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AP's 308th book of science// Conjecture, Sphere is maximum volume given specific surface area, while Torus is reverse-- maximum surface area given specific volume.

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Archimedes Plutonium

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Nov 26, 2023, 1:58:07 AM11/26/23
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Now I am sure the sphere is maximum volume given a specific surface area is proven a long time ago, and have to see how that proof goes.

But I am conjecturing that AP is the first to notice that the counterpart to a sphere of volume to surface area is the Torus. Which is inverse or reverse of sphere so that the Torus when given a specific volume, that the torus figure maximizes the Surface Area from that volume.

It is the torus I make this conjecture and make this my 308th book of science, if not already proven true.

I have plenty of time to research this as my queu line of book writing is currently at 263, so sometime next year in 2024 will get to this idea.

AP, King of Science

AP's 308th book of science// Conjecture, Sphere is maximum volume given specific surface area, while Torus is reverse-- maximum surface area given specific volume.

Archimedes Plutonium

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Nov 26, 2023, 2:05:35 AM11/26/23
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Immediately I can recognize what is going to hold this conjecture up or even defeat it.

In my living room I have a electric space heater shaped like this ||||||| with ribs. So the volume inside this heater is a small number but the surface area with all those ribs are enormous in comparison to volume.

So unless I can well define a torus opposed to a space heater, I have no conjecture.

What can I say about a torus that eliminates the space heater as a figure of contention??

AP

Archimedes Plutonium

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Nov 26, 2023, 3:53:52 AM11/26/23
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Yes, ribs are a minor problem easily dispensed with by a well defined definition of Solid Figure versus Ribbed Figure. So my electric space heater is a ribbed rectangular box with 7 ribs and so I define the Solid Rectangular box that the heater can fit into and then I define a ribbed rectangular box. Same thing for Sphere-- the solid sphere and a ribbed sphere. But then what happens is the ribbed Torus beats out the other ribbed figures. So no problem with ribs.

And I should detail some special ribbed figures, the Saddle shaped figure is actually the intersection of two ellipsoids or ovoids or a mixture of ellipsoids and ovoids that are then ribbed. None of this fancy Lobachevsky and negative curvature need be applied or thought of.

But having solved the rib problem and dismissing it be saying the ribbed Torus beats out its rivals, I have a new problem to tackle. And I wonder why I never tackled this before. I need to well define the associated sphere to a given torus. I suppose it is to enclose a torus inside a sphere and then pick off what the torus r radius and R radius are from the enclosing sphere. There should be ample new insights and new discoveries here in this undertaking.

So yes, this conjecture is going to be new and sweet.

AP

Archimedes Plutonium

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Nov 26, 2023, 4:23:11 PM11/26/23
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The problem of a unique enclosed in a sphere is easily solved when we demand the torus be 840 rings each separated from its neighbor rings by an angle of 360degrees/840 = 0.42 degrees.

--- quoting from my 205th book of science ---
Faraday Law is inverse projective-geometry of Coulomb-gravity Law//Physics-Math
by Archimedes Plutonium

Preface: This book discusses the symmetry of the 4 differential laws of Electromagnetic theory, the Faraday law, Ampere-Maxwell law, Coulomb-gravity law and the Transformer law. This book also dives into the numbers of importance of physics and math, the 1/137, the pi, the pi subtract 2.71... and much more.

Cover Picture: Is my iphone photograph of 840 windings of slinky toy to form a torus that is the proton torus of physics of its 840MeV with a muon stuck inside at 105MeV doing the Faraday law.
--------

If true, my conjecture then places the meaning of the Fine Structure Constant, -- its meaning is geometrical, in that the fine-structure-constant is a measure of how many windings you need to make a torus, that allows the Faraday law to be carried out as freely in a torus, as the Faraday law is carried out in a cylinder.

The 840 windings is the minimum number of windings that creates a angle so that the band that represents the muon thrusting through the torus, that the muon band is still almost touching the torus walls but moves freely inside the torus to travel round and round in the Faraday law.

So now, what sort of angle are we talking about in this Conjecture? Well a circle or torus has 360 degrees and thus we have 360/840 = about 0.428 degrees. We are talking about a small angle of 0.428 degrees that gives a torus the ability to move a band-width-muon through the torus, as freely as if it moves through a cylinder.

The angle of 0.428 degrees in radian measure is 0.0072, the Fine Structure Constant of physics.

So, this is an exquisite idea of the physics Fine Structure Constant. We knew its algebraic numeric value for a long time. Just today, we are learning its Geometry meaning.

Its geometry meaning is that give me a circle, any size of circle you want. Call it the unit circle. Then take 840 of those unit circles and make a slinky toy torus out of those 840 circles as 840 windings. It is the Minimum torus for that given size circle for which the muon electron as bar magnet, fits inside that slinky toy torus and does the Faraday Law producing Maximum Electricity.

As Wikipedia defined the Fine Structure Constant -- "fundamental physical constant which quantifies the strength of the electromagnetic interaction between.." elementary Dirac magnetic monopoles. AP is adverse to "charge" in physics for that is a fake concept and the true concept is magnetic monopoles.

Yes, the Maxwell Equation of Gauss that says no magnetic monopole exists is a false law. Magnetic monopoles exist every time one finds a 0.5MeV particle.

Of course, we never see the 1/137 or the 137, but we do see the square root of 1/137 as that of 0.084 which is the prefix of the rest-mass of true proton 840MeV, and the true electron is the muon at 105MeV, stuck inside the proton torus doing the Faraday law with the proton.
--- end quoting from my 205th book ---

Only the number of windings of 840 in a Faraday coil combined with 360 degrees gives the Fine Structure Constant of physics.

AP

Archimedes Plutonium

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Nov 26, 2023, 4:37:26 PM11/26/23
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In my 205th book of science I discover a way of producing four important constants of physics and math-- pi, Fine Structure, 840, and 360. One of those is a angle measure-- the 360 for a full revolution.

In an experiment described in that 205th book, I took 840 identical circles and crafted them into a perfect torus. To find the angle of separation I did a 360/840 and found an angle of 0.428 degrees. Look up what 0.428 degrees is in radians-- for it is the Fine Structure constant of physics 0.0072. Then in that experiment I measured the diameter and radius of the torus I constructed from 840 identical circles placed with a angle 0.428 degree separation. Measuring the diameters of circles and of the donut hole, the diameter of the donut hole of 840 circles forming a ratio which is pi = 3.14159....., specifically, in my experiment --
--- quoting my book ---

Alright, well it is easy to see that 210/65 is 3.230... So I went back to the lab and measured again and it was actually 205mm to 65mm. That gives me 205/65 = 3.15 and compared to pi 3.15/3.14 is a 0.3% sigma error, so close that it is automatic we announce that given any circle, and if you have 840 of those circles wound around into a torus, that the donut hole diameter divided by given circle diameter is pi number.
--- end quote ---

This is amazing for in a torus construction of precisely 840 circles, specifically 840, it cannot be any other number. That in the construction of 840 falls out pi = 3.1415... falls out that a full circle revolution must be the number 360 degrees and falls out the Fine Structure Constant in radians 0.0072.


My 205th published book of science.



Faraday Law is inverse projective-geometry of Coulomb-gravity Law//Physics-Math

by Archimedes Plutonium (Author) (Amazon-Kindle edition)

Preface: This book discusses the symmetry of the 4 differential laws of Electromagnetic theory, the Faraday law, Ampere-Maxwell law, Coulomb-gravity law and the Transformer law. This book also dives into the numbers of importance of physics and math, the 1/137, the pi, the pi subtract 2.71... and much more.

Cover Picture: Is my iphone photograph of 840 windings of slinky toy to form a torus that is the proton torus of physics of its 840MeV with a muon stuck inside at 105MeV doing the Faraday law.

