Archive for the ‘Math’ Category

Sam and Polly and How I Know the Answer.

Sunday, November 23rd, 2008

So anyway, I’m too lazy to write an original post, so I’ll just copy a draft of my report that I turned in for this project.

The Problem:

An alien appears to two mathematicians, Sam and Polly.  The alien says, “I’m thinking of two numbers, X and Y with 3 <= X <= Y <= 97.  I will tell their sum to Sam and their product to Polly.”  The alien does this and then disappears.  The following conversation occurs:
Sam:  You don’t know what X and Y are.
Polly:  That’s true, but now I do.
Sam:  And now I do too.
Find X and Y.
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It’s a mathstorm

Tuesday, October 14th, 2008

Well it’s been a long time since I’ve had any real time to do much posting here.  The senior design course I’m in is sucking up a large proportion of my time.  I don’t see this changing any time soon, either.

Anyway, as of Monday, we were assigned projects in the Number Theory class I’m taking.  I chose to solve a “difficult” problem.  It goes like this:

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Cones, Planes, and Automobiles…err…Spheres

Wednesday, May 14th, 2008

Ohh, something new? Well, seeing as I was out of town with essentially nothing to do, I do what I always do: math.

I was looking at an image posted earlier by KC LC with shadows cast by a sphere sitting on a plane. Something stuck out at me which I guess would only stick out if you’ve been staring at conics as long as I have (keep in mind, I’m in aerospace and conics show up big time in orbital mechanics). I noticed that the point of contact between the sphere and plane was pretty damn close to the focus of each of the conic section shadows (I explained the reason for them being conics in a previous comment). While it was easy to demonstrate that the shadows themselves were indeed conics, it’s not nearly as obvious that the point where the sphere touches the plane is a focus of the resulting conic.

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Anaclastic lenses

Wednesday, April 23rd, 2008

It all began as a daydream. I was bored in class and for some reason got to thinking of melting things (in particular this eraser I was playing with). I devised a contraption composed of a large parabolic reflector and a set of small lenses or other mirrors that would focus the light from the reflector into a death beam. But I realized that to get a high quality beam, I’d need a lens that would have no geometric aberrations (i.e. a convergent lens which focuses all rays parallel to its focal axis to the same point). There are two (simple) geometric cases I could consider here but I’ll only do one (something magic will happen to take care of the other).

Positive Meniscus Lens

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