Flat Land

A world or domain existing entirely in two dimensions.

From a work by (Edwin A. Abbott 1838-1926)
   A romance of many dimensions
   With Illustrations by the Author, A SQUARE
One of many copies online: http://downlode.org/etext/flatland/

--- Closing words of "Flatland"

That is the hope of my brighter moments. Alas, it is not always so. Heavily weighs on me at times the burdensome reflection that I cannot honestly say I am confident as to the exact shape of the once-seen, oft-regretted Cube ; and in my nightly visions the mysterious precept, " Upward, not Northward," haunts me like a soul-devouring Sphinx. It is part of the martyrdom which I endure for the cause of the Truth that there are seasons of mental weakness, when Cubes and Spheres flit away into the background of scarce-possible existences ; when the Land of Three Dimensions seems almost as visionary as the Land of One or None ; nay, when even this hard wall that bars me from my freedom, these very tablets on which I am writing, and all the substantial realities of Flatland itself, appear no better than the ofispring of a diseased imagination of the baseless fabric of a dream.

I was introduced to the inhabitants of FlatLand as a student in high school algebra (early '60s). The reactions of those in the class spanned a spectrum of "huh?" to "AhHa!" and pretty much everything in between. I have since been haunted by the thought of encountering a being whose world was rendered as 4 physical dimensions or for whom the behavior of time was that of "just another direction" like "up" in our world.

Even so, I have found the metaphor useful as a thinking tool when dealing with people for whom some concept is completely alien.

I find myself, occasionally, having to explain, nearly always in metaphor, some concept of programming or communications to someone whose experience has no foundation on which to build the needed understanding.

Likewise I often find myself trying, vainly, to grasp some concept for which my education ill prepares me. Sometimes I'm able to construct a question or establish a metaphor that then serves as a vehicle for communication of such a concept.

Constructing "LearningMetaphors?" for one's own benefit is something of a trick, since you're trying to establish a scenario to illustrate a framework, within a framework you already don't understand.

This is exacerbated by the (frequent) impatience of an elected source of such new knowledge, resulting in such non-communications as "Swizzler & Feebusch wrote the definitive work on this: RtFm." Another tricky bit, in this same vein, is convincing one's elected source that, inability to understand Swizzler & Feebusch notwithstanding, one is not actually a moron, and just needs a better metaphor (that word again).

So, does any of this resonate with others here?

-- GarryHamilton

It's a good book to know about. Most any book that talks about the 4th dimensions references FlatLand. The parallel is seeing how 2d would perceive 3d (in slices), then extrapolating that to see how 3d (us) would see 4d (also "slices") -- LayneThomas

Don't confuse living in a 3D world with seeing in 3D. Because we don't. Humans see a 2D surface embedded in 3D, usually called 2+1D or 2 1/2D. I wish we could see in 3D, it might even make it easier to think in 4D. And 4 is the magic number of dimensions.

And some humans are monocular. To them, every movie is as real as a 3D movie to binocular folks.

It doesn't make a difference. Monocular people see almost the same thing as binocular people, just a bit more flattened because they miss out on some depth cues. The differences between 2D, 2+1D, 3D are far, far more significant than between monocular and binocular.

The raw images from our eyes differ from the holographic perception created in our minds. We would perceive a 4d HyperSphere? passing through 3d space as a sphere growing then shrinking in size - at least intellectually. Visualizing a rotating HyperCube? is difficult, and FlatLand helps show how 3d beings could think 4th dimensionally. FlatLand is the canonical 4th dimension metaphor. -- LayneThomas

Holographic perception? What in bloody freezing hells are you talking about?!

At the lowest level, our eyes provide two 2D FLAT surfaces. These images are then processed into a single 2D CURVED surface embedded in 3-space with the following constraint: every ray starting at the origin intersects the surface once and only once. This is called 2+1D.

The fact is, we do NOT think in 3D. If human beings were capable of visualizing in 3D then we would be able to visualize the backside of an object at the same time as the front side, at the same time as every other side, at the same time as the INside.

Have you got that? We do NOT think in 3D! Is that really so hard to understand??

The reason why visualizing 4D is impossible isn't because it's a step beyond 3D. No, it's because it's a DIMENSIONAL step beyond what we can visualize. We perceive 2D and our minds reconstruct 2+1D. But taking the dimensional step up of going from 2+1D to 3D is something our minds are simply incapable of. If we could visualize 3D then pretty much automatically, we would be able to visualize 3+1D, but again this would NOT be 4D.

The fact is, human beings are NOT able to visualize or imagine higher dimensions than what they see. We see 2D and that's exactly what we can visualize and imagine. And that's IT. No amount of empty blathering is going to make even a single person on this entire planet able to visualize higher dimensions.

Again, we do NOT, can NOT, and NEVER WILL be able to visualize or in any way, shape or form, think in 3 dimensions. So 3+1D is completely out of the question!

(Deducing some of the properties of 4D by visualizing choice projections isn't anywhere close to visualizing or thinking in 4D. In fact, you lose so much information in the projection that it's ridiculous.)

