Friday, April 6, 2012

Project Idea: The Wave - The Golden Mean

Project Idea: The Wave, The Golden Mean.

Note: Knowledge and experience in analytical geometry, origami, and computer programming are required for this project.

Purpose

To determine if there is any possible relationship between the origami fold known as the ''wave'' and the Golden Mean or Ratio, the numerical value of which is  This will be accomplished by determining the equation of the wave's spirals.

Materials Needed

* several sheets of 1-foot-(30-cm-) square origami paper (or more)
* 1 sheet of 20-inch-(50-cm-) square origami paper
* ruler
* 2 sheets of graph paper
* personal computer with 2megabytes of memory
* software for writing a program that will create and calculate geometric and trigonometric figures and calculations

Experiment

The origami paper will be folded into patterns known as "waves." Each pattern will use a different number of divisions to demonstrate the effects of this on the resulting model. A geometric analysis will be performed to derive equations that describe characteristics of the folded models. Measurements of the models will be taken to be used as data to be elated by polar equations. A basic program will be written on a personal computer with the use of software, to draw polar spirals of the logarithmic or equiangular variety, which will be adjusted to match the spirals found in the wave.
Procedure

1) Fold a smaller sheet of paper following the instructions in the diagram, each time using 3, 6, and 12 divisions. Observe the effect this has on the resolution and shape of the model. Try folding another small sheet with an uneven number of divisions (for example, 16ths at the point, moving to 8ths at the outer edges). Record your observations.

2) Carefully examine the structure of the resulting folds. Derive equations for the angles between consecutive secants of each spiral, as a function of the number of the division from the tip of the "wave," taking the tip to be fold #0. Try to use these equations to determine a polar equation describing the spirals.

3) Practice folding the "wave" a few more times. Then, fold a model using 32 divisions with the larger paper. If necessary, use a ruler to make straight creases.

4) Create a polar axis in the center of one of the sheets of graph paper. Draw a radius every .25π radians of rotation. Taking the point of the "wave" as the origin, trace the outline of the outer spiral of the "wave" onto the paper. Measure and record the radius length along each of the radii previously drawn.

5) Graph the measurements, using the angle as the x axis and the radius as the y axis, on the graph paper. Using the resulting graph, try to fit the results into an equation. Test the equations by graphing them in the same fashion, or by graphing them as polar equations and comparing the results to the actual spiral of the "wave."

6) Write a program with software that draws polar spirals, with user-definable constants. Include logarithmic spirals as an option. Using this program, adjust the constants to determine the equation that most closely matches that of the spiral of the ''wave.''

7) Express the best equation in logarithmic form: In(r) kθ and r = c°, where k and c are constants.

8) Through observation and experimentation, determine the angular rotations between the three spirals.