Loopy curvy deliciousness
July 17, 2009 10:00 PM   Subscribe

Does the intersection of a torus and a gravity-bent plane always result in a bezier curve?

I don't know what I'm talking about. I did math up through linear algebra and 3 semesers of calculus -- but -- -

I feel that there is an interesting idea space that combines mandelbrot, bezier, mobius, escher, and fluid dynamics. This is a wholly intuitive notion, coming from a very analytic person.

Cut me down, call me nasty names. I can't follow the math anymore. Please let me know if there are connections here, or if I should just go make pasta.

[tangential: I'd love to see a PhD thesis on the topography of pasta]
posted by yesster to Science & Nature (14 answers total) 2 users marked this as a favorite
No. A gravity-bent plane does not necessarily have to be linear.
posted by torquemaniac at 10:20 PM on July 17, 2009

posted by torquemaniac at 10:27 PM on July 17, 2009

No. Take the limit of either very low gravity or a very small torus, so the "gravity bent plane" will look locally flat. Intersecting that with a torus at normal angles gives you two circles, which are not "a Bézier curve".

Or maybe you want to formalize and restrict a bit.
posted by themel at 10:40 PM on July 17, 2009

Response by poster: I'm unable to formalize, any more than to say that I'm visioning realspace, not alternate worlds/non-euclidian space, etc, but I'm curious if the bezier notion is inherently recursive, and if there is a fractal representation, perhaps torii anchored, that can be explained in interesting ways without too many fomulae. The bending of a weighted branch of a flower approximates a bezier; this repeats at levels of scale. That bezier curve /can/ be expressed as, or perhaps define, an intersection of a planar-like thing with a vessel-like structure. Is there a reiteration of this that is coherently presentable in 3d, so I can see it?
posted by yesster at 11:00 PM on July 17, 2009

Bézier curves can be defined recursively, but they aren't conic sections.
posted by hattifattener at 11:36 PM on July 17, 2009

Response by poster: I should have said "topology of pasta" rather than "topography." My bad.
posted by yesster at 11:47 PM on July 17, 2009

Best answer: The following sentence: I feel that there is an interesting idea space that combines mandelbrot, bezier, mobius, escher, and fluid dynamics indicates that pasta-making is your best bet in this situation.

I don't mean to be rude, but it's clear that you have a lot of ideas and not enough formal training to express them or analyze them precisely. If you're interested in this sort of geometry, and if you have a decent grasp of multivariable calculus and linear algebra (or if you could pick it up again without too much trouble) then you might like to learn about differential geometry. It won't tell you about fractals or bezier curves, but it will help you think about stuff like tori and spheres and related objects.

I recommend John Lee's Introduction to Smooth Manifolds for its wealth of pictures and wordy exposition.
posted by number9dream at 12:24 AM on July 18, 2009

Best answer: I'm with number9dream here. I see this a fair amount, where people have heard of a bunch of disparate mathematical ideas which they think are cool (and they are cool! All of those ideas listed are neat things!), and try to jumble them together in some way without really understanding what it is that mathematicians do.

I don't mean to dissuade you at all; far from it. You have enough of a background that there is no reason that you can't learn more and properly understand these ideas, but you have to be willing to put in the time and effort. From the list of topics, differential geometry seems a good bet. I'd consider John Milnor's "Topology from the differentiable viewpoint", although it might be a little more advanced than I'm thinking.

Anyhow, as to the topics you've listed. There might be some neat relations between them, but the only way for you to find out is to study them. Study them hard.
posted by vernondalhart at 2:36 AM on July 18, 2009 [1 favorite]

Best answer: It's my understanding (as a sometimes graphics programmer) that Bezier curves aren't actually a discovered mathematical relation, but rather an engineer's construct for numerically analyzing curves. So, saying that a torus and a plane intersect in a Bezier curve is like saying that three eggs and some cheese intersect in a frying pan. Maybe true, but totally missing the point.

