Naked Science Forum
On the Lighter Side => New Theories => Topic started by: Mirkin on 27/10/2010 17:59:23
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The universe is said to have three (at least) space dimensions. My point is this:-
When an object moves it can go in ANY direction, not just one of three. Secondly, at the very smallest scale it can only move in a straight line. A curve is merely a series of extremely small straight lines that when placed side by side give the impression of one. If this is true then every proton, neutron or quark in an object can be imagined to be making Planck length steps as the object as a whole moves through space. As stated, each of these steps must be a straight line. The motion of the object is the sum total of these small steps, which we can never hope to measure because the length of each one is incredibly small. All we can see from our perspective is something moving smoothly. If there were three dimensions the particle would be forced to move on a grid which not only makes no sense it can be shown to be impossible. This leads to the conclusion that there are no actual dimensions at all, just a void, in which all particles are free to move about without constraint.
The reason we perceive the universe as having three dimensions is because it’s the absolute minimum number needed for our brains to define the size and shape of an object. Two leaves us with something that can’t physically exist (in our universe) and four is one more than is necessary. This leads me to believe that the three dimensional world around is an illusion created by our brains.
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The universe does indeed have three dimensions. The definition of a dimension is complicated, but its easy to see why we need three of them: if I ask you to tell me where objects in a room are located with respect to you, you'd have to tell me how far they are to your left/right, how far they are up/down, and how far they are front/back of your position. These are three numbers, and so we live in three spatial dimensions. If we lived in two dimensions, you would only be able to use two number to specify where an object was with respect to you, which wouldn't represent reality.
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Can you point me to somewhere that explains a dimension?
I'm still struggling with the concept. In my mind if we only have three dimension then we can only move in one of three directions which is patently not true.
Thinking about what I wrote before it might even be better to suggest that there are an infinite number of dimensions. [???]
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There's some information here: http://mathworld.wolfram.com/Dimension.html
If that's confusing/not enough, it has a bunch of sources at the bottom. The book, Flatland, that gets cited is a fairly good introduction for the lay-person.
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..... tell me how far they are to your left/right, how far they are up/down, and how far they are front/back of your position. These are three numbers, and so we live in three spatial dimensions......
But can there be an up/down, left/right or back/forward in space? Surely there can only be a position relative to a n other position, which would be just one dimension. What say you JP?
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True, you can describe how to get from one object to another with only one number (kind of). You still have to give a direction, which is a vector consisting of three numbers. In other words, you could tell me "go 50 meters straight ahead to get to the sheepy farm." But by telling me that, you've implied that I go 0 meters left/right and 0 meters up/down.
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According to our own perception and senses mechanism, dimensions may be only an illusion. But we still live our lives in a minimum of 4 dimensions...
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True, you can describe how to get from one object to another with only one number (kind of). You still have to give a direction, which is a vector consisting of three numbers. In other words, you could tell me "go 50 meters straight ahead to get to the sheepy farm." But by telling me that, you've implied that I go 0 meters left/right and 0 meters up/down.
Three dimensions is certainly a convenient way of navigating in space, but it's really just a human convention.
We could actually define a position with two numbers instead of three. One number would be a direction and the other number would be a distance in that direction. The direction number would define a direction superimposed on a spherical surface relative to the centre of the sphere. The combination of the two numbers would represent a sort of spatial vector.
The other way of looking at this is to think of each possible direction as a dimension, so you end up with as many dimensions as you need to obtain the required resolution, and a single number to represent the distance in that dimension.
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I've had a ponder on what you say JP and something still troubles me. Don't get me wrong, I do understand what you are saying, but, as Geezer has mentioned, it all rather relates to the human conception of direction.
We know our left from right, up from down and back from front. Well, most of the time anyway! (https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fbestsmileys.com%2Fdrinking%2F2.gif&hash=8973674950c1fe1630ba7ecd2283843c) But imagine you are a perfect sphere, with no poles or equator, no top or bottom, no left or right etc etc. You are within another (much larger) sphere which also has no poles etc etc. (Please don't take this as an interpretation of the universe being a sphere or any other particular shape, or to have any defined boundaries).
You wish to move from your current position to another position. Since, as a perfect sphere, you have no conception of left/right, up/down or back/forward, you will move in one plane and one plane only. Therefore, can vector really be a factor in your movement?
If you don't quite understand what I mean here, please don't ask me to expand or elucidate, 'cause I'm none too sure I know what the hell I'm jabbering about either! (https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fbestsmileys.com%2Ftalking%2F1.gif&hash=22594bd001c27972987297d5fbc1ee2e)
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Don - the way i see it for your example is as follows; you can walk in one direction (ie one foot directly in front of another) sooner or later you realise one of two things or both. there is stuff beside you that you do not reach if you walk in a straight line and that eventually you realise you are re-tracing your steps.
You walk endlessly in your straight line but never get closer to that pub you notice on every trip around the globe and is becoming increasingly alluring. you try something new and put each foot slightly to the left of the previous footfall and you come to understand that you have a new direction and potentially many more to explore.
But once you get to the pub - and you can almost taste the beer - you see a stile in a fence blocking your path, you try the old faithful trick of one foot directly in front of the other (no beer yet!) and the new paradigm of each step slightly to the side of the last (still no beer); it is only in the dire straits of beer deprivation that you try one foot above the other and enlightenment and best bitter becomes a reality.
No matter how much you move in one direction on a globe - something that was on your left on the first rotation will stay there; and no amount of walking straight or in a curving course on the flat will increase your height above the plane you are walking on. If your house, the cashpoint, and the pub are at the points of a triangle then NO straight course will let you walk from home to the wateringhole via the hole in the wall.
I always find beer examples help me understand things - and I normally only have trouble putting one foot after the other once I have been to the pub!
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Geezer - I am not sure we can define a position in 3-ball with two simple numbers.
One number would be a direction and the other number would be a distance in that direction. The direction number would define a direction superimposed on a spherical surface relative to the centre of the sphere. The combination of the two numbers would represent a sort of spatial vector.
The only way I can see one number uniquely defining a direction from centre of sphere to the surface is if that surface has been divided and each section labelled with a number. As this numbering would be arbitrary I could not envisage how this could be used mathematically - with polar coordinated or cartesian coordinated we can determine distances and relations between points through their description, and these coordinate systems allow for transformation.
The other way of looking at this is to think of each possible direction as a dimension, so you end up with as many dimensions as you need to obtain the required resolution, and a single number to represent the distance in that dimension.
this bit I agree with entirely - a comboined unit vector and magnitude
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Geezer - I am not sure we can define a position in 3-ball with two simple numbers.
One number would be a direction and the other number would be a distance in that direction. The direction number would define a direction superimposed on a spherical surface relative to the centre of the sphere. The combination of the two numbers would represent a sort of spatial vector.
The only way I can see one number uniquely defining a direction from centre of sphere to the surface is if that surface has been divided and each section labelled with a number. As this numbering would be arbitrary I could not envisage how this could be used mathematically - with polar coordinated or cartesian coordinated we can determine distances and relations between points through their description, and these coordinate systems allow for transformation.
I'm with Matthew on this one. If you want to specify a position on the surface of a sphere with perfect resolution, you'd need two numbers. I think mathematically, the explanation is that if you're divide the surface into boxes, you can also arrange those boxes in a line, so you're actually in one dimension. If your surface is continuous, i.e. it has infinite resolution, you can't arrange it in a line.
The other way of looking at this is to think of each possible direction as a dimension, so you end up with as many dimensions as you need to obtain the required resolution, and a single number to represent the distance in that dimension.
this bit I agree with entirely - a comboined unit vector and magnitude
Yes, but you can't think of each possible direction as a dimension unless you're using the term very loosely.
