I was just pointing out that making ellipses as a practical exercise is so easy, a Greek philosopher could do it. (That is, does not need XY coordinates, algebra solutions, etc.)
Yes, imagine the side view of the double cone, essentially an X. For simplicity, let’s describe it as vertical, so the axis of the cones is the vertical (y) axis in the plane view. We will describe lines that define the edge-on view of a plane.
A line that is parallel to one arm of the X represent a plane that will only go through one cone represents (unless it is overlaid on the edge, in which case it’s the line solution)
A line perpendicular to the axis of the cone describes plane that defines a circle. Again, only ever goes through one cone of the pair.
If the line is not parallel to a side of the cone it defines a plane with one of two cases:
It eventually intersects both cones - the line is vertical or with slope greater than the side of the cone - it intersects both cones, the plane it defines describes a parabola.
It only intersects one cone - if the line defining the plane has a slope less than the slope of the sides of the cone, then it will only intersect one cone and the cross section defines an ellipse.
Why would a conic section ellipse way down on the cone where the radius is very large, define a very narrow elliptical shape? It may be counter-intuitive, but think of it not unlike taking a cheese slicer to a sausage. You will get a very thin, very elliptical slice.
(A cylinder is not unlike a cone/sausage where the angle of the cone approaches infinity, so it never comes to a point…)
I don’t know; I can barely visualize 2 dimensions myself! :smack:
Especially amazing is Archytas’ doubling of the cube (impossible with just straightedge and collapsing compass) 150 years before the time of Archimedes and Apollonius.
The citizens of Delos consulted the oracle at Delphi in order to learn how to defeat a plague (or political turmoil?) sent by Apollo. The oracle responded that they must double the size of the altar to Apollo, which was a regular cube. The answer seemed strange to the Delians and they consulted Plato, who was able to interpret the oracle as the mathematical problem of doubling the volume of a given cube, thus explaining the oracle as the advice of Apollo for the citizens of Delos to occupy themselves with the study of geometry and mathematics in order to calm down their passions.
To get a semi-circular function out of the equation for a circle, you take the positive square root instead of the negative square root. (All positive quantities have two square roots, e.g. the square roots of 9 are 3 and -3). Or vice versa: either of them is a semicircular function.
If you want to see visually what a hyperbola looks like, this is independent of conic sections, but you can graph the function y = 1/x. (Be careful to include negative values of x too- that’s how you get the two branches of the hyperbola). That’s a special case of the hyperbola: other hyperbolas resemble that one transformed by shifting the origin, rotating it, squeezing it, or all of the above.
When the Delians, circa 370 B.C., suffering the ravages of a plague, were directed by an oracle to increase the size of their temple’s altar, Plato admonished them to disregard all magical interpretations of the oracle’s demand and concentrate on solving the problem of doubling the cube. This is one of the earliest accounts of the significance of pedagogical, or spiritual, exercises for economics.
Some crises, such as the one currently facing humanity, require a degree of concentration on paradoxes that outlasts one human lifetime.* Fortunately, mankind is endowed with what LaRouche has called, ``super-genes,‘’ which provide the individual the capacity for higher powers of concentration,…* [ital added]
FTR–and regarding the quality of the poster’s excerpt in our context, “FTR” is not snark–this excerpt from a long article “Riemnann for Anti-Dummies,” which references "Spaceless-Timeless Boundries [sic] in Leibniz," by Lyndon Larouche Jr.
:smack: (And sorry for not acknowledging the Ninja – I was skimming and missed it.)
I ended up at that site by using Google Images to find the best pictures of Archytas’ construction. I did notice the site was … weird … but the page did have interesting comments about conic sections I’d not seen before.
Yes, my recollection is that a plane parallel to the side of the cone will only intersect one side (branch? limb?) of the cone and yield a parabola.
If the angle of the plane relative to the axis of the cone is greater than that of the cone’s side, it will intersect one side of the cone and form an ellipse (or a circle, a special ellipse).
If the angle of the plane relative to the axis of the cone is less than that of the cone’s side, it will intersect both sides of the cone, forming a hyperbola.*
*IIRC, hyperbolae are symmetric, just as are ellipses. I suppose this is for similar reasons?
I just want to add one observation. For any non-generate parallel, the two arms are asymptotically parallel. For example the curve y = x^2 describes a parallel and the slopes of the two arms approach infinity. If you don’t like that argument, then graph y^2 = x which has two branches y = sqrt(x) and y = -sqrt(x), x >= 0. asymptotically, both branches approach slope 0 so they are asymptotically parallel. For any positive constant c, y cx^2 is also a parabola. As c --> 0, the parabola gets flatter and flatter, but the two arms are still asymptotically parallel. But let c = 0 (this corresponds to letting your plane go through the vertex of the cone) and suddenly the graph reduces to the x-axis and the two arms are no longer asymptotically parallel. Well I guess they both have slope 0, but that is not the same as infinite slope. So the slopes change discontinuously. Amusing example of failure of continuity of derivatives.