Product details
• ASIN ‏ : ‎ B0BG46FS62
• Publication date ‏ : ‎ September 21, 2022
• Language ‏ : ‎ English
• File size ‏ : ‎ 370 KB
• Text-to-Speech ‏ : ‎ Enabled
• Screen Reader ‏ : ‎ Supported
• Enhanced typesetting ‏ : ‎ Enabled
• X-Ray ‏ : ‎ Not Enabled
• Word Wise ‏ : ‎ Not Enabled
• Sticky notes ‏ : ‎ On Kindle Scribe
• Print length ‏ : ‎ 35 pages
• Best Sellers Rank: #1,557,587 in Kindle Store (See Top 100 in Kindle Store)
◦ #195 in One-Hour Science & Math Short Reads
◦ #3,692 in Physics (Kindle Store)
◦ #4,382 in Mathematics (Kindle Store)

Archimedes Plutonium

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Nov 26, 2023, 11:09:51 PM11/26/23
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Now that is an interesting question and one of the problems solved in this book. The question of the horse saddle shaped geometry of Old Math geometry. Of course they thought it to be an example of hyperbolic geometry, or Lobachevsky geometry of negative curvature.

But the big problem is-- how do you pin a Volume quantity to a saddle shaped figure????

In New Math there exists not 3 geometries in the world, just 1 and that Euclidean is the only geometry, but you can separate Euclidean into two parts, a positive curvature along with a negative curvature, only because you separated them. A circle enclosed in a square, and cut out the circle and what remains is 4 pieces of what is called hyperbolic triangles. This is not a separate new geometry but cut-aways of Euclidean geometry.

Back to the horse saddle shape. A true geometry would have volume, not just area. But you cannot place a volume into a saddle shape.

And this then brings our attention to the idea that the Saddle Shape Geometry is merely the intersection of 2 ellipsoids or 2 ovoids and to make ribs of those 2 ellipsoids or combinations of ellipsoids and ovoids.

The Ribs are at a perpendicular to one another.

And then the question of the volume of the Saddle Shape is the volume of the 2 ellipsoids or ovoids.

AP

Archimedes Plutonium

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Nov 27, 2023, 12:54:11 AM11/27/23
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Alright, time for some calculations, and I harken back to a experiment I did in 2022, where I bought enough slinky toys to sort of guide me in what a 840 windings of a torus geometry produces in numbers. Here is a view of that experiment. What I want to do is given a sphere of radius 10, diameter 20 compute the radius and diameter of a torus composed of 840 windings inside the sphere and tangent at the equator line. Compute the r radius, not the big R radius, and diameter of this smaller circle that composes the torus of 840 windings, being sure the ratio of donut hole diameter is that of 3.14159....


On September 17, 2022 in sci.math,sci.physics, plutonium-atom-universe, I wrote a corrected copy:
A fabulous discovery of science physics.

Alright, curiosity in my lifetime has been indefatigable. I wanted to get a rough estimate of the donut hole of 840 windings so I bought 9 more slinky toys to combine with my 2 already owned ones. And I measured what a 840 winding torus donut hole was. My torus of 840 windings has a donut hole diameter of 205 mm and has a slinky toy diameter of 65mm. That would be a total diameter of torus as 65 + 205 +65 = 335mm with the donut hole diameter 205mm.

Now I play with those numbers and see what becomes of them for the Conjectures I placed so far. The most important being the idea that 840 windings is the physical geometry of the Fine Structure Constant as a torus the produces Maximum Electricity in the Faraday law.

Alright, well it is easy to see that 205/65 is 3.15... And I went back to the lab and measured again and it was actually 205mm to 65mm. That gives me 205/65 = 3.15 and compared to pi 3.15/3.14 is a 0.3% sigma error, so close that it is automatic we announce that given any circle, and if you have 840 of those circles wound around into a torus, that the donut hole diameter divided by given circle diameter is pi number.

Now also, looking at the angle which 840 circles create in a torus is 360/840 = 0.428 degree.

Amazingly a angle of 0.428 degree is in radians that of 0.0072 or the physics Fine Structure Constant. Here I have an angle for a torus which allows for a free thrusting of a muon inside the torus, as if the muon is in a cylinder in the Faraday law. And this angle of 0.428 allows the muon to freely thrust without bumping into the torus walls.

But the amazement does not stop there, for where is the number 0.42 degree angle come up in Physics constants? Well if you take pi 3.14 and subtract from "e" 2.71 you get 0.42.

AP

Archimedes Plutonium

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Nov 30, 2023, 3:14:46 PM11/30/23
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I am starting all over in this proof of the Conjecture. I found too many Flaws of Logic while using Old Math Geometry.

