We have said that we would begin by considering how patterns can be created and that we would start with a pattern called the spiral. We said that when considereing how we as artists, how nature, and how mathematics make spirals, that we would do psychological research to see what makes spirals beautiful. Such psychological experiments are called psychophysics. While we study the beauty of the spiral, we should not lose sight of the fact that spirals are not static objects, but rather are dynamical growth processes.
STEP 1: Draw a Spiral
Each person draws a pretty sprial in a given circle (7.875 inch diameter) on a letter-size white paper using standard marker pens of a variety of common colors. (No other instructions or materials are used).
STEP 2: Rank the Beauty of the Spirals Individually
Each person rank orders the resulting picutures in terms of how beautiful they are.
STEP 3: Rank the Beauty of the Spirals as a Group
The rankings by individuals are combined into overall scores of beauty for the whole group, and the group order of the beauty of the rankings is determined. This is done by giving the highest score to first place given to the picture by any individual, the next highest score to every second place picture, and so on for all the ranks and all individuals doing the ranking and all the pictures. Then the scores for each picture are added up. The picture with the highest score gets first place, the picture with the next highest score gets second place, and so on, ranking all the pictures.
STEP 4: Determine the Relationship of Rank to Beauty
Determine if these aesthetic rankings can be related to any of some suspected properties of the spirals:
- Type of spiral (Archimedean or logarithmic).
- Number of turns in spiral.
- Consistency of spaces in the spiral.
- Number of lines spiraling.
- Number of colors.
- Color properties: hue (wavelength), saturation, brightness (amplitude).
Graphs may be used to determine these, but a casual inspection would likely lead one to reject that any of these will show a clear relationship to the aesthetic rankings.
What is wrong? That's right. Too many things are changing. The stimuli (the spirals that we drew) are varying in too many ways at once. In a psychophsical experiment, if you want to see the relationship of judgments to a physical property of the stimuli, you must vary the properties you are interested in and control or keep constant the other properties. That means we can't just draw the spirals any old way we want, we have to draw them more carefully.
STEP 5: Draw Some New Spirals Varying Chosen Properties Using Mathematical Construction.
Let us pick two properties from the above list: the type of spiral (Archimedean versus logarithmic) and the number of turns in the spiral. Both of these are aspects of the rate of growth of the spiral. These can be done by drawing on graph paper or by using a computer to draw for us. We will start with drawing, then try the computer. Details of the mathematical construction are given separately.
STEP 6. Repeat the Psychophysics Experiment with the Mathematically Constructed Stimuli
We could try three Archimedean spirals with 1, 2, and 3 turns, and three logarithmic spirals also with 1, 2, and 3 turns, and get friends or relatives to be subjects and rank order them. Then bring the individual results back to school and determine the group results from the indiviudual subjects' rankings as we did before.
STEP 7: Find Spirals; Repeat the Experiment
Find examplary objects or pictures with spirals in nature or art, and repeat the experiment with them. For example, we could use 5 or 6 beautiful spiral sea shells. Can we list or find many different spirals in nature?
STEP 8: Repeat the Psychophysical Experiment to study Fractal Dimension
The spiral stimuli can be constructed as fractals varing the fractal complexity (fractal dimension), creating again 1, 2, and 3 turn stimuli, logarithmic, but varying in fractal dimension (1.2, 1.4, 1.6 for example).
STEP 9: Can we define pattern and spiral>?
See examples of definitions. Can we improve on these? If so we will add new definitions to our glossary.
Questions to ponder:
Why are some things more beautiful? Why does nature make things mathematically? Does what we think is beautiful have to do with the fact that nature makes them that way? If so, is it because our eyes and our brains have evolved with built-in capacity to see them better, or is it becuase they we have learned that they are familiar?
Be sure to use the resources of the school, including the library, the art rooms, the science rooms, the computer labs, (ask the teachers), as well as outside resources (parents, techers, book stores, art stores, libraries, museums, and the World Wide Web.). One of our resources will be a bookshelf in the school library. Ask Ms. Hood for help if you don't spot what you need right away.
The Golden Mean:More on Mathematics and Aesthetics |
Spiral
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Syn-Dyn |
