle larger than the largest angle in the 3D object before the cut. This conjecture would be somewhat like a Conservation of Physics law.

And I hope to show in Conic sections a Special Relativity principle emerge from cuts.

AP

For those that tuned in late, I am researching again, and again the Conic Sections. Now some may say the Conic Sections is trifle mathematics. And much of it was trifle until Newton came along and found that the orbit of planets and the force of gravity needed the conic curves, and thus spotlighting that Ancient Greek mathematics of Conic Sections.

However some important major mistakes were made in Apollonius Ancient Greek Conic Sections. The slant cut in cone is not a ellipse but rather a Oval. They missed the Oval in Ancient Greek mathematics. But worse yet, the study of Conic Sections-- whether Apollonius did it or a more recent mathematician did it-- perhaps even Newton, placed the conics as apex to apex >< when the placement should have been base to base <>.

I am opening up all the books on Conic Sections for they need a thorough scrubbing, cleaning up the mistakes and a entirely new outlook of what is important about Conic Sections.

What is mostly driving me in this research is the lack of Closed Loop Circuit that is essential to Physics. For the Primal Axiom over all of Physics is -- All is Atom, and Atoms are nothing but Electricity and Magnetism. You cannot have electricity and magnetism with a circuit. And all the mathematicians of the past-- never did the Circuit, the closed loop required for Physics.

And to start with the Circuit in mathematics, we start with conic sections.

No longer is Conic Sections a archaic and remote and fringe math, no, it is of central importance in math.

The Conic Sections not only well defines the Oval, but brings us to the well defined parabola, hyperbola, even causes us to correct the definition of Function, so that a Closed Loop Circuit is a Function in New Math.

Now I started this research again, several weeks past, and have strewn across my posts many unanswered questions-- such as whether the Stadium figure has a 78.5% inscribed "other figure" since a circle has a 78.5% inscribed square for area. Many many items opened for question but my lack of answering them, for I can only hope I remember all the open questions.

But for the bigger picture, that of Circuit in geometry math needs a firm foundation which at the moment has nothing about circuits in mathematics.

And it is possible that when I get finished or partially finished, we begin to teach geometry in schools starting with Cuts into 3D objects to obtain all the other geometry objects in 2D. But being the teacher I am, there is one caveat to that idea, in that youngsters taking knives to objects for geometry and end up cutting themselves. So maybe do conic sections only in senior year High School.

AP

Now let me spend a little time on a logical pointer. The 4 points to make a closed loop such as quadrilaterals. And why the Cone sticks out in 3D as the start of sectioning geometry.

Now a triangle is a closed loop, and the right triangle when spun on one of its legs forms a Cone. Now if we put 2 cones apex to apex we end up with 5 points >< but if we put 2 cones base to base <> we end up with 4 points in the silhouette cut.

In base to base cones we turn parabolas and hyperbolas into closed loop circuit figures.

So why would Physics want right triangles when spun around on a leg of a right triangle becomes a cone??? Because physics electromagnetism is based on perpendicular that a right angle offers and we can think of 1 right triangle as a monopole but 2 right triangles base to base spun on one leg forms the cones base to base.

So the reason we start Math Geometry from cones is because right triangles form cones by spinning them. And the reason we have the cones base to base, is because all cuts end up as closed loop circuits. We can say in this framework that a parabola of Old Math is a 1/2 of a closed loop and a hyperbola of Old Math is a 1/2 closed loop. This is the mistake of Old Math of having cones apex to apex.

AP

Alright so let us suppose we have a cube or rectangular box in 3D.

And we view it from the side face as this

/___________/
|\...................| ..cut 1
|...\.................|
|....................|
|____________|/

Now we have a planar cut parallel to the bottom base of cube or rectangular box and it delivers us a square if it is a cube equal to any one of the 6 faces. And delivers us a rectangle equal to the base rectangle of the box.

But what happens if we make a planar cut 1 but angle it downwards instead of parallel to base. We get a rectangle in both cube and box and a rectangle that preserves the 90 degree angles because the side walls keep the 90 degree angles in place.

But what happens if we slant the cut as in cut 2 with the slant cut \, although I wanted a shallow angle not steep but ascii art only has a steep angle. Does the cut deliver a new rectangle so that all four angles are 90 degrees.

Certainly one of the lengths of the new rectangles is longer than any rectangle in the box or longer than any square in the cube.

What I am getting at here is a Special Relativity Principle embedded in geometry.

