rogerstein
New member
- Joined
- Apr 13, 2008
- Messages
- 16
I wanted to draw attention to an extraordinarily stark illustration of the strengths and weaknesses of calculus vs. geometry, by way of a problem posted on this site,"Ratio of radii of tennis ball behind beach ball in corner", and a brilliant reply by TchrWill. It also illustrates another, perhaps even more important point: you can sometimes attain a dazzingly simple solution by completely recasting the problem--retaining its quintessence but putting it in a totally different context.
So, the beach ball problem was this: You have a large ball hiding a small ball in the corner of a room--both balls are tangent with the floor and two walls and with each other. What is the ratio of their radii? I succeeded in solving the problem with calculus, and as no-one had posted a solution, I did (post #3, rogerstein). In the last paragraph of the lengthy post in which I laboriously detailed my solution, I speculated about a simple geometric approach. Calculus is perfectly designed for the crude rough-and-tumble of the real world, the complex, gnarled, disheveled, asymmetrical Hunchbacks of Notre Dame. Geometry prefers ideal forms, and when, rarely, we're confronted with a Catherine Zeta-Jones of a problem, geometry can embarrass calculus with what it can do. I had a feeling Cathy was in the building and the mentally agile TchrWill had more than a feeling. He instantly recognized that the beach ball/tennis ball problem was contained in a seemingly different problem, and THAT problem can indeed be solved geometrically. TchrWill nimbly sets forth the very thing I had wistfully conjectured about--please go to "Ratio of radii,etc"post #4 TchrWill-- and even though, happily, it confirms the correctness of my answer, it simultaneously shows how incredibly blunt and awkward a tool calculus is in idealized situations compared with geometry. When I looked at the fire-breathing, earth-shaking behemoths of equations that have to be solved in my calculus solution (not to mention the chore of devising them), and contrasted them with the cuddly little teddy bear of an equation in the geometric solution, I said, "Calculus, I love and admire you (though I'm just a neophyte), but oh that Geometry!!!"
The second point to be taken from TchrWill's wonderful contribution is how taking the essence of a problem and putting it in a new setting can dramatically enhance our prospects of a simple, perhaps even beautiful solution. For example, there have literally been hundreds of different proofs of the Pythagorean Theorem, but they all have one thing in common--they are merely proofs of the Pythagorean Theorem. Then Einstein came along and (reportedly at age 12!!) devised a proof unlike any other, one where the Pythagorean Theorem is just one special case of a much larger, deeper idea. Einstein's approach is thrilling in its brilliance --it presages his later work in that he went conceptually to a more profound level that somehow escaped Fermat, Euler, Gauss, Riemann, Cantor, et. al. He recast the problem in a way so surprising that when you first see it you say, "This is madness!", but of course it wasn't. Instead, it provided a solution breath-taking in its elegance. In a similar, though of course less dramatic way, TchrWill provides a new context for the beach ball/tennis ball problem that allows for the simple geometric proof.
I'd be very interested in readers of this post offering other examples of the Calculus/Geometry Wars or of how reframing a problem can make an unwieldy situation laughably easy to deal with.
So, the beach ball problem was this: You have a large ball hiding a small ball in the corner of a room--both balls are tangent with the floor and two walls and with each other. What is the ratio of their radii? I succeeded in solving the problem with calculus, and as no-one had posted a solution, I did (post #3, rogerstein). In the last paragraph of the lengthy post in which I laboriously detailed my solution, I speculated about a simple geometric approach. Calculus is perfectly designed for the crude rough-and-tumble of the real world, the complex, gnarled, disheveled, asymmetrical Hunchbacks of Notre Dame. Geometry prefers ideal forms, and when, rarely, we're confronted with a Catherine Zeta-Jones of a problem, geometry can embarrass calculus with what it can do. I had a feeling Cathy was in the building and the mentally agile TchrWill had more than a feeling. He instantly recognized that the beach ball/tennis ball problem was contained in a seemingly different problem, and THAT problem can indeed be solved geometrically. TchrWill nimbly sets forth the very thing I had wistfully conjectured about--please go to "Ratio of radii,etc"post #4 TchrWill-- and even though, happily, it confirms the correctness of my answer, it simultaneously shows how incredibly blunt and awkward a tool calculus is in idealized situations compared with geometry. When I looked at the fire-breathing, earth-shaking behemoths of equations that have to be solved in my calculus solution (not to mention the chore of devising them), and contrasted them with the cuddly little teddy bear of an equation in the geometric solution, I said, "Calculus, I love and admire you (though I'm just a neophyte), but oh that Geometry!!!"
The second point to be taken from TchrWill's wonderful contribution is how taking the essence of a problem and putting it in a new setting can dramatically enhance our prospects of a simple, perhaps even beautiful solution. For example, there have literally been hundreds of different proofs of the Pythagorean Theorem, but they all have one thing in common--they are merely proofs of the Pythagorean Theorem. Then Einstein came along and (reportedly at age 12!!) devised a proof unlike any other, one where the Pythagorean Theorem is just one special case of a much larger, deeper idea. Einstein's approach is thrilling in its brilliance --it presages his later work in that he went conceptually to a more profound level that somehow escaped Fermat, Euler, Gauss, Riemann, Cantor, et. al. He recast the problem in a way so surprising that when you first see it you say, "This is madness!", but of course it wasn't. Instead, it provided a solution breath-taking in its elegance. In a similar, though of course less dramatic way, TchrWill provides a new context for the beach ball/tennis ball problem that allows for the simple geometric proof.
I'd be very interested in readers of this post offering other examples of the Calculus/Geometry Wars or of how reframing a problem can make an unwieldy situation laughably easy to deal with.