The answer to all questions is No. In fact, even the right reaction to the first sentence - that the Planck scale is a "discrete measure" - is No.
The Planck length is a particular value of distance which is as important as $2\pi$ times the distance or any other multiple. The fact that we can speak about the Planck scale doesn't mean that the distance becomes discrete in any way. We may also talk about the radius of the Earth which doesn't mean that all distances have to be its multiples.
In quantum gravity, geometry with the usual rules doesn't work if the (proper) distances are thought of as being shorter than the Planck scale. But this invalidity of classical geometry doesn't mean that anything about the geometry has to become discrete (although it's a favorite meme promoted by popular books). There are lots of other effects that make the sharp, point-based geometry we know invalid - and indeed, we know that in the real world, the geometry collapses near the Planck scale because of other reasons than discreteness.
Quantum mechanics got its name because according to its rules, some quantities such as energy of bound states or the angular momentum can only take "quantized" or discrete values (eigenvalues). But despite the name, that doesn't mean that all observables in quantum mechanics have to possess a discrete spectrum. Do positions or distances possess a discrete spectrum?
The proposition that distances or durations become discrete near the Planck scale is a scientific hypothesis and it is one that may be - and, in fact, has been - experimentally falsified. For example, these discrete theories inevitably predict that the time needed for photons to get from very distant places of the Universe to the Earth will measurably depend on the photons' energy.
The Fermi satellite has showed that the delay is zero within dozens of milliseconds
http://motls.blogspot.com/2009/08/fermi-kills-all-lorentz-violating.html
which proves that the violations of the Lorentz symmetry (special relativity) of the magnitude that one would inevitably get from the violations of the continuity of spacetime have to be much smaller than what a generic discrete theory predicts.
In fact, the argument used by the Fermi satellite only employs the most straightforward way to impose upper bounds on the Lorentz violation. Using the so-called birefringence,
http://arxiv.org/abs/1102.2784
one may improve the bounds by 14 orders of magnitude! This safely kills any imaginable theory that violates the Lorentz symmetry - or even continuity of the spacetime - at the Planck scale. In some sense, the birefringence method applied to gamma ray bursts allows one to "see" the continuity of spacetime at distances that are 14 orders of magnitude shorter than the Planck length.
It doesn't mean that all physics at those "distances" works just like in large flat space. It doesn't. But it surely does mean that some physics - such as the existence of photons with arbitrarily short wavelengths - has to work just like it does at long distances. And it safely rules out all hypotheses that the spacetime may be built out of discrete, LEGO-like or any qualitatively similar building blocks.
There is quite a lot to discuss around this topic, but I'll do my best to keep it brief. While we can't really say what energy structures at that scale "look like," we have some sense of what may be happening at that scale, and how those energy structures may be arranged.
First of all, let's get the scale straight. The Planck length has some interesting properties when it comes to energy. At this length, the energy of a single wavelength of light obeys the Schwarzschild condition, meaning that if we take the Compton wavelength of a Planck particle or Planck scale object (even a Photon), at this scale this wavelength is equal to the Schwarzschild radius. This means that it obeys the conditions of a tiny black hole. This in turn makes it's mass the Planck mass.1,2
While this might sound ridiculous, remember that what we're dealing with are the quantum dynamics at the smallest scale we currently theorize about, and what we know of the quantum vacuum at this scale is that it should be full of incredible amounts of energy. While calculated to be gargantuan (10113 joules per cubic meter),3,4 it seems that somehow this energy is so well balanced that it only shows up in what quantum electrodynamics calls "virtual particles" popping in and out of existence.
So let's get to the point. How is the energy so well structured that it seems like empty space?
Buckminster Fuller gives us a clue. His studies in architectural and energetic tensegrity showed that energy that is triangulated or tetrahedrally arranged can establish a perfect equilibrium. This means that regardless of the energy level, the energy is stable when in coherence and geometric alignment.
This means that if there are immense energies active at this scale, they must be tetrahedrally aligned in 3D, or triangulated in 2D to maintain equilibrium. Buckminster himself left us a very interesting quote foreshadowing this connection:
"Gravitational Field: Omnidirectional geodesic spheres consisting exclusively of three-way interacting great circles are realizations of gravitational field patterns... The gravitational field will ultimately be disclosed as ultra high-frequency tensegrity geodesic spheres. Nothing else." -RBF Definitions5
In my research, I found that you could easily begin to map gravitational topography of the spacetime fabric using this means, simply by following the rules of Bucky's domes. Where there is an energy point missing (radiated or compressed) from the hexagonal structure created by triangulation, there is a curvature formed (usually around pentagonal structures initially).
Loop Quantum Gravity also initially used triangulation to map quantum gravitational structures, though now it tends more towards "spin foam."6
Quantum Gravity and the Holographic Mass7 is an incredible peer-reviewed paper that applies these geometries to linking gravitational properties in cosmological scales to the energetic properties of the proton, and provides an elegant geometric solution for quantum gravity.
So what do things looks like at this scale? Well, if you took that fractal image and added a geometric lattice inside it, that might be close, because we see self-similar geometries all the way up to the macrocosm. Looking a 2D version of a lattice of gravitational curvature, it might look more like this:
Frequencies of Dome Structures - Chosen for Scale Relationship Visibility http://theblackharbor.com/wp-content/uploads/2011/09/dome-peace_0131.jpg
If we want to see it in 3D, maybe something like this:
(source: netdna-cdn.com)
or like this if we're dealing with Planck Spherical Unit intersections:
(it won't let me post more images or links so here's the URL:
https://s-media-cache-ak0.pinimg.com/236x/7e/44/e3/7e44e3e970c51cc539c6981623d87f41.jpg )
Great question, and I hope this gets you started in further exploration and research! There is certainly a need for better illustrations of the theories and principles at this scale.
Highest Regards,
Adam Apollo
:::
References:
Michel M. Deza; Elena Deza. Encyclopedia of Distances. Springer; 1 June 2009. ISBN 978-3-642-00233-5. p. 433.
"Light element synthesis in Planck fireballs" - SpringerLink
Peter W. Milonni - "The Quantum Vacuum"
de la Pena and Cetto "The Quantum Dice: An Introduction to Stochastic Electrodynamics"
Synergetics Dictionary Cards - RW Gray Projects
Loop Quantum Gravity (PDF Article) - IGPG Gravity PSU - Carlo Rovelli
Quantum Gravity and the Holographic Mass (PDF Paper) - SDI - Nassim Haramein
Best Answer
The Standard Model and General Relativity are both successful in appropriate limits, but they cannot be consistently combined for scales below $\sim\ell_P:=\sqrt{\dfrac{G\hbar}{c^3}}$ for various reasons. (By $\sim$, I mean "give or take a multiplicative constant that's besides the point here and may be hard to compute".) For example, what happens if you try to probe such length scales with a photon? How will its wavelength compare to its Schwarzschild radius?
When you ask about the physical meaning or significance of such short length scales, that's where it gets contentious. I'll try to summarise the range of views on this, but I'll probably fudge or simplify a few details: