This is another wonderful visual demonstration from Jason Davies: a combinatorial bracelet generator.
http://www.jasondavies.com/necklaces/
Combinatorics is the mathematics of counting things, and one of the classic “advanced” counting problems is this: given a certain number of beads of various colors, how many different bracelets can you make?
The problem may seem easy enough, but it becomes quite difficult when you start to understand what “different” really means.
For example, if you turn one bracelet into another by rotating it, then those two bracelets aren’t different. Even more complicating is that if you can obtain one bracelet from another by flipping it over, then they are also the same!
This visualization can really help develop a sense of the complicated symmetries at work here.
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This sculpture by Charles Sowers functions simultaneously as a stimulating piece of art, a representation of data, and an illustration of vector fields.
http://goo.gl/WoXu3
Windswept is a giant billboard covered in little aluminum arrows that twist and turn in the wind. The arrows can be thought of as bits of data, namely, the direction of the wind that point. Taken together, and dynamically, they give a sense of the complicated nature of swirling and changing winds.
Just like this wonderful wind map, this sculpture represents mathematical ideas in a beautiful and thought-provoking way!
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This is a beautiful representation of ocean currents around the world:
http://www.flickr.com/photos/gsfc/7009056027/
Put together by the NASA/Goddard Space Flight Center Scientific Visualization Studio, this short video circles the digital globe, showing the relative strengths and directions of ocean movement.
Watching this allows one to see some of the basic mathematics of fluid flow, like tendency toward rotation and how fluid behaves at boundaries. In addition, global phenomena like the jet stream and trade winds can also be perceived.
This dynamic representation of data is similar to this wind map in how it brings to life the ideas of vector fields and flow lines.
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This is a stunningly beautiful visualization of wind patterns in the US:
http://hint.fm/wind/
Not only is this a functional and immediately accessible representation of data, but it also brings to life the mathematical concepts of vector fields and flow lines.
Apart from atmospheric science questions like “Why is this area windier than others?” are purely mathematical questions like “Which location is the calmest?” and “Which location is most volatile?”
And if you enjoy this, be sure to check out this visualization of the world’s ocean currents!
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World of Technology has several great GIF animations demonstrating some fundamental mechanics:
http://goo.gl/cHblY
Seen at right is the radial engine. The constant velocity joint is my favorite, but it’s also great to learn how a sewing machine really works!
Some great visualizations of interesting and intricate 3D geometry and engineering.
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This is another wonderful visual demonstration from Jason Davies: a combinatorial bracelet generator.
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