"Benford-FourthDimension" - читать интересную книгу автора (Benford Gregory)

The reality of three dimensions we take for granted, but for us, what is the
reality of two dimensions? Would flatlanders have physical presence in our world
-- that is, could we perceive a two-dimensional universe embedded in our own?
Could we yank them up into our world?

Flatlanders could be as immaterial as shadows, mere patterns in our view. If an
isosceles triangle soldier cut your throat it would not hurt. Abbott did not
consider this in his first edition, but in the second he says that A Square
eventually believes that flatlanders have a small but real height in our
universe. A Square discusses this with the ruler of Flatland:

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I tried to prove to him that he was "high," as well as long and broad, although
he did not know it. But what was his reply? "You say I am 'high'; measure my
'highness' and I will believe you." What could I do? I met his challenge!

If flatlanders were even quite thick, they would not be able to tell, if in that
direction they had no ability to move or did not vary. Height as a concept would
lie beyond their knowable range. Or if they did vary in height, but could not
directly see this, they might ascribe the differences to qualitative features
like charisma or character or "presence." There would be rather mysterious
forces at work in their world, the Platonic shadows of a higher, finer reality.

If a flatlander soldier of genuine physical thickness attacked, it would cut us
like a knife. Otherwise, it could not impinge upon us. We would remain oblivious
to all events in the lesser dimensions.

In a sense, a truly two-dimensional flatlander faces a similar problem if it
tries to digest food. A simple alimentary canal from stem to stem of, say, a
circle would bisect it. To keep itself intact, a circle would have to digest by
enclosing whatever it used for food in pockets, opening one and passing food to
the next like a series of locks in a canal, until eventually it excreted at the
far end.

This is typical of the problems engaged by thinking in another dimension. Not
until 1910 did artists respond to non-Euclidean spaces, with Cubism and its
theories. Mute image and poetic metaphor, they said, were ways of perceiving
what scientists could only describe in abstractions and analogies.

They were right, and many, including Picasso and Braque, struggled with the
problem. Looking downward at lower dimensions is easy. Looking up strains us.

Visualizing the fourth dimension preoccupied both mists and geometers. A cube in
4D is called a tesseract. One way to think of it is to open a cubical cardboard
box and look in. By perspective, you see the far end as a square. Diagonals (the
cube edges) lead to the outer "comers" of a larger square -- the cube face
you're looking through. Now go to a 4D analogy. A hypercube is one small cube,
sitting in the middle of a large cube, connected to it by diagonals. Or rather,
that is how it would look to us, lowly 3D folk.