\omega_n + \omega_m = \omega_k
they can generate new standing wave configurations, forming nested shells of constructive interference. These shells naturally form spiraling density patterns, mirroring the logarithmic geometry of the Nautilus shell.
This recursive nesting follows a generalized logarithmic spiral function:
r(\theta) = r_0 e^{k\theta}
where:
is a scaling constant,
encodes the "pitch" of the spiral (determined by mode spacing and initial conditions),
is an angular coordinate mapped to either space or mode phase.
In 3D spacetime, these spirals translate into nested spherical wavefronts intersecting with other modes, producing filaments and walls at predictable locations. Such geometries have echoes in observed cosmic web topology, including:
