http://www.aes.org/e-lib/browse.cfm?elib=5755

I have been communicating with Adrian Freed at the Center For New Music and Audio Technologies at UC Berkley about our research. He pointed me to the paper linked above

links for Adrian Freed:

CNMAT website: http://cnmat.berkeley.edu/people/adrian_freed

blog: http://cnmat.berkeley.edu/user/adrian_freed/blog

Manipulate[
where = PolyhedronData[“Tetrahedron”][[1, 1]];
tet = PolyhedronData[“Tetrahedron”][[1]];
oct = PolyhedronData[“Octahedron”][[1]];
octn = {RGBColor[0.6, 1, 0.2],
Rotate[oct, ArcTan[Sqrt[2]], {0, 1, 0}]};
octn2 = Scale[octn, 2 {1, 1, 1}, {0, 0, 0}];
d2 = Map[Translate[{tet}, #] &, a  where];
d3 = Map[Translate[{octn, d2}, #] &, b 2  where];
Graphics3D[{d3, octn2}, Boxed -> False, SphericalRegion -> True,
ViewPoint -> {-5, 20, 5}, ViewAngle -> Pi/60],
{{b, 1, “divide tetrahedron b”}, 0, 2},
{{a, 1, “divide tetrahedron a”}, 0, 2}, TrackedSymbols -> {b, a}]

Here is the Mathematica code modeling of 2D vibrations of the rectangular region excited by a sine function.

k[x_, y_] = Sin[x + y]

Sin[x + y]

a[n_, m_] =
4 / (l1 l2) Integrate[
Integrate[
k[x, y] Sin[n Pi x / l1] Sin[m Pi y / l2], {x, 0, l1}], {y, 0, l2}]

-(4 (-2 n \[Pi] Cos[2] Sin[m \[Pi]] +
n \[Pi] Cos[n \[Pi]] (m \[Pi] Sin[4] + 2 Cos[6] Sin[m \[Pi]]) –
4 m \[Pi] Cos[4] Sin[n \[Pi]] +
8 Sin[6] Sin[m \[Pi]] Sin[n \[Pi]] +
m \[Pi] Cos[
m \[Pi]] (n \[Pi] Sin[2] – n \[Pi] Cos[n \[Pi]] Sin[6] +
4 Cos[6] Sin[n \[Pi]])))/((-4 + m^2 \[Pi]^2) (-16 +
n^2 \[Pi]^2))

Manipulate[
Plot3D[Sum[
Sum[a[n, m] Cos[
c Sqrt[((n Pi x / l1)^2) + ((m Pi y / l2)^2)] t] Sin[
n Pi x / l1] Sin[m Pi y / l2], {n, 1, B}], {m, 1, M}], {x, 0,
10}, {y, 0, 15}, PlotRange -> {-10, 10}]
,
{t, 0, 10},
{B, 1, 10},
{M, 1, 10},
{l1, Exp[-25], 12},
{l2, Exp[-25], 10},
{c, 0, 5}
]

So I had an idea after some coffee and reading through Sapoval’s article as well as a bunch of related stuff. Can we possibly guide the light, or electromagnetic wave in general, in the direction we need using some 3D structure, for now not necessarily a fractal, or alter the waveform in some other way? That is basically what we are shooting for, but I just browsed through general info on density of states, which led me to photonic crystalls, and so I was thinking about applying the related principle to our project’s light part. As I understand it, highly ordered structure of photonic crystalls together with the principles of diffraction, energy levels, and DOS allow, say, opal crystalls to work the way they do: cancel some wavelength of light, enhance other ones, giving us that pretty blinkage (couldn’t find a better word, pardon my English:). My question is then, since according to the property of scale  invariance of the electromagnetic fields, and hence waves, does it make any sense to assume that certain fractal structure we are yet to construct/find/copy would cancel out, say, wavelength in the microwave range could be applicable to reducing noise in antennas maybe, or if we create the structure small enough to work with visible light wavelengths, could we possibly get some artistic effects out of the cancellation/enhancement effects of the structure? This is just a flow of thought I did not want to lose in the morning, so it is quite disordered. The basic idea would be, I guess, could we apply principles of photonic crystall structure to the analysis and maybe construction of fractally shaped wave cancellation setup, or maybe wave altering setup?

Links:

Density of states

Photonic crystalls

The original publication on vibrations of fractal generators (the shape of the drum boundary generated)

Periodic orbit theory of fractal drums

Vibrations of fractal drums (another Sapoval publication)

Anderson localization (Wiki article): a phenomena that depicts chaotic behaviour somewhat similar to frequency distribution of fractal drum

Study in Acoustic Field of Fractal Boundaries using Cellular Automata

The general info on Koch Snowflake

Infinite Series – Koch Curve generation