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I've looked over the internet, and everyone seems to agree that the $4^\text{th}$ dimension is a chaotic place.(who can blame them? a $2$D being would think the $3$D world may look chaotic) but i wonder: what do $3$D figures look like when they're given this extra degree of freedom? They will seem to rotate oddly, a fact I mentioned in one of my questions, where the hypercube and other polychora rotate with oddness. My question is, what do some well-known objects, like the Klein bottle and torus, look like in $4$D?

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  • $\begingroup$ If you really want fun, going backwards, have a look at en.wikipedia.org/wiki/Flatland and read the novel and enjoy it ! $\endgroup$ Apr 28, 2017 at 12:31
  • $\begingroup$ The Klein bottle is intrinsically a 4-D shape, of course. $\endgroup$
    – Joffan
    Apr 28, 2017 at 12:56
  • $\begingroup$ @ClaudeLeibovici this flatland thing is very interesting! $\endgroup$ Apr 28, 2017 at 23:23
  • $\begingroup$ @Joffan it's already 4D!!!!???? $\endgroup$ Apr 28, 2017 at 23:24
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    $\begingroup$ @AlexanderDay yes, the "true" Klein bottle does not intersect itself. Only the 3-D projection of a Klein bottle does that. $\endgroup$
    – Joffan
    Apr 28, 2017 at 23:30

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I think you might have a better view of $3D$ objects if you lived in a $4D$ world. Think of a $2D$ shape, like a circle: if you were in a $2D$ environment, supposing you are outside the circle, you'd see it similar to a segment in front of you. If you had stereo view, the segment's midpoint would appear closer to you and the extremities farthest. But, since you live in a $3D$ world, you have a much clearer understanding of the overall shape and structure of the circle. I guess something similar would happen if you had a chance to look at, say, a cube or a torus in $4D$. Moreover some shapes that actually cannot exist in $3$ dimensions and can only have projective representation, like the Klein bottle you mention, may be available in $4$. Just as a cube cannot really exist in $2D$, it may only be represented under some perspective.

EDIT: I can better explain why, in a $4D$ world, $3D$ objects could be seen more clearly. The process of viewing, in any number of dimensions, consists in a projection on the hypersurface of a virtual hypersphere centered at the observer's position. You are in fact "losing" one dimension. This is why in our environment, up to $2D$-figures can be seen without the need of overlapping parts of them. Obviously you have to be correctly positioned to see them at best (you can look at a circle with your eyes lying on its same plane, so you wouldn't benefit from the $3^{rd}$ dimension you have). In $4D$ you could make sure to look a up-to-$3D$ object from a perspective that lets you see it entirely, with no parts of it covering other parts, exept for the opposide "faces" of what you are seeing (even in our world you cannot see at the same time both sides of a playing card). You have to make sure that the $3D$ object is perpendicular to the line joining you and it in the $4D$ world. It "lacks" one dimension of the $4$ available, and this dimension has to be aligned with the projection direction, so that the figure essentially lies entirely on the hyperplane it's projected on.

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  • $\begingroup$ because a 3D being sees 2D, and a 2D being sees 1D (from the flatland movie), and a 1D being sees a point, and a 0D being should be blind $\endgroup$ Apr 29, 2017 at 21:35
  • $\begingroup$ A 1D being should be able to see 2 points, one at its "left" and one at its "right" (or one "in front" and one "behind", this names wouldn't really make sense in such a world). He probably wouldn't be able to assess the distance of other 1D beings, unless you think something like measuring the travel time of a light pulse back and forth... @Alexander $\endgroup$
    – lesath82
    Apr 29, 2017 at 21:45
  • $\begingroup$ since they cannot triangulate the position, being shackled to the confinements of two directions, and they cannot see anything outside their realm. $\endgroup$ Apr 29, 2017 at 21:58
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There are multiple ways to come at your question:

  1. If you were your ordinary three-dimensional self lifted into four-dimensional space, and your insides were somehow prevented from blowing out like a soap film spanning a wire in space on a windy day, there are a couple of possibilities:

    (a) If your retinas could receive light from a four-dimensional cone, then familiar spatial objects would look more-or-less the way they do in three-dimensional space, though there would be wonky parallax (proposed technical term) as you moved in their vicinity.

    (b) If your retinas could only receive light from the three-dimensional slice of four-space that contains your eyeballs, then familiar spatial objects would, from most perspectives, look sliced into planar sections, analogous to the description of A. Sphere in Flatland.

  2. If instead you were a four-dimensional being whose eyes had three-dimensional retinas, then spatial objects would look "flat" in the same sense that a plane curve floating in space looks flat.

    ::rummage:: ::rummage::

    Compare this image from Definition of a shadow in space, and how to derive a shadow for a given shape; a four-dimensional being would see a torus as looking roughly analogous to the way we perceive its shadow on a plane.

An object and its shadow

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