Here is a partial list of Old Math Geometry flaws.

1) How can we do surface area of curved figures and end up with measuring them in square-units. Specifically a sphere surface area is counted in square units, when what is logically needed is circular units.

2) What is a square that has no top or bottom just 4 sides?

3) Does a cylinder really need a top and bottom that is included in surface area?

4) Does surface area really include no volume? To make a biology analogy, the volume of a porcupine when it is agitated?

5) Old Math Geometry never included electricity and magnetism as the basic foundation of geometry figures. For electricity moves in Space, with magnetic Lines of Force.

6) As we move from the astronomical or even Earth sized figures move to the microscopic figures, does our volume and surface area concepts remain the same? I say no, and that this has been a logical fallacy of Old Math Geometry to assume they remain the same with the same formula of sphere surface area and volume applied to large objects and to tiny microscopic spheres.

7) The break through figure is the Cylinder which is really just a planar sheet wrapped around into a circular planar sheet. Uncurl the planar sheet that makes a cylinder and hinge it in four equal line segments and then curl them to form a cube, a cube without top and bottom but still a cube.

8) Old Math never had a valid proof that Spheres maximize volume given a surface area. For the cylinder Maximizes volume given surface area. A contradiction to Old Math Geometries alleged proof that the sphere maximizes volume given a specific surface area. No, Old Math Geometry is wrong, the cylinder maximizes volume given a surface area, for you need no top or bottom of a cylinder. And there is no discarding any of the sphere surface area, but you can always discard cylinder top and bottom.

So I need to rephrase the title of this book to read.

AP's 308th book of science// Conjecture, Cylinder is maximum volume given specific surface area, while Torus is reverse-- maximum surface area given specific volume.

AP, King of Science, especially physics and logic

Archimedes Plutonium

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Nov 30, 2023, 4:32:15 PM11/30/23
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On Thursday, November 30, 2023 at 2:08:45 PM UTC-6, Archimedes Plutonium wrote:
> I am starting all over in this proof of the Conjecture. I found too many Flaws of Logic while using Old Math Geometry.
>
> Here is a partial list of Old Math Geometry flaws.
>
> 1) How can we do surface area of curved figures and end up with measuring them in square-units. Specifically a sphere surface area is counted in square units, when what is logically needed is circular units.
>
> 2) What is a square that has no top or bottom just 4 sides?
>
> 3) Does a cylinder really need a top and bottom that is included in surface area?
>
> 4) Does surface area really include no volume? To make a biology analogy, the volume of a porcupine when it is agitated?
>
> 5) Old Math Geometry never included electricity and magnetism is the basic foundation of geometry figures. For electricity moves in Space, with magnetic Lines of Force.
>
> 6) As we move from the astronomical or even Earth sized figures move to the microscopic figures, does our volume and surface area concepts remain the same? I say no, and that this has been a logical fallacy of Old Math Geometry to assume they remain the same with the same formula of sphere surface area and volume applied to large objects and to tiny microscopic spheres.
>
> 7) The break through figure is the Cylinder which is really just a planar sheet wrapped around into a circular planar sheet. Uncurl the planar sheet that makes a cylinder and hinge it in four equal line segments and then curl them to form a cube, a cube without top and bottom but still a cube.
>
> 8) Old Math never had a valid proof that Spheres maximize volume given a surface area. For the cylinder Maximizes volume given surface area. A contradiction to Old Math Geometries alleged proof that the sphere maximizes volume given a specific surface area. No, Old Math Geometry is wrong, the cylinder maximizes volume given a surface area, for you need no top or bottom of a cylinder. And there is no discarding any of the sphere surface area, but you can always discard cylinder top and bottom.
>
> So I need to rephrase the title of this book to read.
>
> AP's 308th book of science// Conjecture, Cylinder is maximum volume given specific surface area, while Torus is reverse-- maximum surface area given specific volume.
>

Alright, here, I start all over again. Too many assumptions and errors in Old Math Geometry. I listed a few above.

The proof of the maximum volume given a specific surface area should start in 2D not 3D.