There's a precise definition, but it takes some effort:
A vector is an ordered set of numbers, for example (1,2,3) is a vector, as is (.333,0,Pi) and (0,0,0) and (1,1,1), etc.
A linear combination of vectors is the result of adding vectors together, where each vector in the sum can be multiplied by a number, for example: 3x(1,2,3)+.2x(0,1,0)+10000x(0,0,0)=(3,6.2,9)
A linearly independent set of vectors is a collection of vectors such that no single vector can be written as a linear combination of the others in that collection, for example: (1,0,0),(0,1,0),(0,0,1).
Once you have these definitions, the dimensionality of your space is given by the maximum number of linearly independent vectors in your space that you can put in a set.
Phew!
The gist of this is, though, that any vector (a,b,c) can be written as a(1,0,0)+b(0,1,0)+c(0,0,1), so that by specifying a,b and c, you specify any vector in 3D space--that's why you can loosely say that the number of dimensions is given by the minimum set of numbers you need to describe a position in space.
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Geezer - I am not sure we can define a position in 3-ball with two simple numbers.
One number would be a direction and the other number would be a distance in that direction. The direction number would define a direction superimposed on a spherical surface relative to the centre of the sphere. The combination of the two numbers would represent a sort of spatial vector.
The only way I can see one number uniquely defining a direction from centre of sphere to the surface is if that surface has been divided and each section labelled with a number. As this numbering would be arbitrary I could not envisage how this could be used mathematically -
Who said anything about this actually being useful? [:D]
But that's my point really. The system we use is based more on it's utility than some fundamental physical truth. Three dimensions happens to be a very good way of defining a point in space. The 3D concept is so deeply ingrained in our thinking that it's hard for us to describe space any other way.
The spatial direction idea is certainly cumbersome, but fundamentally I don't see why chopping a surface into tiny little pieces and assigning each piece an arbitrary address is really so different from doing the same thing with a line. Of course, there would be a few minor details to work out like defining an entirely new math, or even maths, but I'm sure these are things that could be left as an exercise for the reader.
Of course, we could take the idea a bit further and reduce everything to a single number by assigning arbitrary addresses for all points in space with various resolutions. That might take a little bit longer of course.
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I'm with Matthew on this one. If you want to specify a position on the surface of a sphere with perfect resolution, you'd need two numbers. I think mathematically, the explanation is that if you're divide the surface into boxes, you can also arrange those boxes in a line, so you're actually in one dimension. If your surface is continuous, i.e. it has infinite resolution, you can't arrange it in a line.
Bbbbbbbbut, there is no such thing as "perfect resolution". You always have to decide how much resolution you need. Admittedly Geezer's Disco Ball model would be an incredible pain in the arm to use (actually you'd need a very large number of disco balls with different resolutions).
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Maybe perfect resolution is the wrong way to phrase it, since that makes it sound like we're measuring things with perfect accuracy.
Distance measurements in space are continuous variables is probably the more correct way to phrase it. There is no way to map a sphere to a line if the possible positions on the sphere are continuous, which appears to be the case for physical objects.
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There is no way to map a sphere to a line if the possible positions on the sphere are continuous, which appears to be the case for physical objects.
Bbbbbbut the possible positions on a line are also continuous, so it follows that there is no way to map a point in space to a line.
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That's true, because a point is zero-dimensional. If you could map it to a line, it would be one-dimensional.
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Hi Guys
I just read all your responses and I have to say that although some of the maths was a little confusing nobody has managed to persuade me that I'm wrong.
I still contest that to get from A to B you need only travel in a straight line. I.E along one dimension. If there are three dimensions then an object only has the option of going along one of three paths and if none of those paths leads to where it needs to be then would be forced to take a series of steps at right angles to each other. That would also mean that there would be areas in between these dimensions that nothing could ever get to.
