Now if Pentcho Valev is a Russian Robot-- I would please like to enlist his help, after seeing his 20 year long history, he has the stamina to be very helpful.
What I like for the robots to do, is list every day, about 4 Colleges ( of the West) math dept, and ask why that math department is teaching false and fake math, and if unable to change to the correct true math, well, simply fire that math department until they can find professors who recognize truth in math from fakery.
-------------------------- 1. Borderline between Finite and Infinite
This is probably the biggest mistake in all of mathematics for it affects all other mathematics.
by Archimedes Plutonium
There is only one way you can have a concept of finite, by having a concept of infinity, and the only way you can have both, is that a borderline exists between them.
I have pinpointed that borderline from tractrix-circle analysis, from algebraic analysis of algebraic completeness, and from angles of regular polyhedra. The borderline in microinfinity is 1*10^-604 and in macroinfinity is 1*10^604.
-------------- 2. Why no continuum and no curves exist in Math, so that the Calculus can exist, and does exist
by Archimedes Plutonium
Calculus is based upon there being Grid points in geometry, no continuum, but actually, empty space between two neighboring points. This is called Discrete geometry, and in physics, this is called Quantum Mechanics. In 10 Grid, the first few numbers are 0, .1, .2, .3, etc. That means there does not exist any number between 0 and .1, no number exists between .1 and .2. Now if you want more precise numbers, you go to a higher Grid like that of 100 Grid where the first few numbers are 0, .01, .02, .03, etc.
Calculus in order to exist at all, needs this empty space between consecutive numbers or successor numbers. It needs that empty space so that the integral of calculus is actually small rectangles whose interior area is not zero. So in 10 Grid, the smallest width of any Calculus rectangle is of width .1. In 100 Grid the smallest width is .01.
But, this revolutionary understanding of Calculus does not stop with the Integral, for having empty space between numbers, means no curves in math exist, but are ever tinier straight-line segments.
It also means, that the Derivative in Calculus is part and parcel of the function graph itself. So that in a function such as y = x^2, the function graph is the derivative at a point. In Old Math, they had the folly and idiocy of a foreign, alien tangent line to a function graph as derivative. In New Math, the derivative is the same as the function graph itself. And, this makes commonsense, utter commonsense, for the derivative is a prediction of the future of the function in question, and no way in the world can a foreign tangent line to a point on the function be able to predict, be able to tell where the future point of that function be. The only predictor of a future point of a function, is the function graph itself.
If the Calculus was done correctly, conceived correctly, then a minimal diagram explains all of Calculus. Old Math never had such a diagram, because Old Math was in total error of what Calculus is, and what Calculus does.
The fundamental picture of all of Calculus are these two of a trapezoid and rectangle. Trapezoid for derivative as the roof-top of the trapezoid, which must be a straight-line segment. If it is curved, you cannot fold it down to form a integral rectangle. And the rectangle for integral as area.
From this: B /| / | A /----| / | | | |____|
The trapezoid roof has to be a straight-line segment (the derivative) so that it can be hinged at A, and swiveled down to form rectangle for integral.
______ | | | | | | ---------
And the derivative of x= A, above is merely the dy/dx involving points A and B. Thus, it can never be a curve in Calculus. And the AB is part of the function graph itself. No curves exist in mathematics and no continuum exists in mathematics.
In the above we see that CALCULUS needs and requires a diagram in which you can go from derivative to integral, or go from integral to derivative, by simply a hinge down to form a rectangle for area, or a hinge up to form the derivative from a given rectangle.
