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2, 3, & 4-D Space: The square, cube, & tesseract.
To a hypothetical 2-D being, the 3rd dimension is that direction in which he cannot point, and for us
the 4th dimension is that direction in which we cannot point. The eyes, or a camera, of a hypothetical 2-D being cannot record a 3-D object, simply because the only light rays which can be focussed by a 2-D camera are those moving in the plane of the paper. Imagine a ray passing through the paper from the 3rd dimension -- it would intersect the paper only at a point and so could not be focussed. By analogy, we cannot see or photograph any 4-D object.
To a hypothetical 2-D being, living in the plane of this paper, the figure below is how we could
explain to him what a cube is: the outer square is in his 2-D world but the inner square is in a parallel 2-D world lying above (or below) his 2-D world, remote from it in the 3rd dimension.
The six enclosures are all squares and all the same size, but are of course distorted by the
perspective of the 2-D drawing.
We can visualise this in 3-D, as a cube - just stare at it for 10 seconds !
A further explanation is that the 2-D figure below must be imagined to have its 4 outer squares
bent up (or down) at right angles out of the 2-D plane of the paper and their 4 furthest edges will then form another square but entirely remote in 3-D space above (or below) the central square.
Similarly, by analogy, 3-D beings (ourselves) can visualise a tesseract, which is the 4-D analogue
of a cube. See the tesseract figure below and imagine a cube is forming inwards into 4-D space from each of the faces of the large outer 3-D cube, and these six new cubes will then form another smaller cube (having 6 faces) totally remote in the 4th dimension. But in a tesseract, all angles are right angles and all 8 cubes it contains are the same size! The need for a perspective drawing distorts the angles and sizes. This is a drawing on 2-D paper of a 4-D tesseract:
Similarly, a 5D analogue of a tesseract is obtained by imagining a tesseract going into 5D space
from each cube of the 4D tesseract, and the far cubes of such a structure then forms another tesseract totally remote in 5D space.
There is no name for such a structure and so the first person to draw one can have the honour of
naming it ! |
