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We consciously know and feel the first three dimensions and with some thinking time as well. But according to literature like String theory etc., we have many many more dimensions. We can readily express them in paper. But why can't we realize them? Why can't we experience them like the first three dimensions?

PS. I was pondering on this when I saw the movie Interstellar where even the fourth dimension 'time' is expressed using the first three dimensions for Cooper. PPS. The first three dimesnions being the X, Y and Z axes.

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    $\begingroup$ Most people aren't even very good with three dimensions. Why not? That's more of a question for biology than physics. Why do humans not need detailed three dimensional mental maps for survival? Probably because we are normally living on a flat surface and understanding the structure of the third dimension above our heads doesn't do much for us, so nature didn't waste neural cells on it. And why not four dimensions? Because it does even less for survival to be able to map objects that do not exist in our immediate reality. $\endgroup$
    – CuriousOne
    Jun 10, 2015 at 9:22
  • $\begingroup$ @CuriousOne So you're saying that survival determines how much we want to perceive right? I agree $\endgroup$ Jun 10, 2015 at 9:39
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    $\begingroup$ Its always my favorite sentence in mathematics that an n dimensional shape is the "shadow" of an n+1 dimensional shape. Ie like that of a point/line/square/cube/tesseract/general hypercube $\endgroup$
    – Triatticus
    Jun 10, 2015 at 11:34
  • $\begingroup$ Wow! Never knew that tesseract was actually something in the academic world. Shadow analogy seems to be working fine till 4th dimension. After that, my mind cannot comprehend $\endgroup$ Jun 10, 2015 at 12:16
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    $\begingroup$ I think it is not impossible to visual a 4th (or even more) dimensions in the case of simpler forms, we only have to encode the extra cordinates into some other attribute (for example, into the color of the pixels). Furthermore, it would require a little bit of learning (for us) to understand such images. In the case of descriptive geometry, we've already learned it long, this is why we can "extrapolate" the 2d image into a 3d object. $\endgroup$
    – peterh
    Dec 2, 2016 at 13:19

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Why can't we experience them like the first three dimensions?

The usual explanation is that these additional dimensions, if they exist, are tightly curled up or compacted. Humans can't move around in them like we can move through the three "normal" spatial dimensions we are familiar with.

Why are we not able to visualize Dimensions beyond 3

Mostly because as our brains develop they don't experience sensory input corresponding to movement in more than three spatial dimensions. Therefore we don't build up the mental structures needed to intuitively comprehend more spatial dimensions.

Also our major senses, our eyes, are intrinsically two dimensional. Information about the third dimension has to be synthesized in our brains. Proprioception is, I suppose, provides three dimensional information - but we rely on that less.

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  • $\begingroup$ Can you please elaborate on 'curled up or compacted'? They are dimensions after all, they should exist endlessly right? Like the space is infinite $\endgroup$ Jun 10, 2015 at 9:36
  • $\begingroup$ @SrikanthGuhan Just so as an example of being curled up a circle can be compact and endless. Now what he means to say is that the dimensions are way too odd for us interact directly with them. $\endgroup$ Jun 10, 2015 at 9:41
  • $\begingroup$ The operative term in the above answer is "if they exist". Nobody knows if that is the case. For myself, all I am going to say is that I take an extremely dim view of "string theory", and leave it at that. $\endgroup$
    – Pirx
    Dec 1, 2016 at 21:51
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First of all, we can project four-dimensional objects onto three dimensions, and then further project those onto 3 dimensions. See https://en.wikipedia.org/wiki/Tesseract The picture shown in this article is itself a 2D image, but you can surely make a 3D version of it.

Second, there is a pretty standard way of graphically representing arbitrary $n$-dimensional real vectors — using a bar chart. The discrete $x$ axis in this case corresponds to coordinate indices, while the continuous $y$ axis represents the values of corresponding components. This would be a picture for a vector in $\mathbb{R}^6$:

enter image description here

You may argue that this representation, unlike drawing an arrow, is very sensitive to basis changes, but this would not be fair, as we are simply more used to arrow rotations.

Importantly, the suggested representation is incredibly powerful in the sense that it immediately allows one to trace the connection between finite- and infinite-dimensional vector spaces. Indeed, if we agree to put all the $n$ components of a vector on the interval $\overline{AB}$ of a fixed length along the $x$ axis, it will become clear how in the limit $n\to\infty$ a finite-dimensional vector turns into a continuous function. In this case, $x$ simply becomes a continuous index! Furthermore, this illustration naturally explains the definition of scalar product of two real functions, $\langle f|g\rangle=\int_{A}^B f(x) g(x)\operatorname{d}x$ it is no more than the generalized component-wise product.

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Actually I recently published a paper on Mathematical Intelligence trying to answering that question

Why are we not able to see beyond three dimensions?

Abstract:

This is perhaps a philosophical question rather than a mathematical one, we do not expect to give a full answer, even though we hope to clarify some ideas. In addition, we would like to provide a new perspective on the subject. We will find curious analogies with the way we perceive color and make some imaginary experiments showing that, even living imprisoned in three dimensions it could be different.

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I don't know why we would consider that other dimensions are curled up and so small that they aren't observable or detectable. We don't think of time like that. It exists at every point in space but doesn't occupy any space. My thought is that 3 dimensions are all that are needed to describe/identify physical objects and their relationship to other objects of mass. Time is a dimension needed to differentiate those objects that are not static. So other dimensions can easily exist at every point in space and time but are not physical like maybe electromagnetic force or gravity os possibly a spirit realm.

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  • $\begingroup$ A "spirit realm"? $\endgroup$ Oct 31, 2021 at 8:34

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