Why in Old Math could no professor of math ever do the Calculus Diagram? Why? The answer is simple, no-one in Old Math pays attention to Logic, and that no-one in Old Math was required to take formal Logic when they attended school. So a person bereft of Logic, is never going to find mistakes of Logic and think clear and think straight.
by Archimedes Plutonium ------------------
Correction of Logic errors by Archimedes Plutonium 3. Logic errors:: otherwise we cannot think clearly and think straight and true History of those pathetic errors::
by Archimedes Plutonium
The 4 connectors of Logic are:
1) Equal (equivalence) plus Not (negation) where the two are combined as one 2) And (conjunction) 3) Or (exclusive or) (disjunction) 4) Implication
EQUAL/NOT table: T = T = T T = not F = T F = not T = T F = F = T
Equality must start or begin logic because in the other connectors, we cannot say a result equals something if we do not have equality built already. Now to build equality, it is unary in that T=T and F =F. So we need another unary connector to make equality a binary. Negation is that other connector and when we combine the two we have the above table.
Equality combined with Negation allows us to proceed to build the other three logic connectors.
Now, unfortunately, Logic must start with equality allied with negation and in math what this connector as binary connector ends up being-- is multiplication for math. One would think that the first connector of Logic that must be covered is the connector that ends up being addition of math, not multiplication. But maybe we can find a philosophy-logic answer as to why Logic starts with equal/not and is multiplication rather than addition.
Here you we have one truth table equal/not whose endresult is 4 trues.
New Logic AND T & T = T T & F = T F & T = T F & F = F
AND is ADD in New Logic, and that makes a whole lot of common sense. AND feels like addition, the joining of parts. And the truth table for AND should be such that if given one true statement in a series of statements then the entire string of statements is true. So if I had P and Q and S and R, I need only one of those to be true to make the string true P & Q & S & R = True if just one statement is true.
The truth table of AND results in 3 trues and 1 false.
New Logic OR(exclusive) T or T = F T or F = T F or T = T F or F = F
OR is seen as a choice, a pick and choose. So if I had T or T, there is no choice and so it is False. If I had T or F there is a choice and so it is true. Again the same for F or T, but when I have F or F, there is no choice and so it is false. OR in mathematics, because we pick and discard what is not chosen, that OR is seen as subtraction.
OR is a truth table whose endresult is 2 trues, 2 falses.
New Logic IMPLIES (Material Conditional) IF/THEN MOVES INTO T -> T = T T -> F = F F -> T = U probability outcome F -> F = U probability outcome
A truth table that has a variable which is neither T or F, but U for unknown or a probability outcome. We need this U so that we can do math where 0 divided into something is not defined.
Now notice there are four truth tables where the endresult is 4 trues, 3 trues with 1 false, 2 trues with 2 falses and finally a truth table with a different variable other than T or F, with variable U. This is important in New Logic that the four primitive connectors, by primitive I mean they are independent of one another so that one cannot be derived by the other three. The four are axioms, independent. And the way you can spot that they are independent is that if you reverse their values so that 4 trues become 4 falses. For AND, reversal would be FFFT instead of TTTF. For OR, a reversal would be TFFT instead of FTTF.
To be independent and not derivable by the other three axioms you need a condition of this:
One Table be 4 of the same One Table be 3 of the same One Table be 2 of the same And to get division by 0 in mathematics, one table with a unknown variable.
So, how did Old Logic get it all so wrong so bad? I think the problem was that in the 1800s when Logic was being discovered, is that the best minds of the time were involved in physics, chemistry, biology and looked upon philosophy and logic as second rate and that second rate minds would propose Old Logic. This history would be from Boole 1854 The Laws of Thought, and Jevons textbook of Elementary Lessons on Logic, 1870. Boole started the Old Logic with the help of Jevons and fostered the wrong muddleheaded idea that OR was ADD, when it truly is AND. But once you have textbooks about Logic, it is difficult to correct a mistake because of the money making social network wants to make more money, not go around fixing mistakes. So this nightmarish mistakes of the truth tables was not seen by Frege, by Russell, by Whitehead, by Carnap, by Godel, and by 1908 the symbols and terminology of the Old Logic truth tables were so deep rooted into Logic, that only a Logical minded person could ever rescue Logic.