Not sure if I’m following you, but I thought when the plane (parallel to the side of the cone) passes through the vertex of the cone the trace in the plane is a line.
To get flatter and flatter parabolas (i.e., c approaches 0) we need to move the plane away from the vertex farther and farther.
BTW - I drew some cones and planes in CAD and I’m playing around with moving the planes and changing their angle and viewing them in 3D to visualize what’s happening with various changes to get the different conics being discussed.
As many of us did, I studied conics in calc mumble mumble years ago, but I didn’t have access to 3D CAD at the time, so it’s fun to re-visit these topics with the ability to draw, manipulate and view them from any angle.
Take a look for yourself; you can download a 3rd-century BCE book on conics in any of several formats, for free:
I have a math degree, but I would hate to have to take a test on that book, and I would estimate that 80% of college-educated people would not be able to get more than a few dozen pages into it without becoming completely lost. It’s just a towering intellectual achievement from over 2000 years ago.
Apollonius is my go-to guy in religious debates, where people try to justify the nursery-school science of Genesis by saying that people back then weren’t capable of comprehending anything more complex. They seem to think that because we have cars and computers, we are smarter than people were thousands of years ago.
We aren’t. We’re just standing on the shoulders of giants, as Newton said. Take a typical kid from 4000 years ago and give him a modern education, and he could do anything the average American could do. But that’s a pretty low bar; he could certainly watch sitcoms and play video games even without the education.
I mean, that’s partly because we are more advanced. Analytic / Cartesian geometry really is a more powerful and better intellectual framework than anything the Greeks thought up, so we don’t generally think about things (including conic sections) in the same way that appolonius did, and his intellectual formalisms would presumably be difficult for us to understand.
[QUOTE=Winston Spencer Churchill]
I had a feeling once about Mathematics, that I saw it all—Depth beyond depth was revealed to me—the Byss and the Abyss. I saw, as one might see the transit of Venus—or even the Lord Mayor’s Show, a quantity passing through infinity and changing its sign from plus to minus. I saw exactly how it happened and why the tergiversation was inevitable: and how the one step involved all the others. It was like politics. But it was after dinner and I let it go!
[/QUOTE]
I’ve had that feeling a number of times, too. But I strongly suspect that it’s illusory. That is, the brain isn’t actually understanding anything, but simply telling itself that it’s understanding (and, since it’s the brain, believing itself).
Such illusions of understanding, when stronger or more frequent, doubtless constitute some form of mental illness.
I have nothing to add to the overall discussion. But I wanted to single this post out as a particularly beautiful explanation.
It’s a fine example of a qualitative argument working from the obvious to the non-obvious by appropriate-sized steps. It neatly bridges the gap between the quantitative facts of ellipse geometry most of us learned and the qualitative understanding of *why *it works. It has that “aha” quality where the key insight (bolded) is obvious once pointed out, but was invisible (to at least some people) before.
This thread happily has caused me to look into an exhilarating book The Shaping Of Deduction In Greek Mathematics A Study in Cognitive History, by Reviel Netz. To me, a tourist, I do not know how well he succeeds in demonstrating his thesis following his intellectual lodestars, Kuhn (The Structure of Scientific Revolutions, 1962) and Fodor (The Modularity of Mind, 1983).
But you won’t think about numbers, letters, and marks on paper (or a mental “visualization” thereof)–and what made “mathematics” a thing–the same again after reading this.
This Churchill fellow should be complimented on “Byss,” capped or not. Although his future as a writer or public speaker might be in doubt if he continues to pepper his work with such archaicisms. All the more so with his bold archaic root form “tergiverse”–to turn backwards, in reverse–the shock of catachresis replaced by historical linguistic wit.
FTR, yet again I remember that Chronos once used this great physics-y word, now forgotten, for “everything that is and which includes the possibility of there being” (not “God”), and what’s worse is I asked him what it was in another thread long after, he came up with a few which didn’t ring anybody’s bells, and then I forgot those. Something with “plenum,” maybe, which is what OED gives “byss.”
So I’m going with this for the time being.
[† byss, n.2 Obs.
[formed by removing the privative ἀ from abyss, Gr. ἄ-βυσσος; cf. Gr. βυσσός ‘depth of the sea, bottom’.]
In the philosophy of Boehme: The opposite of abyss or void; plenum, substance, ground of attributes.
