Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


Math Forum
»
Discussions
»
sci.math.*
»
sci.math
Notice: We are no longer accepting new posts, but the forums will continue to be readable.
Topic:
3)MIT math dept all fired if unable to teach correct math Conic section = Oval, never ellipse//clamors Alouatta
Replies:
5
Last Post:
Oct 3, 2017 6:14 AM




3)MIT math dept all fired if unable to teach correct math Conic section = Oval, never ellipse//clamors Alouatta
Posted:
Sep 30, 2017 4:53 AM


3)MIT math dept all fired if unable to teach correct math Conic section = Oval, never ellipse//clamors Alouatta
Is this Alouatta forewarning mathematicians they will be fired if they do not straighten up??
On Friday, September 29, 2017 at 5:09:25 PM UTC5, alouatta....@gmail.com wrote: > > It lasts because it's the TRUTH, Archy. Now I know you wouldn't recognize the truth if it bit you in the ass, but that doesn't matter. Anyone with any math ability has no trouble following the Dandelin proof, but you can't. You think you see an error in it, but IT'S NOT THERE! Show us the error, or even better, provide a proof of your own. No other (actual) mathematicians will do either of those things, because they're not true.
Geometry proofs that Cylinder section= Ellipse// Conic section= Oval
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 saywhen the circle plate is parallel with base, it is a circle but rotate it slightly in the cylinder or cone and determine what figure is produced. When rotated at diameter, the extra area added to the upper portion equals the extra area added to bottom portion, 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 on the slanted walls of the cone upward. Rotate as far as you possible 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, 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.
Array:: Analytic Geometry proof that Cylinder section= Ellipse//Conic section = Oval, never ellipse Alright, I gave a proof, a Synthetic Geometry proof that the Conic Section was a Oval, never an ellipse and the Cylinder section turns out to be the ellipse.
A Synthetic geometry proof is where you need no numbers, no coordinate points, now arithmetic, but just using concepts and axioms. A Analytic Geometry proof is where numbers come into the picture.
So, here is a Analytic Geometry proof that the Cylinder Section is a Ellipse, and a Conic Section is a Oval, never an 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 piston 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 con. 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 basecircle 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 AngleSideAngle 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 conefor 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.
QED AP
Minicozzi, William Singer Professor of Mathematics Geometric Analysis, PDEs
Moitra, Ankur Rockwell International Career Development Associate Professor of Mathematics Theoretical Computer Science, Machine Learning
Mossel, Elchanan Professor of Mathematics Probability, Algorithms and Inference
Mrowka, Tomasz Professor of Mathematics Gauge Theory, Differential Geometry On Leave Fall and Spring semesters
Munkres, James Emeritus Professor of Mathematics Differential Topology
Negu?, Andrei Assistant Professor of Mathematics Algebraic Geometry, Representation Theory On Leave Spring semester
Pixton, Aaron Class of 1957 Career Development Assistant Professor Algebraic Geometry On Leave Spring semester
Poonen, Bjorn Claude Shannon Professor of Mathematics Algebraic Geometry, Number Theory
Postnikov, Alexander Professor of Applied Mathematics Algebraic Combinatorics
Rigollet, Philippe Associate Professor of Mathematics Statistics, Machine Learning
Rosales, Rodolfo Professor of Applied Mathematics Nonlinear Waves, Fluid Mechanics, Material Sciences, Numerical pde
Saccà, Giulia Assistant Professor of Mathematics
Sacks, Gerald Emeritus Professor of Mathematical Logic Mathematical Logic, Recursion Theory, Computational Set Theory
Seidel, Paul Levinson Professor of Mathematics Mirror Symmetry On Leave Fall and Spring semesters
Sheffield, Scott Leighton Family Professor of Mathematics Probability and Mathematical Physics
Shor, Peter Morss Professor of Applied Mathematics Applied Mathematics Committee Chair Quantum Computation, Quantum Information
Singer, Isadore Emeritus Institute Professor Differential Geometry, Partial Differential Equations, Mathematical Physics
Sipser, Michael Dean of School of Science Donner Professor of Mathematics MacVicar Faculty Fellow Algorithms, Complexity Theory
Speck, Jared Cecil and Ida B. Green Career Development Associate Professor of Mathematics Analysis related to Mathematical Physics, General Relativity, PDEs
Staffilani, Gigliola Abby Rockefeller Mauze Professor of Mathematics Analysis: Dispersive Nonlinear Partial Differential Equations On Leave Fall and Spring semesters
Stanley, Richard Professor of Mathematics Algebraic Combinatorics
Stark, Harold Emeritus Professor of Mathematics Number Theory
Strang, Gilbert MathWorks Professor of Mathematics
Stroock, Daniel Emeritus Professor of Mathematics Probability, Stochastic Analysis
Tabuada, Goncalo Associate Professor of Mathematics Algebraic Topology, Noncommutative Algebraic Geometry
Toomre, Alar Emeritus Professor of Applied Mathematics Astrophysics, Stellar Dynamics, Fluid Mechanics
Vogan, David Norbert Wiener Professor of Mathematics Group Representations, Lie Theory
Zhang, Wei Professor of Mathematics
Zhao, Yufei Assistant Professor of Mathematics Combinatorics