by Archimedes Plutonium
3.1 The "and" truth table should be TTTF not what Boole thought TFFF. Only an utter gutter mind of logic would think that in a series of statements, that AND is true when all statements are true, but to the wise person-- he realizes that if just one statement is true, the entire series is true, where we toss aside all the irrelevant and false statements --(much what life itself is-- we pick out the true ones and ignore all the false ones). 3.2 The error of "if-then" truth table should be TFUU, not that of TFTT 3.3 The error of "not" and "equal", neither unary, but should be binary 3.4 The error that Reductio Ad Absurdum is a proof method, when it is merely probability-truth, not guaranteed 3.5 The error, the "or" connector is truth table FTTF, not that of TTTF
------------------- 4. Many Errors of what Numbers exist and how to represent Numbers.
4.1 Why no Irrationals exist-- lowest terms, anthyphairesis
Why No Irrationals exist, and why pi and 2.71? are rational numbers-- as easy as Decimal Number representation-- they have a denominator power of 10 by Archimedes Plutonium
Why No Irrationals exist, and why pi and 2.71? are rational numbers
Old Math, and their "Lowest Terms Error" although don't tell them-- proved that 1/2 is irrational Re: analyzing why the Ancient Greek proof that sqrt2 is irrational is flawed
Alright, let me get started on the proof that 1/2 is irrational number using the invalid method of Ancient Greeks that sqrt2 is irrational, only because, the method is invalid.
Earlier I showed how a definition of Lowest Term for p/q needed to be extended to include a number in Rationals in decimal representation. So, what is the Lowest Term for 1/2 in 10 Grid, for it would be .1/.2 and then the next lowest is .2/.4, etc etc.
So, let us run through a proof that 1/2 is a Irrational number using the proof method of Ancient Greeks.
Proof:: Suppose 1/2 is Rational. And now, put 1/2 in Lowest terms and it is thus, in lowest terms. But now, taking 2 and dividing it into 1 __________ 2| 1.00000.... = .50000.....
and then dividing 2 by 2 _________ 2|2.00000..... = 1.0000.....
And now, we have 1/2 in Lowest terms as .5/1.
But now, hold on a minute, let us divide .5 by 2, then 1 by 2, giving us .25 and .5 respectively.
Since we can never get a Lowest Term for the Rational number 1/2, means a contradiction, hence 1/2 is irrational.
So, of course the above is flawed and flawed in the same way the method was used to prove sqrt2 is irrational, when truly sqrt2 is rational.
What went wrong? What went wrong is a bad definition-- Lowest Terms.
The proof that sqrt2 is Rational, simply involves observation for that
In 10 Grid, sqrt2 = 1.42 X 1.42 = 2.0 (oh, you question the 2.0164, you question the "164", well in 10 Grid, the only digits that exist are the ten place value and that is 2.0.
In 100 Grid, sqrt2 = 1.415 X 1.415
In 1000 Grid, sqrt2 = 1.4143 X 1.4143 and on and on.
Sqrt2 and all sqrt root numbers are Rationals. Even pi and 2.71.... are rational numbers.
Anthyphairesis Re: Stillwell gave another phony proof sqrt2 irrational Re: analyzing why the Ancient Greek proof that sqrt2 is irrational is a flawed
On Sunday, October 8, 2017 at 6:06:01 PM UTC-5, Archimedes Plutonium wrote: > On Sunday, October 8, 2017 at 3:50:43 PM UTC-5, Archimedes Plutonium wrote: > > > > > That is the only one proof in all of mathematics-- an argument based on a definition of Lowest Terms. > > Apparently there is a second proof of sqrt2 irrational. A far more challenging proof to see if phony. >
Apparently there was a second proof, but whether it was known by Euclid, by Archimedes, I rather doubt it.
