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In this issue of Math in the News we look at 3D printing technology. In particular, we look at Nike's recent unveiling of a new running shoe developed using 3D printing. Mathematically, we look at recursive functions, since the 3D printing technology was used for rapid, iterative prototyping of the new shoe. We look at an example of a recursive function, using the Babylonian method to calculate the square root of a number.

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- 1. The Wonders of 3D Printing Math in the News Issue 87
- 2. 3D Printing Nike recently announced the release of a new running shoe for use by NFL players. It was developed using 3D printing technology. Watch this video to learn how they did it. http://youtu.be/qWlWStIVbBg
- 3. 3D Printing 3D printing relies on a computer model similar to the kind you might see in a CGI movie. Such models consist of a network of (x, y, z) coordinates that form a mesh to define the 3D figure.
- 4. 3D Printing When seen from different angles, the 3D mesh creates an endless number of 2D blueprints. These blueprints are used by the 3D printer to create a real object. Watch this video to learn how this works. http://youtu.be/dnIVrLqrEI8
- 5. 3D Printing What Nike did with the 3D printer was to create an iterative process so they could test different prototypes to find the ideal shoe configuration that met their requirements. 3D printing allowed for rapid prototyping.
- 6. Recursive Functions An iterative process in design and engineering is usually based on the mathematics of recursive functions. In a recursive function, an input value results in an output value, which is then fed back into the function repeatedly.
- 7. Recursive Functions Recursive operations are nearly as old as civilization itself. The Babylonian Method for calculating a square root is an example. Let’s use it to estimate 73 . 1 S xn+1 = (xn + ) 2 xn
- 8. Recursive Functions Since 73 falls between perfect squares 82 and 92, let’s make our initial guess 8.1 and use that as the input value x0. x0 = 8.1 1 73 x1 = (8.1+ ) » 8.556 2 8.1
- 9. Recursive Functions Use the result from the previous calculation to refine the calculation. Compare the result to what you get using a calculator. x0 = 8.1 1 73 x1 = (8.1+ ) » 8.556 2 8.1 1 73 x2 = (8.556 + ) » 8.544 2 8.556
- 10. Recursive Functions Here are some additional examples done on a spreadsheet. In each case the recursive nature of the calculation resulted in a more precise value. Square root of this number: 599 1 2 3 4 5 Actual 5,914,789 20 Guess Iteration 2945 30 2000 24.975 24.47949199 24.47447701 24.4744765 24.4744765 64.08333333 55.01956003 54.2729893 54.26785445 54.2678542 2478.69725 2432.473158 2432.033962 2432.033922 2432.033922 24.4744765 54.2678542 2432.033922
- 11. Recursive Functions This is the graph of the iterative solution for one of the square roots. In each case the subsequent iterations are more precise. Babylonian Method: Square Root of 2,945 70 60 50 40 30 20 10 0 x0 x1 x2 x3 x4 x5
- 12. Recursive Functions Nike’s engineering challenge was to find the ideal cleat that maximized traction for improved power and speed. It became a geometry problem solved through iteration of different designs, using 3D printing.

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