http://www.Media4MathPlus.com 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.

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.