EXPERIMENT:: take a given length of chain, a straightline chain, which maximizes area? The square or the circle?

So I have a chain that is 64cm long, now I bend it into a square with side 16. And this square would have area 16^2 = 256.

Now I bend that same chain of length 64 into a circle. And its radius ends up being (pi)x diameter = 64. Diameter = 20.37. So the radius is 10.18. And thus the area of the given length of 64, turned into a circle of radius 10.18 is area = pi(r^2) = 3.14 (10.18^2) = 3.14(103.75) = 325.78.

Compare the area enclosed by the same length of 64, that the square is area 256, while the circle from that same length is 325.78.

Now, I have the proof method that will prove the Cylinder, missing its top and bottom maximizes volume given surface area.

In fact we can draw up the proof already.

Statement:: Given a specific surface area, the Cylinder maximizes the volume of that area because the cylinder needs no top and bottom, while the sphere has to use up all that surface area to form a sphere. Proof:: Apply the length argument above in 2D and it holds in 3D.

AP

Archimedes Plutonium

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Dec 1, 2023, 1:49:38 AM12/1/23
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Archimedes Plutonium<plutonium....@gmail.com>
Nov 29, 2023, 2:13:06 AM (yesterday)



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Alright I started to look what Old Math gave as a proof that the sphere when given a specific surface area gives the maximum volume. And found this.

Alright, I noticed this in Quora and my attention is perked whenever Fundamental Theorem of Calculus is mentioned.


--- quoting Quora ---
So, you noticed that if you take the formula for the volume of a sphere:

𝑉=43𝜋𝑟3
V
=
4
3
π
r
3


and differentiate with respect to the radius 𝑟
r
, you get:

𝑑𝑉𝑑𝑟=4𝜋𝑟2
d
V
d
r
=
4
π
r
2


which is the surface area of the same sphere.

Good job on noticing this; noticing things is a major part of mathematics. Yes, there is a significance. But before I explain, let’s notice some other things:

If you do the same thing with the area of a circle

𝐴=𝜋𝑟2
A
=
π
r
2


you get:

𝑑𝐴𝑑𝑟=2𝜋𝑟
d
A
d
r
=
2
π
r


which is the circumference of the circle!

How about this: define the “radius” of a cube to be the distance from the center to one of the faces. The volume of a cube is:

𝑉=8𝑟3
V
=
8
r
3


The derivative with respect to 𝑟
r
is:

𝑑𝑉𝑑𝑟=24𝑟2
d
V
d
r
=
24
r
2


which sure enough, is the surface area of the cube!

What’s going on here?

N... hinted at the meaning in his answer. Let me see if I can clarify a bit more.

Imagine a slowly expanding sphere. As the radius increases by a tiny amount, how much does the volume change? It changes by the volume of the spherical shell that was just added. That volume is roughly the area of that shell times its thickness. As we take the limit, and imagine the sphere to be increasing by an infinitesimal amount, the rate of volume change is actually the surface area of that spherical shell. Which is equal to the present surface area of the sphere.

The rate of change is the surface area. And there we have our connection with derivatives.

There’s actually something very deep here. The derivative of the sum of the volumes of the infinitesimally thin spherical shells is the surface area of the spherical shells. This can be generalized to certain infinite sums in general: The derivative of the sum of infinitesimally thin pieces under the graph of a continuous function is the function itself. This result is known as The Fundamental Theorem of Calculus, and from the name, you can see that we think it’s pretty useful.
--- end quoting Quora ---

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Archimedes Plutonium<plutonium....@gmail.com>
Nov 29, 2023, 1:53:16 PM (yesterday)



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From the looks of it, Old Math never had a good valid proof of the theorem:: Sphere maximizes volume given any particular specific surface area. I looked and found nothing that is persuasive.

And what mathematics needs now, is an Archive in which they storage all important theorems along with their proofs. So that I and anyone else, wanting to know a specific theorem proof can go to and not have to sift through thousands of "hack notioned proofs". There is a math archive for sequences and series called if memory is correct called AEOIS, something along that line. So if there is a question on a sequence we go to AEOIS and pinpoint the known literature of that sequence. Same thing needs to be done with important theorems of math, an archive where if you want to see the literature proof of that theorem, you go there. The trouble with a online search for math proofs is that we now have AI, some like to call it artificial intelligence but it is really "artificial ignorance" a vast sea of a new type of spam. So that if you want to see the Old Math proof of Sphere maximizes volume, you have to trudge through hundreds of worthless AI robots and their stupid prattle on the subject.

