## Thursday, December 10, 2009

### Angle & Frequency

Here's a good example of where our ET Math might seem obfuscatory, whereas the design actually holds a lot of water: what's all this ruckus about "angle vs. frequency"?

When you think "frequency", you may immediately think of a radio dial, maybe a car radio console with buttons. You may think of a microwave oven, or of cell phone towers. If you're science-minded, you know the whole visible spectrum constitutes a thin band of frequencies detected by the naked eye, with 99+% of transmissions going on outside that range.

Integral to the above concept are both time and size. Frequency has to do with periodicity, intervals, the tick tick of a clock, a beating heart. Frequency also has to do with vast time range orbitings, such as of moons around planets, planets around suns (some say Jupiter was an "almost sun"). Time and size go together, and frequency partitions same into smaller and smaller units, starting with Universe as a whole ("Universe" is a proper place name here, more like "Narnia").

When you think of "angle", you should think of something's shape, irrespective of how it shrinks or grows relative to the surroundings. Alice in Wonderland takes the red pill and gets smaller, but all the central and surface angles stay the same, such that she remains ever similar to herself. In many a state standard, it's a requirement to explain how Alice's volume is changing as a third power of her linear rate of fluctuation, whereas her surface area (skin surface) is fluctuating as a second power rate. Her shape, on the other hand, is held constant, thereby dramatizing the difference between... Angle and Frequency (shape and size).

So why didn't we just go with Shape & Size in the first place? We could, and we will, but Angle & Frequency are well chosen candidates for this distinction, precisely because of the daydreaming each might inspire. Saying "size" wouldn't get most people outside of "shoe size" and into some awareness of the electromagnetic spectrum and its photochemical interactions with luminous surfaces, excited atoms, radiating eye-tunable frequencies. Saying "shape" might keep one fixated on someone "shapely", not a bad thing in itself, but we need that "compass look", the simple V shape, and, very important, triangles (named for their three angles). Saying "omni-triangulated" so often (part of our shop talk) means harping on "angles" a lot.

In sum, there's a good case to be made that we're dealing with a consciously well-designed namespace, conducive to inventive yet realistic thinking of a scientific nature. And we're not just talking "pretty prose" in that we have these hooks into "frequency to a power" i.e. exponential rates of change relating linear, areal and volumetric rates of change.

These tie to "concrete activities" such as scripting a few lines in Python or other logic with runtime to generate a numerical growth pattern. Veteran gnu math teachers will recall how we often start with the CCP's 1, 12, 42, 92.... (cubic closest packing, check On-Line Encyclopedia of Integer Sequences).

The cumulative number of spheres will increase as a 3rd power, whereas successive layers describe a 2nd power rate of growth. Or think of other exhibits (after all, you're their teacher, so that's your job).