math in the news: issue 87

Math in the NewsIssue 87

The Wonders of 3D Printing

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

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.

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

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.

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.

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 .

Recursive Functions

Since falls between perfect squares 82 and 92, let’s make our initial guess 8.1 and use that as the input value x0.

Recursive Functions

Use the result from the previous calculation to refine the calculation. Compare the result to what you get using a calculator.

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 2945 5,914,789

Guess 20 30 2000Iteration

1 24.975 64.08333333 2478.69725

2 24.47949199 55.01956003 2432.473158

3 24.47447701 54.2729893 2432.033962

4 24.4744765 54.26785445 2432.033922

5 24.4744765 54.2678542 2432.033922

Actual 24.4744765 54.2678542 2432.033922

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.

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.