The notion that we need three numbers to describe where two points are relative to each other also seems to be an illusion. This system only works if we use a third point as an origin. Without it all we can do is say how far apart the two objects are. Again this can be measured in a straight line and one dimension is sufficient. On a cosmic scale its impossible to now the actual position of anything relative to anything else anyway because everything in the universe is in motion and the universe itself is expanding. What point would we use as the origin?
I think my lack of formal mathematical training is hampering me with understanding some of what you other guys are discussing but on the other hand I might be a little freer in my thinking because of it.
Just one other final point if somebody wants to have a go at telling me how this is wrong.
In a theoretical one dimensional universe as described in flatland (it was reading this book that set me thinking) is it not possible for the one dimension to be curved? Even loop back on itself to form a circle? Or for that matter loop randomly around until it forms a kind of cotton wool ball that touches itself in every direction. In that kind of universe an object could move in any direction and still only travel along one dimension!
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There is no way to map a sphere to a line if the possible positions on the sphere are continuous, which appears to be the case for physical objects.
Bbbbbbut the possible positions on a line are also continuous, so it follows that there is no way to map a point in space to a line.
Pretty good Geez. Did you learn that from the back of a Beano? http://en.wikipedia.org/wiki/The_Beano
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Mirkin
I still contest that to get from A to B you need only travel in a straight line. I.E along one dimension.
Yes A to B is a straight line.
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If there are three dimensions then an object only has the option of going along one of three paths and if none of those paths leads to where it needs to be then would be forced to take a series of steps at right angles to each other. That would also mean that there would be areas in between these dimensions that nothing could ever get to.
... NO! Dimensions "describe" position and movement they do not constrain it (well maybe they do - but not in the way you are giving). You may be at any position and move in any direction - and we can mathematically describe that. If you take a glass plate box on the table in front of you with a fly inside - the fly can be anywhere and move any direction inside. We can describe where the fly is by describing how far along (your left to right) it is along the table, plus how far away from the front of the case, plus how high he is off the table. The more accuracy you want - the more careful you are in your measurement. But the manner of our description of the fly does not stop the fly doing its thing (until it flies into the wall).
The notion that we need three numbers to describe where two points are relative to each other also seems to be an illusion. This system only works if we use a third point as an origin. Without it all we can do is say how far apart the two objects are. Again this can be measured in a straight line and one dimension is sufficient. On a cosmic scale its impossible to now the actual position of anything relative to anything else anyway because everything in the universe is in motion and the universe itself is expanding. What point would we use as the origin?
This is correct - there is no all-powerful origin. We choose one and if we are consistent in that choice then we can use the coordinate to find out physically useful facts. As an example - if we were to model a ball thrown and the effect of gravity - we would choose a coordinate system such that the horizontal direction of the throw was along one axis and up/down was along another. We could choose a totally separate set of axes - the results would be the same but would be significantly harder to visualize, draw, or calculate. Three dimensional description allow comparison of different positions and different movements in real space- whilst you are right that the first straight line between 2 points can only really be described with a distance; if we place a 3 dimensional grid onto our space every subsequent straight line can be related and compared with every other one
In a theoretical one dimensional universe as described in flatland (it was reading this book that set me thinking) is it not possible for the one dimension to be curved? Even loop back on itself to form a circle? Or for that matter loop randomly around until it forms a kind of cotton wool ball that touches itself in every direction. In that kind of universe an object could move in any direction and still only travel along one dimension!
I thought flatland was 2-d (its been a long time) surely main character is a square? Yes a one-dimesional object can be curved - this is a one-d object that exists in higher dimensional space. To a fair approximation we live on the 2-d surface of a 3-d ball (although of course we do live in a 3-d world. If your cotton just touches itself - then yes you are correct; the distance needed to be travelled from point to point might be huge, and I guess there would have to be gaps (although I cannot visualize wrapping up a one-dimensional string).