> It is seen in Stillwell's Mathematics and Its History, 3rd ed. 2010, page 45. In the same book, page 12 is the Lowest Terms phony proof. > > Now looking at that alleged proof on page 45, it says and I quote. > > " We notice that the rectangle remaining after step 2, with sides sqrt2-1 and 2-sqrt2 = sqrt2(sqrt2-1), is the same shape as the original, though the long side is now vertical instead of horizontal. It follows that similar steps will recur forever, which is another proof that sqrt2 is irrational, incidentally." > > Does Stillwell expect readers to "read his mind". Why would a recurrence ever make Stillwell think that was a proof of sqrt2 is not able to be P/Q where P and Q are Counting Numbers. Why? Is it because two rational sides would cancel out in a square further down the line? And, if so, then the reason this proof is nonrecurring is only because, well, you use a symbol of sqrt2 that cannot commingle with actual numbers. If you call a number a symbol, call it S, call it Y, obviously you cannot get rid of it. > > Now this one is going to be challenging for me to show it is phony. But it is easy if we demand sqrt2 be written as a number, not some abstract symbol. Once we demand that a number in decimal representation or in fractions be forced upon rather than a "just a symbol sqrt2", then the phoniness of the proof is immediately apparent. Because, that forcing demands sqrt2 be written as 1.42 = 142/100 in 10 Grid or written as 1.415 = 1415/1000 in 100 Grid, etc. Writing sqrt2 in a number, then it behaves like all other Rationals, for it is a rational. > > You see, the rub on sqrt2 that Old Math installed is the same mistake they made with 1/3. They want 1/3 be .33333....., when, if called to be logical, 1/3 is .3333...33(+1/3) what Newton called the Compleat Quotient. >
nice proof that no irrationals exist, simple fact that all numbers are Decimal represented and thus a denominator of power of 10 Re: analyzing why the Ancient Greek proof that sqrt2 is irrational is a flawed
Now, here is a Commonsense proof that No Irrationals exist. It is not formal, it is not flowery or pilfered with abstractions. It is a proof that an old grandma or grandpa would understand and recognize, even if starting to slow to think in old age. It is a proof that young kids would be proud of owning. For it is a proof that since 3000 years ago, humanity has thought there was something known as "irrational number" and only now, today, realizes that there are no irrational numbers. That irrational numbers was the grand fake of fakeries.
Theorem Statement:: Rational numbers exist, but Irrationals do not exist.
Proof Statement:: Once we are able to have a Decimal Number system we can build all the numbers via Grids and using a math-induction element and adding that element successively to build the numbers. They are all Decimal numbers, meaning that their place-value is established. So that say for instance .003, or 3.14159..... are all rational numbers because, depending on what place value you want to talk about, it is 3/1000 or 314159/100000. In other words, writing a number in Decimal Representation alone, proves the number is a Rational for the denominator is always a power of 10. And since decimal numbers is ALL POSSIBLE DIGIT ARRANGEMENTS, means that all numbers are a Rational. QED
Now, there is one possible exception to this rule or proof. The imaginary number of square root of -1.
Is it even a number? I am going to say it is not a number, because all numbers have to come from Math induction on a induction element, be it 1 for Counting Numbers, be it .1 for 10 Grid, or .01 for 100 Grid, etc etc. So where does that leave us with sqrt -1. I suggest that i is not a number but an angle, a symbol for an angle. What angle is it? Not 90 degree for that is +1. I suggest i = sqrt-1 is the angle 180 degrees that lies in 2nd and 3rd quadrants.
4.2 Completing a Division correctly such as 1/3 = .3333..33(+1/3)
By Archimedes Plutonium
Newton, way back in the 1600s called it "Compleat Quotient", but that was some 400 years ago, and do you mean to tell me, that in 400 years time no-one had a good enough logical mind since Newton, that everyone since Newton was a failure of Logic when it comes to division?
Everyone gets this much ______ 3| 10000 = 3333+1/3
and then, everyone falls to pieces, into some pit of stupidity on this ______ 3|1.0000 = .3333(+1/3)
They fall to pieces, because they think, in their stupid little minds that ______ 3|1.0000 = .3333?.. and forget about any remainder
So that truly, 1/3 as a decimal is not, is never .3333..... but rather
1/3 = .33333..33(+1/3)
Where we always realize a remainder in division must always be tacked on.