Richard, I am intrigued by this assertion that we cannot visualize in 3D. I believe I understand your point that what we see is always 2D, and I accept that. Do you agree, however, that our internal modelling is in 3D? To me the term "visualization" is more involved with the manipulation of that internal model than with the specific concept of vision. It is perhaps an unfortunate use of the term, but "manipulate" no longer simply means "to adjust with the hands" (this may be a bad example, please don't be distracted by it - you can probably find a better one of your own anyway). Similarly "visualization" does not always mean "produce in one's mind what one would see". My usage of the term "visualization" is completely compatible with 3D because it's the internal model I'm dealing with, not the specific resulting image. Does that make sense?

No, I don't agree with anything you say. If the term "model" refers to something completely abstract then saying it is 3D is BS. If it is concrete enough to be a visualization in your head ("in" consciousness) then it's simply false that it's 3D. So take your pick: BS or false.

A disconnected series of 2D projections doesn't add up to a single 3D model. And even if it did, it wouldn't add up to a 3D model in your head. Rather, the 3D model would be abstract.

The number of times my mind stretched enough to visualize something in 3D can be counted on the fingers of one hand. Missing most of its fingers.

It's unclear to me how your inabilities allow you to assert impossibility. The dichotomy you create leads me to suspect that I don't know what you mean by "completely abstract". I have in my mind a model of a coffee mug. It's not a disconnected series of 2D projections, it is a single, coherent model. I claim that my model is sufficiently complete to allow me to make complex predictions about things that happen in the real world. I thereby claim that the model is 3D. If you don't agree with me then I suggest we make this science rather than philosophy. Can you suggest an experiment whereby we get one outcome if I can visualize in 3D and another if I can't? If you can't suggest such an experiment then I stand by my claim based on my introspection and abilities that I can and do think in 3D.

Sigh. What do you mean by "model"? And what do you mean by "in my mind"?

Can you see in your mind simultaneously all sides of the coffee mug, without having to rotate, move it, or change it in any way? Does your model have all of the symmetries of the real object at all times?

Firstly, do you really think that all of that is necessary to claim that I can "visualise" in 3D? Secondly, yes, I can, and it does.

Yes, it is absolutely necessary.

If a spoon is in the mug, can you see every detail of the spoon as well as the outside surface and the inside surface of the mug? Simultaneously, with no rotation, translation, moving or change of any kind?

Is the mug transparent, translucent or solid? What colour is it?

Take a ray from your POV through the middle of the mug and the spoon. The only distinguishing factor between these points is their depth, they are at the exact same horizontal and vertical position in your visual field. How many of the points along that ray do you see simultaneously? How many are solid? What colours do they have? How many colours can you perceive simultaneously at the exact same point in your visual field?

You see, I suspect that by "model" you mean an incomplete wireframe diagram or a superimposed translucent cloud. In both cases, you have a single-valued discontinuous depth map as opposed to the multi-valued continuous depth map of a real object.

The problem is that while models aren't completely abstract, they aren't first-class visualizations either. They're strictly second-class and don't have all of the properties of first-class visualizations. Properties like continuity and completeness. 3D models aren't real 3D mental objects. They're heavily hacked up optimizations. And it's not possible to squeeze any more optimization to create a 4D model.

So, we see in 2D, we visualize in 2+1D, we model in 3D, and that's it. And we can only say that by using an extremely narrowly defined meaning of model, one that falls far, far short of what most people expect.

Adding to this, note that under some circumstances you can see one object directly behind another, thanks to the different views your eyes give you. But most if not all people have a difficult time understanding such sensory data; if you set it up on a stereoscope, it breaks the illusion of depth. That strongly suggests our models of reality can't handle two points in the same line of sight, so are strictly 2+1D.

Thank you for that explanation. It is clear to me that you use the term "visualise" in a sense at odds with how I use it, and how I expect most people use it. You use the term very narrowly, not going beyond the concept of "to picture explicitly", and while that is a perfectly valid point of view, in my experience it doesn't match common usage. This means that to use it without explaining it first is to court misunderstanding.

I also feel that people can "model" 3D and 4D quite well. In my case I can model 4D well enough to make correct predictions in non-intuitive cases. This is in part due to my training as a mathematician specializing in low-dimensional topology. Again, I feel that your bald statements along the lines of

"...taking the dimensional step up of going from 2+1D to 3D is something our minds are simply incapable of."
"We do NOT think in 3D!"

are misleading. Whether you mean it to be deliberately so is open to speculation. I wouldn't expect that to be the case, but I would also not expect you to misunderstand the implications of your words, so I'm somewhat at a loss. I feel that I do think in 3D, so I refute your assertions soundly with the robust rejoinder of "Piffle!"

Indeed. It's obvious they haven't read the book, thereby missing the entire point of the book - and the lesson.

Flatland doesn't even remotely suggest how to think in 4D. It tells us how a 3D object might look to a 2D creature, and so by analogy how a 4D object might look to a 3D creature - but not how it should look to a 4D creature. Also, if you talk about the entire point, remember the book is about more than just geometry.