AP

On Tuesday, December 27, 2022 at 3:26:20 PM UTC-6, Archimedes Plutonium wrote:
> Alright so let us suppose we have a cube or rectangular box in 3D.
>
> And we view it from the side face as this
>
> /___________/
> |\...................| ..cut 1
> |...\.................|
> |....................|
> |____________|/
>
> Now we have a planar cut parallel to the bottom base of cube or rectangular box and it delivers us a square if it is a cube equal to any one of the 6 faces. And delivers us a rectangle equal to the base rectangle of the box..
>
> But what happens if we make a planar cut 1 but angle it downwards instead of parallel to base. We get a rectangle in both cube and box and a rectangle that preserves the 90 degree angles because the side walls keep the 90 degree angles in place.
>
> But what happens if we slant the cut as in cut 2 with the slant cut \, although I wanted a shallow angle not steep but ascii art only has a steep angle. Does the cut deliver a new rectangle so that all four angles are 90 degrees.
>
> Certainly one of the lengths of the new rectangles is longer than any rectangle in the box or longer than any square in the cube.
>
> What I am getting at here is a Special Relativity Principle embedded in geometry.
>

Alright, if the slant cut into the box is itself angled so that the cut no longer uses the side walls to preserve the 90 degrees we end up with a figure that is no longer a rectangle.

I am thinking back to my High School and College education of Conic Sections and thinking how terribly difficult this study is now. Yet in schools, these Conic Sections are taught as if they are a breeze. This is probably because no-one spends much if any time on them.

As for a Special Relativity in Physics, that is because the Faraday Law in EM theory has to be the very same law whether the bar magnet is stationary and coil is moving or whether the bar magnet is moving and coil is stationary, and to cover the situation when both are moving. Many people get bent out of shape when doing physics and coming to the Special Relativity, only because they have few logical marbles.

In conic sections, we see that a angle and a length measure can be altered. In EM theory the length contracts and the time dilates, both shortened. In the slant cut of a box, the length is extended but the angle is shortened.

AP

On Tuesday, December 27, 2022 at 3:26:20 PM UTC-6, Archimedes Plutonium wrote:
> Alright so let us suppose we have a cube or rectangular box in 3D.
>
> And we view it from the side face as this
>
> /___________/
> |\...................| ..cut 1
> |...\.................|
> |....................|
> |____________|/
>
> Now we have a planar cut parallel to the bottom base of cube or rectangular box and it delivers us a square if it is a cube equal to any one of the 6 faces. And delivers us a rectangle equal to the base rectangle of the box..
>
> But what happens if we make a planar cut 1 but angle it downwards instead of parallel to base. We get a rectangle in both cube and box and a rectangle that preserves the 90 degree angles because the side walls keep the 90 degree angles in place.
>
> But what happens if we slant the cut as in cut 2 with the slant cut \, although I wanted a shallow angle not steep but ascii art only has a steep angle. Does the cut deliver a new rectangle so that all four angles are 90 degrees.
>
> Certainly one of the lengths of the new rectangles is longer than any rectangle in the box or longer than any square in the cube.
>
> What I am getting at here is a Special Relativity Principle embedded in geometry.
>

Alright, if the slant cut into the box is itself angled so that the cut no longer uses the side walls to preserve the 90 degrees we end up with a figure that is no longer a rectangle.

I am thinking back to my High School and College education of Conic Sections and thinking how terribly difficult this study is now. Yet in schools, these Conic Sections are taught as if they are a breeze. This is probably because no-one spends much if any time on them.

As for a Special Relativity in Physics, that is because the Faraday Law in EM theory has to be the very same law whether the bar magnet is stationary and coil is moving or whether the bar magnet is moving and coil is stationary, and to cover the situation when both are moving. Many people get bent out of shape when doing physics and coming to the Special Relativity, only because they have few logical marbles.

In conic sections, we see that a angle and a length measure can be altered. In EM theory the length contracts and the time dilates, meaning by dilates, is that time is extended. In the slant cut of a box, the length is extended but the angle is shortened. So here in geometry we can picture length as time in physics, and picture distance in physics as a angle of geometry.

AP

Marvelous progress, and for those wanting to see all posts go to my newsgroup--Read my recent posts in peace and quiet.
https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe  
Archimedes Plutonium

Empty food boxes have never been more important to me than now. Where the small ones I can use a heavy duty scissors to make planar cuts. Or to be on the safe side, just draw a line in a magic marker where the cut takes place.. I prefer the small boxes like soap boxes.

The Ancient Greeks probably had clay and a sword to make cuts. Perhaps if they had more material to experiment, the Ancient Greeks would have realized slant cut of Cone is Oval, never ellipse.

