"Flatland" nitpick: How can A. Square, even in Spaceland, see anything but line segment?

[BrainGlutton]

In Flatland, by Edwin Abbott Abbott, the Sphere tries to explain 3-dimensional space to A. Square. He just doesn’t get it, so the Sphere pops A. Square out of Flatland, into Spaceland, and all of a sudden he can see in 3D – well, that is, he sees two-dimensional images from which 3D space can be inferred, as you and I do. (We have the added advantage of stereoscopic vision, but it’s still just 2D images we see.)

But, how is that possible? A. Square is still a two-dimensional creature with a two-dimensional eye that, as has already been explained, is designed to perceive nothing but line segments, i.e., to see in one dimension – just as a Linelander can see nothing but points to either side. It seems to me all A. Square would see in Spaceland is a jumble of incomprehensible 1D images.

[RealityChuck]

I believe he describes how things look in his world; you determine the shape of the line segment by the shading. A line at 90 degrees to you would be one shade, but in general things don’t approach that way, so if it were in any other configuration, you’d see one end darker than the other. A triangle would show two line segments darker on one end. A circle would have a more gradual shading, etc. While there are problems with anything that had a single side facing you (say the hypotenuse of a triangle), they were finessed away.

[BrainGlutton]

@RealityChuck

I believe he describes how things look in his world; you determine the shape of the line segment by the shading. A line at 90 degrees to you would be one shade, but in general things don’t approach that way, so if it were in any other configuration, you’d see one end darker than the other. A triangle would show two line segments darker on one end. A circle would have a more gradual shading, etc. While there are problems with anything that had a single side facing you (say the hypotenuse of a triangle), they were finessed away.

Yes – but how would things look any differently to A. Square when he leaves his world? His environment has changed, but he has not – he still has a two-dimensional eye.

[Shawn1767]

In Flatland, this has been my contention: while in Flatland, the Flatlanders would see nothing but line segments because they are seeing everything edge on. A circle would look like the edge of a circle and a square would look like an edge of a square. In Cosmos, Carl Sagan uses this analogy to say that if an apple appeared and intersected the plane of Flatland, the Flatlanders would see increasingly larger cross sections of the apple. I say that’s not possible, they would merely see the edge of the cross section that is intersecting the plane at the time. The edge lines would go from small to large then disappear.

However, if the square were knocked into a third dimension, it would give him a different perspective, however, would his 2-dimensional brain be able to understand what he was seeing?

[BrainGlutton]

@Shawn1767

In Flatland, this has been my contention: while in Flatland, the Flatlanders would see nothing but line segments because they are seeing everything edge on. A circle would look like the edge of a circle and a square would look like an edge of a square. In Cosmos, Carl Sagan uses this analogy to say that if an apple appeared and intersected the plane of Flatland, the Flatlanders would see increasingly larger cross sections of the apple. I say that’s not possible, they would merely see the edge of the cross section that is intersecting the plane at the time. The edge lines would go from small to large then disappear.

That’s what Sagan was saying.

@Shawn1767

However, if the square were knocked into a third dimension, it would give him a different perspective, however, would his 2-dimensional brain be able to understand what he was seeing?

The main problem is not his brain, it is his eye. His retina-analogue is a semicircle, not a hemisphere like your retina or mine. How can it get any images that are more than line segments?

[Peremensoe]

@BrainGlutton

The main problem is not his brain, it is his eye.

Is the text unambiguous that he is seeing an extra dimension?

[BrainGlutton]

@spark240

@BrainGlutton

The main problem is not his brain, it is his eye.

Is the text unambiguous that he is seeing an extra dimension?

Yes.

“Fool! Madman! Irregular!” I exclaimed; “never will I release thee;
thou shalt pay the penalty of thine impostures.”

“Ha! Is it come to this?” thundered the Stranger: “then meet your
fate: out of your Plane you go. Once, twice, thrice! 'Tis done!”

SECTION 18 How I came to Spaceland, and what I saw there
An unspeakable horror seized me. There was a darkness; then a dizzy,
sickening sensation of sight that was not like seeing; I saw a Line
that was no Line; Space that was not Space: I was myself, and not
myself. When I could find voice, I shrieked loud in agony, “Either
this is madness or it is Hell.” “It is neither,” calmly replied the
voice of the Sphere, “it is Knowledge; it is Three Dimensions: open
your eye once again and try to look steadily.”