Now the above is important in that it eliminates the obnoxious idea put forth by half=brains in math that 1 = .9999?. The number 1 never equals .9999?. but it does equal .9999..99(+9/9). So, half brains of math, time to run for the hills.
Explaining why most modern mathematicians are logically brain-dead-- simply because in modern day times, students are not forced to take logic-- to learn how to think straight and think clearly. If I had my way. Every Freshmen at College is required to take Introduction to Logic, for, it is only commonsense that Colleges and Universities do see that thinking straight and thinking clearly is top priority. And, if I had my way, the science majors all have to take a second year of logic called Symbolic Logic, because every day as a -- scientist -- the most important tool is logic
-- Archimedes Plutonium
------------ 5. Sine & Cosine are semicircle waves, not sinusoidal
by Archimedes Plutonium
They come to math, and physics, but they come without Logic, barren of logic, deplete of logic, never any logic in their tools of the trade.
They define sine as opposite/hypotenuse. Good so far.
They know of the unit circle with hypotenuse as 1. Good again.
They then blunder, so pitifully, so badly, so poorly, and so early on. I mean even a child can understand the first few steps. And they blunder badly for they spuriously assign 180 degrees to be 3.14.... Why? Why assign 180 degrees as 3.14... when you already defined sine as opposite/hypotenuse with unit circle forcing 180 degrees to be 2, since 90 degrees is 1 of unit circle.
You see what happens when you do science without logic-- you become a village idiot fool.
If you had had just a gram of Logical intelligence could see that the unit circle forces sine to define 180 degrees as being a diameter of 2. Thus making the sine graph and cosine graph to be a SEMICIRCLE Wave graph.
-- Archimedes Plutonium
SECOND PROOF THAT SINE AND COSINE ARE SEMICIRCLE WAVES::
This proof has a hands on experiment involved. Take a close look at a screen door spring, and verify it is wound up circles per wind.
EXPERIMENT:: make a 2nd dimension graph of semicircle wave. Cut out the semicircles but leaving them in one piece so you can bend and fold. Now, fold the sheet of cut out semicircles to begin to approach a spring of circle windings. Now, do the same with the idiotic Old Math's sinusoid shape wave. Can you form a spring, without vertices, a vertex at each joint and which those joints are physics vulnerable to cracking and breaking apart.
Theorem Statement:: A spring in mathematics is a winding of semicircle waves and is the sine function and cosine function wound from 2nd dimension into 3rd dimension.
Proof Statement:: Only a semicircle wave can be wound from 2nd dimension into 3rd dimension and be free of vertices, (weak spots). Only a circle is free of vertices when attaching half waves.
------------------ 6.1 Conics = oval, 4 Experiments 4th experiment Re: -World's first proofs that the Conic section is an Oval, never an ellipse// yes, Apollonius and Dandelin were wrong
by Archimedes Plutonium > > > > 1st EXPERIMENT:: Fold paper into cone and cylinder, (I prefer the waxy cover of a magazine). Try to make both about the same size, so the perspective is even. Now tape the cone and cylinder so they do not come undone in the scissor or paper cutter phase. A paper cutter is best but dangerous, so be careful, be very careful with paper cutter. Make the same angle of cut in each. and the best way of insuring that is to temporary staple the two together so the angle is the same. Once cut, remove the staples. Now we inspect the finished product. Hold each in turn on a sheet of paper and with a pencil trace out the figure on the flat piece of paper. Notice the cylinder gives an ellipse with 2 Axes of Symmetry, while the conic gives a oval because it has just one, yes 1 axis of symmetry. > > > > That was my first experiment. >
Easy and fast experiment, and gets the person able to make more cones and cylinders in a rush. Only fault I have of this experiment is that it leaves a scissors mark-- a vertex so to speak. But it is fast and easy. The proof is in the comparison. Now the cut should be at a steep enough angle. If you cut straight across, both will be circles, so make a steep cut.