Mandlebrot sets aren't, I believe, related at all to Bezier curves. Somebody may prove me wrong, but I don't think they have anything to do with one another at all. Thinking about it, I don't even think you could express either as the other. There's a reason it's called a Mandlebrot set--the definition is of a set of points satisfying an iterative condition, not for the geometry of the curve itself. Indeed, as I think about it, solving for the subset of points on the border of the in and out sets would stump me pretty immediately.

Most of Mobius' neat work was in number theory. The Mobius strip is pretty much just a gimmick; like the Klein bottle. Pretty cool, but not especially interesting other than as a degenerate case.

Escher? Who's Escher? M.C. Escher? He drew stuff. His work, while visually accurate, was not mathematically rigorous. I like and respect him very much, but I don't think he belongs on the same list as Bezier, Mandlebrot and Mobius.

I'm with the you-need-training crowd. Please note, I'm not calling you an idiot. There's no reason you can't learn this stuff. But, you absolutely can't learn math without actually doing it--no number of popular math histories will teach you math, no matter how interesting they might be. I recommend you start taking some classes at the nearest university.
posted by Netzapper at 3:44 AM on July 18, 2009

Response by poster: thank you, esp. for being polite
posted by yesster at 7:07 AM on July 18, 2009

while it isn't everything, there is a certain utility to being precise. for example there are lots of torii in the world that don't necessarily look like a donut. i'm still trying to figure out what you mean by a gravity bent plane, perhaps something like a catenoid. in which case you could interpret your musings as being about the intersection of a minimal and a constant mean curvature surface...

a bezier curve is just a polynomial curve (i.e. can be parameterized by polynomial functions) with certain properties generated by an algorithm...so a mathematican might ask: is the intersection of a minimal and a constant mean curvature surface a polynomial curve? but that sounds, perhaps, boring...
posted by geos at 7:19 AM on July 18, 2009

I disagree with the notion that loading yourself down with all manner of formal mathematics is a prerequisite for the kind of thinking yesster wants to do.

For example, according to a biographical sketch I read of Murray Gell-Mann some years ago (perhaps by Jeremy Bernstein?), Gell-Mann was well into formulating relationships between elementary particles which became the theory of quarks when a colleague pointed out that he was essentially in the process of re-inventing group theory, a branch of mathematics with which he was largely unfamiliar at the time.

And the celebrated discoverers of Buckminsterfullerene supposedly never heard of Fuller or geodesic structures until after they had constructed a model of their famous molecule.

In fact, I think it's very common for capable people in physics to get themselves so fueled up with mathematics they are never able to take off and do the actual physics.
posted by jamjam at 11:01 AM on July 18, 2009

jamjam: I disagree with the notion that loading yourself down with all manner of formal mathematics is a prerequisite for the kind of thinking yesster wants to do. etc.

That's a fair comment, but the examples you cite of Gell-Mann is a bit of a misleading one; Gell-Mann did not do his work by having read a variety of pop-science books about maths and physics. He did it by learning, nuts and bolts, the physics. Then he came 'round to the mathematics.

So if your interest is to understand relations of fundamental ideas in physics, then by all means, learn physics. But as with the advice about mathematics, be prepared to work for it. Same goes with mathematics.
posted by vernondalhart at 6:21 AM on July 19, 2009 [1 favorite]

Response by poster: Thank you for being so polite with me.

Admittedly I shouldn't post questions when I'm drunk.

But I want to explain what my tangent was:

there has to be a relation between fluid dynamics and bezier curves; a bezier curve is a natural recursive way of approaching a dimunition of force or a divertion of momentum (non- bezier curved road structures result in more accidents than bezier-compliant curves/ cite required, I know). There is an established technology regarding fractal structure of antennae, which we all have in our current cell phones, which is why they don't have four foot long antennae sticking out of them.

the pattern is recursiveness; the limiting factors are the laws of physics/electromagnetism/fluid dynamics.

Escher presents a multi-modal understanding of this pattern, albeit math-free.

so consider a fractally-structured model of turbulence.

That's as far as I got. Then I went and did this.

I love metafilter. Thank you again for being kind to me.
posted by yesster at 6:29 PM on July 23, 2009

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