I looked, and, behold, a new world! There stood before me, visibly
incorporate, all that I had before inferred, conjectured, dreamed, of
perfect Circular beauty. What seemed the centre of the Stranger’s form
lay open to my view: yet I could see no heart, lungs, nor arteries,
only a beautiful harmonious Something–for which I had no words; but
you, my Readers in Spaceland, would call it the surface of the Sphere.

Prostrating myself mentally before my Guide, I cried, “How is it, O
divine ideal of consummate loveliness and wisdom that I see thy inside,
and yet cannot discern thy heart, thy lungs, thy arteries, thy liver?”
“What you think you see, you see not,” he replied; “it is not giving to
you, nor to any other Being, to behold my internal parts. I am of a
different order of Beings from those in Flatland. Were I a Circle, you
could discern my intestines, but I am a Being, composed as I told you
before, of many Circles, the Many in the One, called in this country a
Sphere. And, just as the outside of a Cube is a Square, so the outside
of a Sphere represents the appearance of a Circle.”

Bewildered though I was by my Teacher’s enigmatic utterance, I no
longer chafed against it, but worshipped him in silent adoration. He
continued, with more mildness in his voice. “Distress not yourself if
you cannot at first understand the deeper mysteries of Spaceland. By
degrees they will dawn upon you. Let us begin by casting back a glance
at the region whence you came. Return with me a while to the plains of
Flatland and I will shew you that which you have often reasoned and
thought about, but never seen with the sense of sight–a visible
angle.” “Impossible!” I cried; but, the Sphere leading the way, I
followed as if in a dream, till once more his voice arrested me: “Look
yonder, and behold your own Pentagonal house, and all its inmates.”

I looked below, and saw with my physical eye all that domestic
individuality which I had hitherto merely inferred with the
understanding. And how poor and shadowy was the inferred conjecture in
comparison with the reality which I now behold! My four Sons calmly
asleep in the North-Western rooms, my two orphan Grandsons to the
South; the Servants, the Butler, my Daughter, all in their several
apartments. Only my affectionate Wife, alarmed by my continued
absence, had quitted her room and was roving up and down in the Hall,
anxiously awaiting my return. Also the Page, aroused by my cries, had
left his room, and under pretext of ascertaining whether I had fallen
somewhere in a faint, was prying into the cabinet in my study. All
this I could now SEE, not merely infer; and as we came nearer and
nearer, I could discern even the contents of my cabinet, and the two
chests of gold, and the tablets of which the Sphere had made mention.

[Alessan]

Does A. Square have binocular vision?

[BrainGlutton]

@Alessan

Does A. Square have binocular vision?

No; but that’s not what I’m talking about. You and I have “three-dimensional” vision in the form of a stereoscopic image the brain composes out of two two-dimensional images, one from each eye – but the result is still a two-dimensional image, i.e., not what a four-dimensional being would see. If A. Square had two eyes, the resulting images he perceived would still be line segments, just with some depth added.

[BrainGlutton]

@BrainGlutton

@Alessan

Does A. Square have binocular vision?

No; but that’s not what I’m talking about. You and I have “three-dimensional” vision in the form of a stereoscopic image the brain composes out of two two-dimensional images, one from each eye – but the result is still a two-dimensional image, i.e., not what a four-dimensional being would see. If A. Square had two eyes, the resulting images he perceived would still be line segments, just with some depth added.

It’s like this: Close one eye. You’ll notice you lose depth perception – but you still have some depth perception, because it comes not only from the stereoscopic effect but from all the subconscious ways you have learned to judge the relative distance of objects from relative size, shading and shadows, perspective, etc. That is why it is possible to paint a picture that looks almost three-dimensional with the proper use of perspective; it is also what makes a lot of optical-illusion puzzles possible.

However, all you’re seeing, with one eye or two, is two-dimensional images.

[BigT]

I noticed this the first time I read it. But I later encountered parts that seem to indicate that the sphere just kept showing him stuff until he got it mathematically, and the he let him FEEL him, which would include feeling from the inside, and thus give him 2D images to work with.

And it seemed that A. Square was picked specifically because he was a a great and gifted mathematician, and would be able to handle it.

So I specifically disregarded anything that said he actually sees in 2D, believing it to be metaphor.