> > > > 2nd EXPERIMENT:: get a Kerr or Mason canning lid and repeat the above production of a cone and cylinder out of stiff waxy paper (magazine covers). Try to make the cone and cylinder about the same size as the lid. Now either observe with the lid inside the cone and cylinder, or, punch two holes in the cone and cylinder and fasten the lid inside. What you want to observe is how much area and where the area is added to make a section. So that in the cylinder, there is equal amount of area to add upwards as to add downwards of the lid, but in the cone, the area upwards added is small, while the area added downwards is huge new area. Thus the cylinder had two axes of symmetry and is an ellipse, while cone is 1 axis of symmetry and is an oval. >
This experiment is the best for it immediately shows you the asymmetry of an axis, where the upward needs little area to fill in any gap and the downward needs an entire "crescent shaped area add-on to the circle lid.
> > > > 3rd EXPERIMENT:: Basically this is a repeat of the Dandelin fake proof, only we use a cylinder. Some tennis balls or ping pong balls come in see through plastic cylinder containers. And here you need just two balls in the container and you cut out some cardboard in the shape of ellipse that fits inside the container. You will be cutting many different sizes of these ellipses and estimating their foci. Now you insert these ellipse and watch to see the balls come in contact with the foci. Now, you build several cones in which the ellipses should fit snugly. Trouble is, well, there is never a cone that any ellipse can fit inside, for only an oval fits inside the cones. > >
This experiment is cumbersome and takes much precision and good materials. It is just a repeat of the Dandelin work on this topic, and one can easily see how the Dandelin fake proof is constructed-- he starts off with assuming the figure is an ellipse. Which tells us, he never had a good-working-model if any at all. For you cannot stuff a ellipse inside a cone. You can stuff a ellipse inside a cylinder. So this suggests the entire Dandelin nonsense was all worked out in the head and never in hands on actual reality. So, in this experiment, we give a proof that Dandelin was utterly wrong and that it is a cylinder that you can stuff a ellipse sandwiched by two identical spheres-- one upper and one lower.
The only amazing part of the Dandelin story is how an utterly fake proof could have survived from 1822, and not until 2017 is it thoroughly revealed as ignorant nonsense. One would think in math, there is no chance such a hideously flawed proof could even be published in a math journal, and if anything is learned from Dandelin, is that the math journal publishing system is a whole entire garbage network. A network that is corrupt and fans fakery.
> > 4th EXPERIMENT:: this is a new one. And I have it resting on my coffee table at the moment and looking at it. It comes from a toy kit of plastic see through geometry figures, cost me about $5. And what I have is a square pyramid and a cone of about the same size. Both see through. And what I did was rest the square pyramid apex on top of the cone apex, so the cone is inside the square pyramid. Now I wish I had a rectangular box to fit a cylinder inside the box. But this toy kit did not have that, but no worries for the imagination can easily picture a cylinder inside a rectangular box. Now the experiment is real simple in that we imagine a Planar Cut into the rectangular box with cylinder inside and the cut will make a rectangle from the box and a ellipse from the cylinder. Now with the cut of the square pyramid that contains a cone inside, the square pyramid is a trapezoid section while the cone is a oval section. If the cut were parallel to the base, the square pyramid yields a square and the cone yields a circle. This experiment proves to all the dunces, the many dunces who think a conic section is an ellipse, that it cannot be an ellipse, for obviously, a cone is not the same as a cylinder. > >
Now this 4th Experiment is a delicious fascinating experiment, for it reveals to us another proof that the conic section is a oval. For the square-pyramid section is a Isosceles Trapezoid, and what is so great about that, is we can take a cone and place inside of the cone a square pyramid and then place a second square pyramid over the cone, so the cone is sandwiched in between two square pyramids.
