Reid Barnes 
Member since Dec 28, 2013



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Re: “Video: 2013's Comet of the Year

To combat the non-Euclidean relativist con game, I have decided to introduce THE CONE GAME! Here is a link to a moving 3-D model of this comet's path, which also shows the path of the planets as the comet comes within earth's orbit as well as the curved path of the comet which is very nearly a conic section as is an ellipse.

ARE YOU READY FOR THE CONE GAME? Given three or four sightings and reported positions of a comet, Gauss was able to calculate the path of the comet and predict future positions. He was able to do this by first recognizing that the path would be a curved path known as a conic section (the curve that would appear where a plane intersects a cone). It is interesting to compare the curves in a moving 3-D model using this idea that the path of the comet and of the planets in the model are conic sections. Can someone by just looking at the 3-D model figure out what type of conic section curve most nearly describes the comet's path? The way I view it is that the top of the cone of an ice cream cone is a circle, which is where a plane cuts across the cone. If you imagine where a parallel plane cuts across the cone further down, the intersection is a smaller circle. Then if you imagine that plane tilted slightly, the intersection with the cone that had previously been that smaller circle becomes slightly elongated, which forms an ellipse. One of Kepler's laws was that the path of a planet orbiting the sun is an ellipse. So ellipses describe the orbits of the planets in the model. Using your imagination, the more you tilt the plane that passes through the cone, the more elongated the ellipse where the plane intersects the cone becomes, until you tilt the plane so far that the plane is parallel to a straight line inside the surface of the cone; the intersection then becomes a U shape that passes up through the top of the cone. The curve with this shape is called a parabola, which is also a conic section. If you keep tilting the plane further though, it becomes a hyperbola: the plane is no longer parallel to a straight line in the surface of the cone, but the intersection of the plane and cone is still a U shape that passes up through the top of the cone.

Of course Gauss couldn't have calculated the path of a comet using non-Euclidean geometry. If you are using coordinates or angle measurements, you are using Euclidean geometry because non-Euclidean geometry, including the Riemannian type, without Hilbert's Theorem 8 [5 in earlier books] does not have coordinates and does not have the SAS triangle congruency theorem. If Gauss thought this could be done with non-Euclidean mathematics (with Theorem 8 in Hilbert's time), then he would have failed to realize that would be based on self-contradicting geometry. And without the coordinates there is no basis for a mapping with a non-Euclidean 'space.' See the Facebook Note:

Posted by Reid Barnes on 12/28/2013 at 4:52 PM

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