[Forget the book. The book doesn't teach you how to model in 4D. All it does is provide some shmarmy pap so that a bunch of idiots can go around saying "see, see, all you have to do is analogize 3D-4D to 2D-3D". The book provides the barest rudiments for laboriously constructing an understanding of 4D on your own, and doesn't acknowledge any of the difficulties of modeling in 4D. It's disgusting is what it is.

Flatland isn't disgusting; in my opinion, at least, it's a good book. It doesn't give any insight into modelling 4D and it wasn't really meant to - A. Square can only see 3D once removed from the plane, and nobody else can. I don't know where people got the idea that the book was about seeing in higher dimensions, maybe they phased out the whole "religion of space" thing.

Now to respond to the intelligent contributor:

Most people use the term visualize to mean "to picture explicitly". The problem is that they believe modeling to be the same thing as visualizing. They don't recognize the difference between a hacked up and almost completely abstract model and a concrete and vivid image of the same thing. As proof of this, I advance the fact that there is no commonly used term that means "to picture explicitly". In fact, visualize is the closest term that means this. Don't mistake incomprehension of a concept for misunderstanding of a term.

Where it isn't completely abstract, most people's thinking is limited by their ability to visualize. Especially thinking about geometric concepts. The term "thinking" itself usually implies an effortless and uncomplicated activity; a natural and untrained ability. Given what you've said about modeling in 4D then even if I were to accept that this is modeling, the extensive and advanced training required knocks it out of consideration for most people.

Finally, keep in mind that when I say above that models are 3D, I'm gritting my teeth and only grudgingly accepting this for the sake of argument, for the sake of differentiating models from visualizations. I don't consider models to be genuinely 3D objects, not for a single moment.]

It is the analogy that is important. There are plenty of books on 4D thinking - FlatLand is referenced by virtually all of them. I'm not willing to debate whether I can think in 3D or perceive 4D - I know I can, and if someone else can't, that's their problem.

FlatLand also had some nice social commentary for those who were too locked into certain modes of thinking. I believe the original intent was a critique of the royal family, but it comments just as well on those who refuse to think beyond their mental locks.

Try actually reading the book before dismissing it
I think we, Richard and I, have found our crux point of disagreement. Where I come from originally, and where I live now, people do not generally use "visualise" to mean "to picture explicitly". They have a much weaker intent. Now having understood your point more clearly I can interpret your original comments. I still think they are too strong, but I can at least appreciate the point you are making. In your sense perhaps no one can visualize 3D. Perhaps that is evidence that your definition is so limited as not to be useful.

For what it's worth, the book does provide effective ways of manipulating 4D "objects" and predicting what the effects will be. It can also help people stop and take stock when their audience is clearly just not getting the point. Analogies and other similar tools and techniques are essential sometimes - this book can be useful if only to teach the use of them.

I haven't here said exactly what I wanted to say, but it's late here and I need to go - I don't have time to word-craft it properly. I hope it conveys the intent.

Richard has very specific opinions on the meanings of words. See the discussion of "representation" on HtmlSucks for another example. (Note: this is not meant as a criticism.)

There are mathematicians who are trained to "visualize 3d", by their study of geometry, and even the most abstract course in differential geometry, will still use 3d visualizations to drive the point home. There are also sculptors who have an amazing ability to think about their creation in 3d, and act upon it. As a mathematician I was always fascinated to see a sculptor in action, especially when cutting stone. Their precision is astonishing, and their actions are irreversible. They do have a pretty strong claim that they do see their work of art in 3d, before even the first cut is made. -- Costin

Sculptors and mathematicians have an understanding of how certain objects work in 3D by keeping several 2+1D projections in mind simultaneously. They know what both the front and back of something should look like, and can slide continuously from one to another, but still can't picture the full assembly in full detail all at once.

As I read through the above ... discussion ... of 2D/3D visualization, I found myself wondering, "what's the big deal?" with regard to visualizing 3D. Then I remembered that I didn't always have this ability. Some several years ago I learned it through deliberate exercise and practice. It didn't come naturally to me, although it does to some.

My brother (a gymnast in his youth) has no difficulty with 3D visualization, and had trouble grasping that not everyone can do it. Interestingly, he was about a year late (in childhood) in beginning to talk. Everyone thought he was handicapped in some way. It rapidly became clear that this was not the case.

I have developed it, but with me it's deliberate. On the other hand, I have no trouble "seeing" things in front of other people, hundreds of miles away, while talking them through a sticky problem on the phone. I remember being surprised that not everyone could do that.

My daughter is able to draw whatever she sees. She doesn't understand why I can't. My son instinctively grasps subtle nuances of animation and composition that elude me. I play chess with people who have to be patient with my inability to "see" several alternative board positions dozens of moves away. My ability to project future positions only goes out about 5 or 6 moves.

And yet I can mentally simulate CPU register and memory states, relationships between buffers, protocol timings, and other digital stuff; temporal sequencing and the forensic analysis of event streams is everyday grist in my mill.

I have long since given up trying to estimate the capabilities of others based on my own limitations.

I'm only guessing here, but I would imagine that there exist among pilots and submariners many for whom 3D is something approaching normal, even natural.

But 4D - that's something I really have to work at.

-- GarryHamilton

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