This week’s entry in the “Wow, the Internet is Awesome!” file:
http://www.youtube.com/watch?v=yVkdfJ9PkRQ
These pendula of staggered lengths are set swinging, and the result is fascinating. At first, the weights seem to trace out sine waves, but they quickly start to cycle through a wide range of behaviors.
At times the pendula appear highly organized, and at times almost chaotic, as they cycle through various patterns. Amazingly, the weights eventually return to their initial state!
A really beautiful mathematical demonstration.
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This is an amazing resource from the British Columbia Institute of Technology: a cross-referenced list of how various mathematical ideas are used in various technology fields.
http://commons.bcit.ca/math/examples/
Interested in how Linear Algebra can be applied to Nuclear Medicine? Or how Logs and Exponentials can be applied to Forestry and Wildlife? Or how Differential Equations can be applied to Mechanical Engineering?
Well, look no further!
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Not only is this visualization of generalized Mobius transformations mesmerizing and beautiful, it is the clearest demonstration of inversion that I have ever seen:
http://www.youtube.com/watch?v=JX3VmDgiFnY
This short movie visually explains all the basic transformations of the plane: translation, rotation, dilation, and inversion. Then it demonstrates how all of these transformations of the plane can really be thought of as simple translations and rotations of the sphere!
If I had seen this video a few decades ago, I might not have given up on topology as quickly as I did.
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This is a great tool for plotting vector fields:
http://dlippman.imathas.com/g1/fullgrapher2d.html
You can also plot functions of all kinds: implicit, explicit, polar, parametric, etc. In addition, you can plot several different graphs simultaneously (say, a vector field and a set of flow lines), and you can modify each graph individually.
I found this very helpful is constructing a recent test!
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