Old Math has no good proof of Sphere maximizes volume. And my proof is along the lines of a concept just recently borne. I created the concept of volume and surface area density.

If we take a cube its volume is side^3 and its surface area is 6side^2. When I divide surface area by volume I end up with 6/side. And I call that the surface area density per volume. If I divide volume by surface area = side/6, then I call that the volume density per surface area.

This book is about proving a conjecture-- the Torus maximizes surface area given a particular specific volume.

It had the initial hurdle of figures with ribs, or grills, or fins. But that hurdle was overcome in that I can make a torus with ribs that outbests a rectangle as ribs or any other figure turned into ribs.

But now I have a challenging rival to the torus that is not so easily dismissed as ribs. It is the tetrahedron or other triangular shaped objects like wedges. Is the Torus the maximal surface area given a volume or does the tetrahedron the maximal?

To explore this question may cause me to write another new book on math. For the volume of the tetrahedron is 1/3 (Area of base triangle)(height from base triangle to apex). This formula is similiar to cone enclosed in cylinder as being 1/3 the volume. And here is the opportunity of uniting figures that are curved with pi and figures that are all straightline figures.

So if I have a cube, can I see where that cube is equal to 3 tetrahedrons composing the volume of that cube? Just as a Cylinder is composed of 3 cones that equals the volume of that cylinder.

And here is the exquisite opportunity of uniting curved geometry with straightline geometry. For the volume density versus surface area density of cone with cylinder may match that of cube with tetrahedron density.

And if all goes well-- a super easy proof of the starting conjecture of this book-- Torus maximizes surface area once given a particular volume.

AP, King of Science
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Archimedes Plutonium<plutonium....@gmail.com>
Nov 29, 2023, 2:25:51 PM (yesterday)



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Alright, a correction already, for it is 3 Square Pyramids, not 3 tetrahedron that equals the volume of a cube. Just saw a Youtube clip of 3 Square Pyramids filling a cube exactly with water.

So I will phase out the discussion of tetrahedron. But I wonder how many tetrahedron associated to a Square Pyramid takes to fill up the Square Pyramid?

One web site said that the smallest number of tetrahedron to fill the volume of a associated cube is 5. But so much on Internet is polluted with AI chatGPT that so much is not trustworthy. AI is not helping education or science but rather spamming and polluting science.

AP
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Archimedes Plutonium<plutonium....@gmail.com>
Nov 29, 2023, 11:52:27 PM (yesterday)



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Now it is a easy observation that if we remove the top and bottom of a cylinder, the floor and ceiling, the circles and speak only of the rolled rectangle that we have a cylinder of Euclidean. Of course we still need to use pi for the circumference of cylinder. But electricity and magnetism do not care about top and bottom. And we can do the same for square pyramid, remove the base square. Same for cone.

And so now we compute surface area for cone without base, cylinder without top and bottom and square pyramid without square.

With the torus, sphere, ellipsoid, ovoid, there is nothing that can be removed. But with the cube, you can remove top and bottom squares and still be considered a cube and now its surface area is 4side^2 instead of 6side^2.

How much does that affect the research of the proof that torus is maximum surface area for a given specific volume.


AP
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Archimedes Plutonium<plutonium....@gmail.com>
Nov 30, 2023, 2:01:31 PM (11 hours ago)



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I am starting all over in this proof of the Conjecture. I found too many Flaws of Logic while using Old Math Geometry.

Here is a partial list of Old Math Geometry flaws.

1) How can we do surface area of curved figures and end up with measuring them in square-units. Specifically a sphere surface area is counted in square units, when what is logically needed is circular units.

2) What is a square that has no top or bottom just 4 sides?

3) Does a cylinder really need a top and bottom that is included in surface area?

4) Does surface area really include no volume? To make a biology analogy, the volume of a porcupine when it is agitated?

