I got some helpful comments to my previous post from others who had struggled with the same concept. Another interesting thought is that you could prove* that a four-dimensional (4D) 'cube' would, if looked at from the correct angle, look like a normal 3D cube, as follows:
1. First, imagine you are looking (with one eye only) at what appears to be a zero-D object, i.e. a point (which is not infinitely distant). The only way to tell whether you are looking at a zero-D object or a 1D object (a line) that is pointing straight at you is to move your head (in any direction except directly towards or away from the point) and look at it from a different angle. If it still looks like a single point, then it is, but otherwise you will tell that it is a line.
2. Then, imagine looking at what appears to be a 1D object (a line). Again, the only way to tell whether it is 1D or in fact you are looking at a 2D object (let's say, a square) edge-on is to move your head relative to the line (in any direction except directly towards or away from it). If it still looks like a line, then the object is 1D, but if you were in fact looking at a 2D object, then you would be able to tell that it is a square.
3. The same applies if you are looking at what appears to be a 2D object, a square. You can only tell for sure whether it is a 2D or a 3D object (a cube) by moving your head, because if you look at a cube from a position exactly perpendicular to the centre of one side, it will look like a square.
4. Therefore, it must follow that if you think you are looking at a 3D cube, you can (assuming that there is such a thing as a fourth dimension) only tell for sure whether it is in fact a 3D cube or a 4D 'cube' by looking at it from a different angle.
So by reverse logic; if you look at a 4D 'cube' from certain angles, it would in fact look like a normal 3D cube.
The example of a 4D 'cube' on Wiki is animated, but really, what you think is 'movement' is in fact the effect of you looking at it from different angles, in the same way as a square only appears to be a perfect square if you look at it from the correct angle; if you look at a square from the side, it looks like a line, and if you look at it from any other angle, it looks like a rhomboid.
* Not really 'prove' as this is all conjecture, but 'prove' in the sense of using apparently logical steps to arrive at what may or may not be a wholly erroneous conclusion.
The age of reason
1 hour ago
11 comments:
A 4-D sphere would be something that arrived from nowhere as a single point (sphere of no dimensions); that immediately started expanding until it reached its maximum size as a sphere; then started shrinking at the same rate until it disappeared up its own central point
its rate of expansion would be greatest initially, tapering off to zero (maximum size) and then going negative (contracting) at an ever greater rate until ... *pufff*
ND, that's cheating as you are using movement (i.e. time) to illustrate your point. Time is another kids of 4th dimension, I am trying to think about a fourth spatial dimension.
A 4D 'sphere' would, using my logic, look like a normal 3D sphere, if you looked at from certain angles.
disagree, Mark
I'm just trying to help you visualize the 4th D as a summing of a series of 3D's (just as an 'ordinary 3D solid' is a summing of a series of 2d planes, bacon-slicer-wise)
a bit like Anon's 'plane of cubes' etc in the comments to your first post
you've already agreed it's helpful to think about looking at your 4D object from different angles - and to do this 'comprehensively', as it were, to view the whole thing in the higher dimension, it needs to be done in a systematic sequence of viewings
which involves time (in a way): one viewing (slice) after another as you shift your perspective and complete the full survey
[there are some thoughts in Kant (and in a more modern exposition, in Strawson) that can also be of assistance here]
I reckon it's more interesting / instructive to think of the 4D analogue of (2D) circles & (3D) spheres, rather than of squares / cubes
i have conjured 4D-sphere out of 'sequence-of-3D-spheres'
you can equally think of a 3-D sphere as a 'sequence-of-2D-spheres' (circles) - and the new dimension you introduce, in which to 'do' this sequence, can be 'time' - if you want to use that nomenclature - or it can be 'just' a sequence along an 'ordinary' spatial axis
ND, I take it all back. That makes much more sense.
So you think it's possible that a 4D sphere would appear larger or smaller depending on what angle you viewed it from (albeit that you might have to move your head in the 4th dimension to see the changes, I'm undecided on this point)?
yup !
another mind-game you can play is this: imagine a creature confined to a 2D world (perhaps an amoeba living in the surface of a pond) that has full movement and perception in that plane, but none outside it
then imagine a 3D sphere being lowered slowly into the water
the first thing the creature observes is a very small (2D) circle that has just appeared in his world. He can go all around it, view it from every angle (within the plane) and establish to his full satisfaction: it's a regular 2D circle, all right !
except that, as it lowers into the water, from his perspective it is growing ! until it passes the point where half of it is in the water and half outside, after which point, as it continues to be lowered, the circle starts to shrink
until, when fully submerged, it no longer presents any part to our surface-bound friend: it has disappeared
(a bit spooky for the little chap)
'time' only plays a contingent role in this explanation
As an introduction to conceiving a fourth spatial dimension, try this http://www.archive.org/stream/flatlandromanceo00abbouoft
It includes Nick Drew's ideas and is a wierd little story (quite short), worth reading in any case.
ND, in other words, if you look at a 4D sphere straight on (i.e. directly towards its centre), it must look bigger than if you look at it from an angle (if your ameoba analogy is correct)?
hmmm
when the amoeba looks at the 3D sphere - as it presents itself in his planar world - he always sees a circle-from-one-side, i.e. a straight line
and if he moves around it, to check it really is a circle, then (provided he always views it from the same distance) from whatever angle, he always sees a straight line of the same size
THAT'S how he identifies it as a CIRCLE !
just as if you walk around the earth, wherever you go, you always see the horizon as being the same (give or take the odd mountain) which is how you identify the earth as a sphere
it won't be looking at it from a different ANGLE that will alter it's size (or shape) - that's the definition of a sphere / circle / whatever
it's looking at it from a different (relative) dimensional POSITIONING - for which either you must move in that other dimension, or it must move
(and being a sphere, its shape will always look the same)
ND, we're agreed on the amoeba example, but the line/circle that the amoeba sees grows and then falls in size as the ameoba looks at it from different positions relative to its own 2 dimensions. (whether the sphere is being dropped into the water slowly or the amoeba is, for once in its life, moving in the third dimension relative to the sphere, i.e. upwards) is neither here nor.
So, going back to my original post, from some angles, a 4D sphere must look like a 3D sphere, but not from others (true or false?) and/or a 4D sphere will appear to change size when we mere 3D humans look at it from different angles/dimensions (your first example, true or false?).
let's move down a level before answering, I find it helpful that way
(1) "from some angles a 3D sphere must look like a 2D sphere, but not from others T/F?"
FALSE. A 3D sphere always looks (to the 2D vision-receptor, = human eye) like a 2D sphere, i.e. a circle.
interestingly, perhaps, a 2D sphere only looks like a circle if the eye is orthogonal to the circle's centre. From other angles it looks like an ellipse, and from sideways on it looks like a line. So the completely re-cast, reversed sentence
"from some angles a 2D sphere must look like a 3D sphere, but not from others T/F?" is TRUE
(this means that for complete pedantry there could be a more nuanced answer required to (1), as follows: FALSE - a 3D sphere always looks like a circle, but never like an ellipse or a line; whereas a 2D sphere sometimes does look like an ellipse ... etc)
(2) "a 3D sphere will appear to change size when we mere 2D amoebae look at it from different angles/dimensions"
from different angles, FALSE (provided we stay at the same distance from it in our plane, natch)
from different dimensions, TRUE
I reckon this answer-set transfers 1 - 1 back to your original questions, mutatis mutandis
ND, agreed to (1) that's the way to look at it, as to (2) I wasn't differentiating between different angles and different dimensions. Hmm.
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