Animations for the fours
Added 2017-05-25 04:25:34 +0000 UTCHello +4's,
The upcoming video is on finding all possible pythagorean triples. Here are a few excerpts from animations I'm working on.
-Grant
Comments
There is something about an animated proof without words that is just simply beautiful. If I were you, during the vid, I'd let the animation speak for itself.
Ankit Agarwal
2017-05-25 14:50:46 +0000 UTCGrant: if you plan to talk about finding pythagorean triples, you should include a discussion of the stereographic projection (“half-angle tangent”). Given coordinates x + iy on the unit circle, this is the value iy / (1 + x), which is equal to the tangent iy' / x' of the complex number x' + iy' such that (x' + iy')^2 = x + iy. In other words, the angle doubling is analogous to the squaring in your video, because squaring a complex number doubles the angle. The stereographic projection is easy to draw a picture of, just connect the point on the circle to the point -1 + 0i, and look where it crosses the imaginary axis. The reason the stereographic projection is nice in general is that it gives a different way of representing orientations or rotations or points on the unit circle, as arbitrary points along an infinite number line (I recommend considering this a pure imaginary number, to correspond with angle measures also being pure imaginary quantities, the complex logarithm of points on the unit circle, and also because that generalizes much better to 3-dimensional rotation, where you can use a pure imaginary quaternion). Rational points on the stereographic projection correspond to rational points on the circle and the map is easy to compute, unlike with angle measures where a transcendental function (log / exp) is required to convert back and forth between angle measures and Cartesian coordinates. It’s pretty straight-forward to put together the formula to compose rotations (i.e. perform complex multiplication) in terms of the projected values. Personally I recommend that people store rotations in computer programs (e.g. for physics simulations or cartography or 3d graphics or whatever) using complex numbers as pairs of floating point numbers internally, and then take the stereographic projection and reduce the precision of the resulting floating point number if the rotations must be compressed for serialization (e.g. saving to a file or database, transfer over the network), and pretty much never use angle measure for anything except possibly interfacing with humans or legacy systems. If you’re still talking about calculus, this is also the source of a cute trick for handling some antiderivatives <a href="https://en.wikipedia.org/wiki/Tangent_half-angle_substitution" rel="nofollow noopener" target="_blank">https://en.wikipedia.org/wiki/Tangent_half-angle_substitution</a>
Jacob Rus
2017-05-25 04:41:01 +0000 UTC