Now the square pyramids are tangent to the cone at 4 line segments, 8 altogether for the two, and what is so intriguing about the tangents is that it allows us to quickly develop a analytic geometry that the cone section must be a oval in order for the two square pyramids to be both isosceles trapezoids as sections.
Synthetic Geometry & Analytical Geometry Proofs that Conic section = Oval, never an ellipse-- World's first proofs thereof by Archimedes Plutonium _Synthetic Geometry proofs that Cylinder section= Ellipse// Conic section= Oval
First Synthetic Geometry proofs, later the Analytic Geometry proofs.
Alright I need to get this prepared for the MATH ARRAY of proofs, that the Ellipse is a Cylinder section, and that the Conic section is an oval, never an ellipse
PROOF that Cylinder Section is an Ellipse, never a Oval:: I would have proven it by Symmetry. Where I indulge the reader to place a circle inside the cylinder and have it mounted on a swivel, a tiny rod fastened to the circle so that you can pivot and rotate the circle. Then my proof argument would be to say--when the circle plate is parallel with base, it is a circle but rotate it slightly in the cylinder and determine what figure is produced. When rotated at the diameter, the extra area added to the upper portion equals the extra area added to bottom portion in cylinder, symmetrical area added, hence a ellipse. QED
Now for proof that the Conic section cannot be an ellipse but an oval, I again would apply the same proof argument by symmetry.
Proof:: Take a cone in general, and build a circle that rotates on a axis. Rotate the circle just a tiny bit for it is bound to get stuck or impeded by the upward slanted walls of the cone. Rotate as far as you possibly can. Now filling in the area upwards is far smaller than filling in the area downwards. Hence, only 1 axis of symmetry, not 2 axes of symmetry. Define Oval as having 1 axis of symmetry. Thus a oval, never an ellipse. QED
The above two proofs are Synthetic Geometry proofs, which means they need no numbers, just some concepts and axioms to make the proof work. A Synthetic geometry proof is where you need no numbers, no coordinate points, no arithmetic, but just using concepts and axioms. A Analytic Geometry proof is where numbers are involved, if only just coordinate points.
Array:: Analytic Geometry proof that Cylinder section= Ellipse//Conic section = Oval, never ellipse
Now I did 3 Experiments and 3 models of the problem, but it turns out that one model is superior over all the other models. One model is the best of all.
That model is where you construct a cone and a cylinder and then implant a circle inside the cone and cylinder attached to a handle so that you can rotate the circle inside. Mine uses a long nail that I poked holes into the side of a cylinder and another one inside a cone made from heavy wax paper of magazine covers. And I used a Mason or Kerr used lid and I attached them to the nail by drilling two holes into each lid and running a wire as fastener. All of this done so I can rotate or pivot the circle inside the cylinder and cone. You need a long nail, for if you make the models too small or too skinny, you lose clarity.
ARRAY, Analytic Geometry Proof, Cylinder Section is a Ellipse::
E __ .-' `-. .' `. / \ ; ; | G c | H ; ; \ / `. .' `-. _____ .-' F
The above is a view of a ellipse with center c and is produced by the Sectioning of a Cylinder as long as the cut is not perpendicular to the base, and as long as the cut involves two points not larger than the height of the cylinder walls. What we want to prove is that the cut is always a ellipse, which is a plane figure of two axes of symmetry with a Major Axis and Minor Axis and center at c.
Side view of Cylinder EGFH above with entry point cut at E and exit point cut at F and where c denotes the central axis of the cylinder and where x denotes a circle at c parallel with the base-circle of cylinder
So, what is the proof that figure EGFH is always an ellipse in the cylinder section? The line segment GH is the diameter of the circle base of cylinder and the cylinder axis cuts this diameter in half such that Gc = cH. Now we only need to show that Fc = cE. This is done from the right triangles cxF and cxE, for we note that by Angle-Side-Angle these two right triangles are congruent and hence Fc = cE, our second axis of symmetry and thus figure EGFH is always an ellipse. QED
Array proof:: Analytic Geometry proof that Conic section= Oval// never ellipse
ARRAY, Analytic Geometry Proof, Conic Section is a Oval, never an ellipse::
A ,'" "`. / \ C | c | D \ / ` . ___ .' B
The above is a view of a figure formed from the cut of a conic with center c as the axis of the cone and is produced by the Sectioning of a Cone as long as the cut is not perpendicular to the base, and as long as the cut is not a hyperbola, parabola or circle (nor line). What we want to prove is that this cut is always a oval, never an ellipse. An oval is defined as a plane figure of just one axis of symmetry and possessing a center, c, with a Major Diameter as the axis of symmetry and a Minor Diameter. In our diagram above, the major diameter is AB and minor diameter is CD.
Alright, almost the same as with Cylinder section where we proved the center was half way between Major Axis and Minor Axis of cylinder, only in the case of the Conic, we find that the center is half way between CD the Minor Diameter, but the center is not halfway in between the Major Diameter, and all of that because of the reason the slanted walls of the cone cause the distance cA to be far smaller than the distance cB. In the diagram below we have the circle of x centered at c and parallel to base. The angle at cx is not 90 degrees as in cylinder. The angle of cAx is not the same as the angle cBx, as in the case of the cylinder, because the walls of the cone-for line segments- are slanted versus parallel in the cylinder. Triangles cAx and cBx are not congruent, and thus, the distance of cA is not equal to cB, leaving only one axis of symmetry AB, not CD.
/ \A x/ c \x B/ \
Hence, every cut in the Cone, not a hyperbola, not a parabola, not a circle (not a line) is a Oval, never an ellipse.
------------------ 7. Fixing the huge math error of gravity in Old Physics
By Archimedes Plutonium
Now let us shift to 2nd dimension geometry for a moment and we have this.
Circle x^2 + y^2 = 1
Ellipse x^2/a^2 + y^2/b^2 = 1
Parabola x^2 - y = 1
Now, in Old Physics, they had gravity as F= Gm1*m2/d^2
They wanted gravity as either circle or ellipse, for they saw planets orbit in closed loops.
Now here is a huge huge flaw of Old Physics, something that even Newton by 1687, himself should have caught and corrected, and if not Newton, surely James Clerk Maxwell by 1860 should have caught the math error. Unfortunately neither caught the huge math error. And why did no-one in the 1900s catch the mistake? Why? I believe even if they caught the huge math error would have been helpless to try to correct for it overturns the whole entire program of Old Physics on their gravity. Now this is a lesson in itself, a sort of like morality lesson or Aesop's Fable lesson, that you cannot find a mistake or flaw of science, if that flaw is going to overturn the entire subject matter. What I mean is say Newton or Maxwell had known that gravity could not be F= Gm1*m2/d^2 but had to be F= kAA/d^2 + jBB/d^2. Suppose they had discovered that, then the problem is, they had nothing in physical reality to give meaning to that math correction. They knew not that Sun was revolving around a galaxy with planets in helical motion, nor did they have any idea that gravity was electromagnetism. So, even if, Newton or Maxwell, realized the math was wrong, they could not link physical reality to a correct math of F= kAA/d^2 + jBB/d^2.
It spoils not only Newton's gravity law but spoils the entire General Relativity.
What I am talking about, is the math of Newton's gravity and General Relativity is a math of just one term kAA/d^2 and that math is a open curve such as a parabola. The math needed for a closed curve for gravity is of at least two terms in the numerator such as (kAA + jBB)/d^2. So that gravity is sufficient to be a closed loop, a circle or a ellipse.
And this is shocking as to how such a math error escaped all physicists and mathematicians until 2016 when I solved it in this textbook.
Gravity that is F= m(a1 + a2 + a3) and not F = ma. Gravity that is F = (kAA+jBB +hCC)/d^2. Gravity that is the same as EM to allow for Solid Body Rotation and V proportional to R, proportional to 1/R and to 1/R^2 and all in between.