But the big announcement for today by me is about Angles.

So, let me define a circle as having a 90 degree angle where the circle intersects halfway on two adjoining sides of a square with circle inside. That would make the angle of a hyperbola be Less Than 90 degrees and the angle of a parabola even smaller than hyperbola.

Now I asked the question before and posed it as a Conjecture-- given any 3D object whose largest angle is K, that no planar cut into this object will yield a angle greater than K. Sounds intuitive, but a little difficult to prove. It takes only one counterexample to defeat this conjecture. But so far it is holding up.

And in that respect if we look at the hyperbola and the parabola and imagine them as straight lines instead of curves, do we not see that they fail to go beyond the 90 degrees that the x to y axis present itself.

What I am saying is that in curved geometry, the circle curve is the maximum angle of 90 degrees and all other curved angles such as hyperbola, parabola are smaller than 90 degrees.

AP, King of Science, especially Physics

Apparently the Conjecture is true-- no planar cut in a 3D object can exceed the largest angle of that 3D object. For the rectangular box, I cannot fetch a planar figure with a angle larger than 90 degrees.

Some progress also on assigning Angles to the curves of parabola, hyperbola, circle, ellipse and oval.

For the circle we assign the largest angle possible for a smooth curve of 90 degrees. We define that angle from the fact that the circle enclosed inside a square touches the midpoint of two adjacent sides and thus the angle 90 degrees is assigned.

Now as for a parabola from a cone. We derive the silhouette of cone as a /\ and that is the angle we assign the parabola.

As for the hyperbola, the 1/x is not a hyperbola but as I described earlier a inverse polynomial curve.

But for true hyperbolas, they are angles steeper than the parabola but smaller than the circle of 90 degrees.

Do I inscribe the hyperbola in a rhombus for the angle? Or inscribe the hyperbola in a Kite for the angle? For the ellipse and oval, for the oval is two different semiellipses joined at widst point. The angle of ellipse and oval is found in the Kite inscribing the semiellipse and semioval.

It is important to give angles to smooth curves, something Old Math was derelict in duty.

AP

Alright, there is a principle that is slowly being formulated on the rectangular box, a principle concerning preservation by being connected with with perpendicular line segments. Apparently any planar cut into a box from any of its faces that uses a perpendicular line segment between side walls will preserve all angles, but if the cut is into a vertex, the angle is shortened. So the perpendiculars act as guide rails for the planar cut. This is good news for the Conjecture-- no planar cut yields a angle greater than the largest angle in the object.

But yesterday I started to define the Angle of Curves. Starting with Circle as the largest angle of a curved circuit being 90 degrees. The next largest angle is from the Inverse Polynomial (for 1/x is not a hyperbola).

If we graph a circle in a 10 by 10 square it meets the midpoint of (0,5) and (5,0) and so we say that the circle is angle 90degrees. What is the angle for Y = 1/x ???? It is close to being a circle angle but is rather 85 degrees. So we graph 1/x in this 10 by 10 square and look to see at what point the intersection is for 1/x and (0,5) and (5,0). Is it at point x = 0.2 and thus we say the angle of Y = 1/x is a 88 degree angle because the circle angle is exactly 90degree.

Now we look at Parabolas and Hyperbolas from base to base <> cones. Here, both parabolas and hyperbolas are Closed Loop Circuit figures, not the Old Math mistaken nonsense of open figures. Old Math just treated 1/2 of the parabola and hyperbola with their mindless apex to apex ><.

The silhouette of the two cones gives us a rhombus <> and the smaller apex angles gives us a clue as to what the parabola angle is.

That leaves remaining the question of the angles for hyperbolas. They are angles smaller than circle 90 degrees and even 1/x angles of about 88 degrees. If we follow the path down the double cones <> base to base about midpoint and build a hyperbola there, we come very close to being a circle of about 85 degrees, smaller than the 1/x angle but approaching it.

AP

So now here, I am holding together two plastic cones, base to base <> and looking for the Hyperbola planar cut that produces a circle. This circle is not a full circle by a Defective Circle by a few points. Because as we put the base to base cones, the equator as I call it, has two identical points of the base where as a true circle has rising, maximum, falling points. At the equator we have rising, then a level point, then another level point, finally falling. So in a sense, we have a circle but whose equator has 2 identical points.

Now where is this Hyperbola Circle located? I estimate it is half way down in distance from apex to equator.

And let that be a Conjecture to prove. That in all Conic Sections, base to base <> the midpoint in distance from the silhouette side of the two cones harbors a Defective Circle as the hyperbola planar cut.

AP