Sunday, December 12, 2021

How bubbles in coffee cup, hyperbolic geometry, and magnetic body could relate?

I received in Facebook from Runcel D. Arcaya a beautiful photo of fractal bubble structure formed at the surface of liquid in coffee cup.

This picture is fascinating and inspires questions. What could be the physics behind this structure? Is standard hydrodynamics really enough?

The bubble structures from the point of view of standard physics

Let us list what looks like standard physics.

  1. In the interior of the liquid bubbles of air are formed: this requires energy feed, say due to shaking.
  2. The bubbles are lighter than liquid and rise to the surface and eventually develop holes and disappear.
  3. The pressure inside the bubble is higher than in air outside and this gives rise to a spherical shape locally. Otherwise one would have a minimal surface with vanishing curvature (locally a saddle). The role of pressure means that thermodynamics is needed to understand them.
What kind of dynamics could give rise to the fractality and the beautiful structure with a big bubble in the center.
  1. The structure brings in mind the illustrations of 2-D hyperbolic geometry and its model as Poincare disk.

    One can also imagine concentric rings consisting of bubbles with scaled down size as radius increases. Atomic structure or planetary system of bubbles also comes into mind!

  2. The so- called (rhombitriheptagonal tiling) of the Poincare disk in the plane comes to mind. Could the bubble structure be associated with a tiling/tessellation of H2 represented as a Poincare disk?
What about TGD description of the bubble structure?

Could one understand the bubble structures in classical TGD in which 4-D space-time surfaces in M4× CP2 would be minimal surfaces with singularities?

  1. Tessellations of 3-D hyperbolic space H3, which has H2 as sub-manifold, are realized in mass shell of momentum space familiar to particle physicists and also in 4-D Minkowski space M4 \subset M4× CP2 as the surface t2-r2= a2,where a is light-cone proper time and identified as Lorentz invariant cosmic time in cosmologies. H3 is central in TGD.
  2. Tessellations of H3 induce tessellations of H2 and they could in some sense induce 2-D tessellations at the space-time surface, magnetic body (MB) of any system is an excellent candidate in this respect.

    This would be a universal process. For instance, genetic code could be understood as a universal code associated with them and genes would be 1-D projections of the H3 tessellations known as icosa-tetrahedral tessellation.

    I have proposed that cell membranes could give rise to 2-D realizations of genetic code as an abstract representation of the 3-D tessellation at MB (see this).

  3. Could these "bubble tessellations" somehow correspond to the 3-D tessellations of H3, not necessarily the icosa-tetrahedral one, since there is an infinite number of different tessellations of H3?
The basic problem is that the Poincare disk has a finite radius and H2 consists of the entire Euclidean plane. Furthermore, the Euclidean plane has vanishing curvature while Poincare disk has a negative curvature. The simplistic attempts indeed fail.
  1. There is no natural map from the hyperbolic plane H2 to the Poincare disk. Projection is impossible since it would require an infinite compression of the Euclidean plane.
  2. What about realizing Poincare plane as 2-surface in M4× CP2 with induced metric equal to the metric of Poincare disk given by ds2P =ds2E/(1-ρ2): here ds2E is the Euclidean metric of the plane and ρ its radial coordinate.
Simple realizations seem implausible. Presumably the negative curvature of H2 in contrast to the positive curvature of CP2 and vanishing curvature of the Euclidean plane is the problem.

The following represents a possible successful attempt.

  1. Could the MB of the system, which can realize the tesselations, somehow induce the discrete Poincare disk based bubble structure as a discrete representation of its 2-D hyperbolic sub-geometry? The distances between the discrete points of the bubble representation, say the positions of bubbles, at MB would have hyperbolic distances and induced correspond to the distances at MB.
  2. There is evidence that in a rather abstract statistical sense neurons of the brain obey a hyperbolic geometry. Neurons functionally near to each other are near to each other in an effective hyperbolic geometry. Hyperbolic geometry at the level of the MB of the brain could realize this concretely. Neurons functionally near each other could send their signals to points near each other at MB of the brain (see this.
For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

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