5) Old Math Geometry never included electricity and magnetism is the basic foundation of geometry figures. For electricity moves in Space, with magnetic Lines of Force.

6) As we move from the astronomical or even Earth sized figures move to the microscopic figures, does our volume and surface area concepts remain the same? I say no, and that this has been a logical fallacy of Old Math Geometry to assume they remain the same with the same formula of sphere surface area and volume applied to large objects and to tiny microscopic spheres.

7) The break through figure is the Cylinder which is really just a planar sheet wrapped around into a circular planar sheet. Uncurl the planar sheet that makes a cylinder and hinge it in four equal line segments and then curl them to form a cube, a cube without top and bottom but still a cube.

8) Old Math never had a valid proof that Spheres maximize volume given a surface area. For the cylinder Maximizes volume given surface area. A contradiction to Old Math Geometries alleged proof that the sphere maximizes volume given a specific surface area. No, Old Math Geometry is wrong, the cylinder maximizes volume given a surface area, for you need no top or bottom of a cylinder. And there is no discarding any of the sphere surface area, but you can always discard cylinder top and bottom.

So I need to rephrase the title of this book to read.

AP's 308th book of science// Conjecture, Cylinder is maximum volume given specific surface area, while Torus is reverse-- maximum surface area given specific volume.

AP, King of Science, especially physics and logic
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Archimedes Plutonium<plutonium....@gmail.com>
Nov 30, 2023, 3:33:42 PM (9 hours ago)



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Alright, here, I start all over again. Too many assumptions and errors in Old Math Geometry. I listed a few above.

The proof of the maximum volume given a specific surface area should start in 2D not 3D.

EXPERIMENT:: take a given length of chain, a straightline chain, which maximizes area? The square or the circle?

So I have a chain that is 64cm long, now I bend it into a square with side 16. And this square would have area 16^2 = 256.

Now I bend that same chain of length 64 into a circle. And its radius ends up being (pi)x diameter = 64. Diameter = 20.37. So the radius is 10.18. And thus the area of the given length of 64, turned into a circle of radius 10.18 is area = pi(r^2) = 3.14 (10.18^2) = 3.14(103.75) = 325.78.

Compare the area enclosed by the same length of 64, that the square is area 256, while the circle from that same length is 325.78.

Now, I have the proof method that will prove the Cylinder, missing its top and bottom maximizes volume given surface area.

In fact we can draw up the proof already.

Statement:: Given a specific surface area, the Cylinder maximizes the volume of that area because the cylinder needs no top and bottom, while the sphere has to use up all that surface area to form a sphere. Proof:: Apply the length argument above in 2D and it holds in 3D.

AP

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Archimedes Plutonium<plutonium....@gmail.com>
Nov 30, 2023, 9:33:29 PM (3 hours ago)



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I wrote on 30Nov2023::
>
> Statement:: Given a specific surface area, the Cylinder maximizes the volume of that area because the cylinder needs no top and bottom, while the sphere has to use up all that surface area to form a sphere. Proof:: Apply the length argument above in 2D and it holds in 3D.
>

So in 3D, given a planar surface area, say given a rectangle of 10 by 100 which is surface area 1000 and would have a volume of a cylinder rolled up as a circle of circumference 100 would be 3.14 x diameter = 100 so diameter is 31.84 and thus radius of 15.92 for circle area of 3.14(15.92^2) = 795.82. So cylinder volume although it has no top or bottom is 10 x 795.82 = 7958.2 volume. The same surface area of 10 by 100 = 1000 to compose a sphere with that surface area is 4(pi)r^2 = 1000, so r^2 = 79.61, and r = 8.92. The volume for a sphere of surface area 1000 is 4/3(pi)r^3, is 4.186(8.92^3) = 4.186(709.73) = 2970.93. So we see the Cylinder volume of surface area 1000 is over twice as large as the sphere created from the same surface area.

And we apply a similar argument for given a specific volume, the torus maximizes the surface area.

AP
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Archimedes Plutonium<plutonium....@gmail.com>
Nov 30, 2023, 10:59:33 PM (2 hours ago)



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So say we have a volume of 50 and want a figure that maximizes surface area. Say we try the regular cube of 6 faces. And so our volume 50 = side^3 or that the side equals 3.684. So our cube has surface area of 6 (3.684^2) = 6(13.572) = 81.43 square units.

Now using the same parameters of volume 50 for a cylinder, a cylinder without a top or bottom. And so we have (pi)r^2 x height = 50. So 3.14(r^2)h = 50. So r^2(h) = 15.92. If I make the height be 4 then the radius is practically 2, diameter 4. This cylinder without top or bottom would be circumference = 3.14(4) = 12.56 and with height 4 would be surface area of 4 x 12.56 = 50.24. But this same cylinder with top and bottom intact would be 50.24 + 2 (3.14)(2^2) = 50.24 + 25.12 = 75.36. Comparing to the cube, not as much area.

For the torus with volume 50. We have.

Surface area = 4(pi^2) R*r = 39.47 R*r
Volume = 2(pi^2) R*r^2 = 19.73 R*r^2

So a given volume of 50 is 19.73R*r^2 = 50. R*r^2 = 2.534. Here I, take R arbitrarily as 2 then r^2= 1.267. So r = 1.12. With R = 2 and r=1.12 and see what surface area yields. Surface area = 39.47(2)(1.12) = 88.41 unit area.

So the regular 6 face cube with surface area 81.43 compared to cylinder with top and bottom as 75.36, compared to torus at 88.41.

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Archimedes Plutonium<plutonium....@gmail.com>
Nov 30, 2023, 11:21:22 PM (1 hour ago)



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I seemed to have forgotten sphere of volume 50 to compare. So we have 4(3.14)r^2 for area and (1.333)(3.14)(r^3) for volume. For volume 50 we have (4.18)r^3 = 50 and so r^3 = 11.96, so r = 2.28. Now plugging that into the surface area equation (12.56)(2.28^2) = 65.29 unit area. Compare that to the cylinder with top and bottom is 75.36.

AP
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Archimedes Plutonium<plutonium....@gmail.com>
12:47 AM (now)



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On Wednesday, November 29, 2023 at 2:11:29 AM UTC-6, Archimedes Plutonium wrote:

(source Quora)

> N... hinted at the meaning in his answer. Let me see if I can clarify a bit more.
>
> Imagine a slowly expanding sphere. As the radius increases by a tiny amount, how much does the volume change? It changes by the volume of the spherical shell that was just added. That volume is roughly the area of that shell times its thickness. As we take the limit, and imagine the sphere to be increasing by an infinitesimal amount, the rate of volume change is actually the surface area of that spherical shell. Which is equal to the present surface area of the sphere.
>
> The rate of change is the surface area. And there we have our connection with derivatives.
>
> There’s actually something very deep here. The derivative of the sum of the volumes of the infinitesimally thin spherical shells is the surface area of the spherical shells. This can be generalized to certain infinite sums in general: The derivative of the sum of infinitesimally thin pieces under the graph of a continuous function is the function itself. This result is known as The Fundamental Theorem of Calculus, and from the name, you can see that we think it’s pretty useful.
> --- end quoting Quora ---


Alright in New Geometry where there are no continuums of numbers, but instead discrete numbers then we have to make Surface Area something different then Old Math.

And we have to apply Physics electromagnetism for a plane, since the capacitor is parallel plate capacitor. In phsysics, the cylinder is still a cylinder even if it has no top and bottom (a pipe). Same thing for a cube is still a cube even if it has no top and bottom face.

But also, in physics, the parallel plate capacitor can be a surface, a surface that itself contains a tiny amount of volume.

So in 100 decimal grid system along the y-axis

0.02 ....................flat straight line
0.01.....................flat straight line
0.0.......x-axis -->

My illustration of New Math Geometry is that the two straight flat lines of Y --> 0.01 and Y--> 0.02 is a surface and it has tiny volume.

Math is often an idealization which dwells too much in ideals and sometimes anthropomorphic dumb ideals. For skin is in biology a surface and surface area but it has a tiny volume. A parallel plate capacitor of physics is surface area but it has a tiny volume between two plates.

The reason I quoted the above is to indicate this layering in geometry that transitions the derivative through sphere volume to land on sphere surface area.

AP
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