Comment by anigbrowl
1 day ago
Pause for a moment to reflect on how outrageous this assertion is. You can’t see into the box at all. Nevertheless, holography says that you can learn exactly what’s happening everywhere in the box without any access to the interior. Observing the surface alone is enough. In this sense, the amount of stuff that fills a box is the same as the amount of paint that covers it. That’s a violation of logic and geometry.
The breathless tone of this article obscures rather than illuminates its subject. It sounds as if the author describes a box with some particles bouncing around inside, and every time a particle bounces off the wall of the box the outside glows briefly indicating the location and intensity of the collision. The author is amazed that by repeatedly measuring this, you can draw inferences about what's going on in the box.
I can't see what's 'outrageous' about this. You need a lower surface which registers information about activity inside a higher-dimensional volume, some way to accurately read that information, and the time/patience to repeat the measurement many times. Isn't this how radar works, or feeling your way around a pitch-dark room using only the 2-dimensional surface of your hands (or shins)? We know from computer science that you can mathematically encode any structure of arbitrary dimension into a binary tree. One might as well ask how it's possible that our complex 3 dimensional world can be contained in the flat surface of a mirror, film strip, or camera sensor.
Indeed. A "violation of logic and geometry"? Reading the author's own description, it's hard to see what the big deal is:
"Put a box around any region of space (space-time, really, but I’m going to drop time throughout this essay for ease of visualization, as physicists often do). The holographic principle asserts that no matter what’s going on inside — from gas molecules pinging around to black holes colliding — you can decipher the entire contents of the box just by repeatedly measuring points on the surface."
Well, if we're bounding a region of space-time then, in a purely classical universe governed by deterministic ODEs or PDEs, the statement reduces to a triviality: having information about the boundary amounts to knowing all boundary conditions. Of course I understand that this isn't really the statement of the holographic principle, but the article's formulation is rather underwhelming.
It’s not just drawing inferences, though; it’s saying you can determine exactly what’s happening in the box. But the pigeonhole principle would suggest there’s far more possible states in the 3-dimension volume than can be unique expressed on the 2-dimensional surface. That’s the part that defies common sense.
A 3d volume can contain a number of states equal to it's 2d dimensional surface.
All the extra states you would assume the 3d volume can contain are actually a single state called Black Hole.
There is a limit to how much you can fill a 3d volume before it's a Black Hole basically.
This is by far the most succinct, understandable, correct, description of the holographic principle. I don’t know all articles don’t just lead with this.
Does that mean that all the black holes of a particular volume have the same interior? Or does it mean that the black hole state is the case where you can’t tell what the interior is?
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> a single state called Black Hole.
Quibble: A black hole still has a charge, mass, momentum vector, and spin vector. Which, yes, still vastly reduces the number of states needed on the surface of the enclosing container.
Is that result proven?
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That's the clearest description of how this dimension collapse via gravity thing works that I've ever heard.
I really think we live on the 3D event horizon of a 4D black hole, folks.
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> All the extra states you would assume the 3d volume can contain are actually a single state called Black Hole.
How can that be? Don't different black holes have different masses?
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2d observed over time is also 3d. 3d over time would be 4d.
You still have less information than if you could observe the full 3d spatial volume over time, because presumably you won't know in perfect detail qnd precision what all the particles are doing internally?
Or does it not work like this?
Is it any weirder than the fact that there is a bijection between the unit interval and the unit square?
Yeah because you can prove that any such bijection cannot be a topological homeomorphism, for example. So necessarily some "nice to have" properties must drop out.
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I'm not sure if I agree, but you certainly articulated it far more clearly than the original writer.
> That’s the part that defies common sense.
Does it? Isn't this basically the same as how cellular companies can figure out exactly where someone is calling from as long as their phone is pinging 3+ towers? Just connect the towers into a box and you have the same result.
Build an object out of legos and then reconstruct the shape of what you built by only looking at it from the sides and above.
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If the events inside the box can’t influence events outside the box directly, only by changing the surface of the box, it’s fair to say they don’t happen at all, and all that happens is the change of the surface.
If you're willing to use continuous dimensions, the surface is the same size as the volume; I don't see why the pigeonhole principle would present any problems.
Interestingly enough, I believe there is a result that space-filling curves cannot be one-to-one, but the implication there is just that, by virtue of the continuity of the one-dimensional curve, it contains more points than the two-dimensional space that it fills.
I remember thinking space filling curves were completely magical when I first learned about them. It feels like there should be some connection with holography although they don't get mentioned in the article.
I feel the most interesting implication of the enclosing surface being a complete description of the enclosed volume and its contents implies the contents don’t need to exist. If you think physics as the set of rules that applied to one full state of reality can completely give it’s next state, then having redundant representations is wasteful and the interior of the enclosed volume, as the interior of the black hole, might not need to exist, or to be represented at all. If you extend this to the full spacetime we inhabit, it all would be the encoded in its boundary and, therefore, only the boundary is needed.
> One might as well ask how it's possible that our complex 3 dimensional world can be contained in the flat surface of a mirror
You could ask, but that isn't possible, so the question is of dubious value. There are many states of the 3-dimensional world that will produce identical images in the surface of a mirror.
The metaphor as written is maybe bad, but the holographic idea really is outrageous and mindboggling. Because according to it the inside might as well be completely empty.
> Isn't this how radar works, or feeling your way around a pitch-dark room using only the 2-dimensional surface of your hands (or shins)?
> One might as well ask how it's possible that our complex 3 dimensional world can be contained in the flat surface of a mirror, film strip, or camera sensor.
No this is different. You can't classically encode an arbitrary 3D world on a camera sensor without losing information. It can only create a 2D projection of surfaces. It can't look inside opaque objects. Countless distinct 3D objects could correspond to one an the same 2D projection. There are actually many examples of optical illusions which show that a 2D image can be ambiguous with regards the 3D space it represents.
>"One might as well ask how it's possible that our complex 3 dimensional world can be contained in the flat surface of a mirror, film strip, or camera sensor."
Space filling curves with three dimensions could potentially provide us with some clues:
https://en.wikipedia.org/wiki/Z-order_curve
"In mathematical analysis and computer science, functions which are Z-order, Lebesgue curve, Morton space-filling curve,[1] Morton order or Morton code map multidimensional data to one dimension while preserving locality of the data points (two points close together in multidimensions with high probability lie also close together in Morton order)."
Also, to answer the question, potentially Octrees might be interesting to research:
https://mathworld.wolfram.com/Octree.html
"An octree is a rooted tree that represents a three-dimensional region by recursively subdividing it into eight congruent, axis-aligned cuboids, usually cubes (Meagher 1982, Samet 1990). Each internal node represents a spatial cell and has eight children, one for each octant of the cell. A tree leaf represents a cell that is not subdivided further."
But, since "a picture is worth one thousand words":
https://www.google.com/search?q=octree&udm=2
Another thing to consider is that according to the Knuth Transform (aka Left-Child Right-Sibling (LCRS) representation), apparently any k-ary tree (Octree = 8-ary tree) can apparently be converted into a Binary Tree without losing information:
https://en.wikipedia.org/wiki/Left-child_right-sibling_binar...
So, is there a 1D (or 2D) representation for 3D reality?
Well, I personally don't know!
What I do know is that a certain popular rock group sang the following lyric in one of their popular rock songs, back in 1999:
"Space may be the final frontier -- but it's made in a Hollywood basement..."
-The Red Hot Chili Peppers, "Californication"
So, you decide! :-)