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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.

Say your boundary is a spherical shell, with a radially constant electric field. Based only on those boundary conditions, you cannot know the distribution of charges inside the shell. It could be a point charge, a solid sphere, a smaller shell, etc.

So classically you cannot decipher the entire contents of the box just from its surface!

No. You are only counting the integral of the field on the surface, but if you take into account the orientation of the field on the surface, you can know the internal distribution of charges.

If the field is always radial, the charge can only be in the center, by symmetry. A different distribution of charges will not produce a radial field on the surface, even if it has the same integral.

You are correct if you mean a 2-sphere, but a spherical shell in space-time would be a 3-sphere, which gives quite a lot more information. To make things a bit simpler we could consider a 2-sphere × an interval -- the boundary of a 4-dimensional cylinder -- which is to say, your spherical shell at all points in time from t_0 to t_1, and also the entire interior of the spatial region at both the initial time t_0 and the final time t_1.

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.

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.

Those drop out in holography too — eg, information local within the interior can get smeared out across the surface.

That is exactly what I was wondering about in my other comment: is physical nonlocality somehow equivalent to topological noncontinuity?

I am not a physicist but i presume that is the hope of some who pursue holographic modes

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.

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.

Why stopping at four dimensions?

Oh sure. The parent universe could be the 4D event horizon of a 5D hole. It’s holes all the way up. If you draw little heads on them I guess they’re like round turtles so if this is true the turtles people weren’t as far off as we thought.

Since all the event horizons “touch” at N-1 dimensions I wonder if this is isomorphic with certain forms of string theory? Maybe there’s 11 levels of holes?

BTW this isn’t my idea. There’s papers on black hole cosmology. You can make the math work but no way to test it. Yet?

Legs, also little legs.

The people at Gillette are working on five.

That's old news. They solved five a while ago.

I heard they were researching up to 18, but most of the blades were curved and really short.

Is that result proven?

Depends on what hypotheses you are willing to accept, but, as far as I know, it is generally accepted by physicists: https://en.wikipedia.org/wiki/Bekenstein_bound

No.

That's a purely mathematical result, the top comment of this thread has a link to the paper

I don't think it has been experimentally verified or anything

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.

> 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.

> 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?

Well the density goes down the larger the black hole so the extra states are still missing.

For exactly how that works, I'm a Comp Sci not a Physicist.

> Well the density goes down the larger the black hole so the extra states are still missing.

Huh? If the correlation between density and mass is perfect, that means two things:

1. There is an upper limit on the mass of the black hole, set by the volume of the box containing it.

2. The black hole's possible mass and density occupy a one-dimensional space and not a two-dimensional space.

Now, it appears to be true that the (possibly nonlinear) correlation between mass and density is perfect; for any mass a black hole can have only one density. But that doesn't make the space of possibilities finite. It certainly won't make it into a single point. And in fact black holes appear to have three independent physical dimensions: their mass, their electric charge, and their spin.

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?

Classical understanding, yes. Black holes have no 'hair'. You can measure charge spin mass that's it

> There is a limit to how much you can fill a 3d volume before it's a Black Hole basically.

Which is exactly the part that "defies common sense" and is intersting

> 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.

Seems simple enough in many cases. But I think the difference is in the generality of it. Being able to do it for any interior arrangement of parts.

Agreed, that was what I was trying to say. You can build a lego structure that you can not infer by just looking at from the top, bottom and sides.

I'm not sure if I agree, but you certainly articulated it far more clearly than the original writer.

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.

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?

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.

> 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.

> 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.

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.

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.

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.

It is not outrageous. For example, conservation of energy tells you that if you observe the surface of a box, you can know things inside the box. If you observe how much energy enters or exits the box, you know the change of energy inside the box. And that's true independently of what is inside the box.

Inside the box, there may be one rat, a galaxy, a nuclear bomb, or whatever thing you can imagine, and still you know exactly the change in stored energy. It only depends on what happened on the surface of the box.

Also, when you solve a differential equation, you need the contour conditions. The contour completely determines what happens inside.

Sure but "total amount of energy evolution in the box" is much, much, lower-grained than "knowing exactly everything about the inside of the box"

>"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! :-)