6)are all German mathematicians like Peter Roquette, Gunther Schmidt Karl-Otto Stöhr- as dumb as Franz, teachi ng a Conic section is ellipse, when in truth it is an oval?
Oct 4, 2017 7:05 PM
Re: 6)are all German mathematicians like Peter Roque tte, Gunther Schmidt Karl-Otto Stöhr- as dumb as Franz, te aching a Conic section is ellipse, when in truth it is an ov al?
Oct 4, 2017 4:29 PM
On Sunday, October 1, 2017 at 11:22:44 PM UTC-7, Archimedes Plutonium wrote: > 6)are all German mathematicians like Peter Roquette, Gunther Schmidt > Karl-Otto Stöhr- as dumb as Franz, teaching a Conic section is ellipse, when in truth it is an oval?
First of all: why are you quoting some random names? You are arguing about the (false) claim regarding conic sections. How does listing some people's names enter into this?
> Alright I need to get this prepared for the MATH ARRAY of proofs,
Not a single proof of your claim exists, let alone any "MATH ARRAY" of same.
> PROOF that Cylinder Section is an Ellipse, never a Oval::
Yeah, let's see your proof. BTW, why didn't you just start your post with it? Why did you post this ludicrous preamble listing some random names and infantile pontification?
Anyway, let's see:
> 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.
What is the rotation axis? Must you always be so diffuse and opaque in your explanations?
> 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
This is not a proof, it's just a (weak) plausability argument. It only shows that a cylindrical cross-section would be some kind of symmetric oval.
> Now for proof that the Conic section cannot be an ellipse but an oval, I again would apply the same proof argument by symmetry.
OK, let's see:
> 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.
Yes. So what? What does that prove? It only proves that your previous method (which "worked" with the cylinder) doesn't work here. Your cylinder method fails for the cone. It doesn't otherwise "prove" or "disprove" anything.
> Rotate as far as you possibly can.
But you've just said you cannot rotate it _at all_ because it gets stuck. So which one is it?
> Now filling in the area upwards is far smaller than filling in the area downwards.
Yes but the entire cross-section shifts sideways (this did not happen in the cylinder case). That's why in the end you get an ellipse regardless.
> Hence, only 1 axis of symmetry, not 2 axes of symmetry.
No, your "hence" is a false implication. And the proof is nowhere near precise or correct.
> Define Oval as having 1 axis of symmetry. Thus a oval, never an ellipse. QED
No, it has two axes of symmetry. There are more subtle reasons for that (to actually see it, you need an actual proof, like Dandelin's: https://en.wikipedia.org/wiki/Dandelin_spheres ) but one thing you overlooked is that the entire cross-sectional area shifts sideways as you rotate the sectional plane.
You assume the cone's centreline must intersect the cross-section in its centre of symmetry but this is false. (It happens top be true in the case of the cylinder which is probably what led you into this trap.)
> The above two proofs are Synthetic Geometry proofs, which means they need no numbers, just some concepts and axioms to make the proof work.
First of all, they are nowhere near being "proofs". Secondly, I have a question: have you actually ever seen any book on mathematics? Do you think math book contain NUMBERS in the proofs?? You think you have here something original because you didn't mention NUMBERS??
> A Synthetic geometry proof is where you need no numbers,
Yeah, BFD. Good grief!
> no coordinate points, no arithmetic, but just using concepts and axioms.
Never seen a topology textbook?
> 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.
What does that have to do with anything? You want to prove the Banach-Tarski paradoxical sphere decomposition with wax paper and nails? How about the h-cobordism theorem in dimensions greater than 4 (this makes it easier)?
> 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.
Very impressive and totally useless.
> 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.
OK, cylinder. Where is the cone? I'm assuming it's coming soon...
> 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 > > | | > | | E > | | > | | > |x c |x > | | > | | > | | > |F | > | | > | | > | | > > > 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).
Wait, you ACCEPT that conic sections can be hyperbolas? But Dandelin's proof applies to that case as well! (It needs a slight modification.) So why do you agree hyperbola is a conic section but not ellipse?
This makes no sense (like everything else you wrote on the subject).
> 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.
No, there is no "hence" again. All you have done is you've shown that a particular method does not detect more than one axis of symmetry. In order to PROVE your claim you have to PROVE there is no OTHER axis of symmetry. Your proof fails to do that, it simply detects one and it does not DISPROVE the existence of any other.
In reality, of course, the cross-section figure does have a second axis of symmetry. Your proof is simply too weak to detect it. > > QED
[skipping another list of random names. Are you going to list organic fruits at a store near you next time?]