One thing my students tend to ask me, is "Mr. V? Where is math in our lives?" Sometimes that answer has clear ties to what we're learning in class, but other times the mathematics are more advanced. Fractals are a great example of advanced mathematics with current application. Some examples follow:
Urban Planning:
http://www.nature.com/srep/2012/120724/srep00527/full/srep00527.html
Natural Patterns:
http://webecoist.momtastic.com/2008/09/07/17-amazing-examples-of-fractals-in-nature/
Chemistry/Physics:
http://www.wired.com/wiredscience/2010/08/superconductor-fractals/
Art
http://www.oneness4all.com/?cat=44
Crop Circles
http://www.cropcircleconnector.com/Bert/3dfractals.html
3D Graphics
http://www.profantasy.com/products/ft.asp
http://specialeffectsmoviesebook.blogspot.com/2013/02/1-fractals-in-special-effects.html
Technology
http://classes.yale.edu/fractals/panorama/ManuFractals/FractalAntennas/FractalAntennas.html
Sound and Music
http://www.tursiops.cc/fm/
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