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!
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.
My latest contribution to the New York Times Learning Network is a collection of teaching and learning ideas that use New York Times resources to explore the mathematics of elections.
The ideas cover both primary and presidential elections, and involve polling, demographics, campaign finance, and even a little graph theory!
Through Math for America, I am part of an on-going collaboration with the New York Times Learning Network. My latest contribution, a Test Yourself quiz-question, can be found here:
https://learning.blogs.nytimes.com/2012/01/25/test-yourself-math-jan-25-2012/
This question is based on the rising number of weight-loss surgeries performed each year in the U.S. How many are being performed on patients under 21 years of age?
This is an amazing application of origami: a steel grocery bag that can fold flat!
http://news.sciencemag.org/sciencenow/2011/03/paper-plastic-or-steel.html
Although you probably wouldn’t want to carry these to the store with you, this result could have real applications in industrial packaging. In addition, it’s another step towards the mathematical-origamist’s dream: designing building that can rearrange and rebuild itself as needed!
A great talk by Erik Demaine on mathematical origami opened my eyes have this amazing application of mathematics, and my students and I have been having a lot of fun with folding ever since!