So, we can define two volumes: one is the volume of the sphere, and the other is a volumetric representation of the different photons that can fit into the sphere. When working in normal space, these two volumes (one of physical space, the other of photon momentums) coincide. And this coincidence is important, because they are mathematically related to each other, which, with a bit of jiggery pokery, leads to Heisenberg's uncertainty principle. In other words, the fact that we can only jointly know the position and momentum of photon to a certain minimum precision, regardless of the measurement technique, is a consequence of the fact that these two volumes are identical to each other.
(For those who enjoy feeling smug when you are ahead of the game, a bit of caution: if the words "Fourier transform pair" sprang to mind, you should now hide your smugness.)
But what if the interior of the sphere is fractal? Then the physical volume is the same as it was before, but the way that photons travel is not. Indeed, one of the points of a fractal is that starting at a position on the surface and traveling in a direction will never ever take you back to the same position. But that's a dimensionless mathematical thing, while photons are extended objects. So, even though they never come back to exactly the same location, they will still meet up with themselves. When they do, the same rules apply, albeit with the caveat that you never get perfectly constructive or destructive interference.
This complicated interior structure ensures that the volumetric representation of the photons that fit on the fractal is different from the volume of the fractal. But the same mathematics apply, so a new "uncertainty" principle can be found. This new uncertainty principle includes the possibility that there is no uncertainty, because the two volumes change semi-independently.
Some of you may be thinking, "but a Fourier transform pair is a Fourier transform pair, therefore the uncertainty principle still holds." The world of fractals holds a surprise in store for you. Apparently, you can't do a Fourier transform in a fractal space. This makes sense when you consider that these transforms rely on the existence of a regular series of periodically repeating waves. On a fractal, there are no regular periodic structures that can exist in a self-consistent manner.
To put this in more concrete terms, in normal space, doing a Fourier transform followed by undoing it returns you to where you started. On a fractal this is not true.
What does this actually mean in terms of the uncertainty principle and real-life objects? No one, not even Akkerman, really knows. The uncertainty principle, apparently, doesn't hold, but this doesn't mean that you can make arbitrarily precise pairs of measurements. Instead, it means we don't know if we can make such measurements.
As for practical examples, Akkerman showed a calculation of a series of waveguides laid out in a Sierpinski gasket pattern (the gasket is pictured above). He showed that, even if the gasket is terminated at just five iterations, the divergence between two volumes (areas in this case, since it is 2D) is large, meaning that we should be able to test his ideas. This is especially true because the waveguide circuit that he showed is well within the realms of fabrication. I suspect we will hear more about this soon.
