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Shapes of things unknown?

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ALanQIMan
599326.  Sat Aug 15, 2009 6:50 am Reply with quote

Second question is about shapes. Every object that we know and see here on Earth has a shape which is composed of straight lines or curves or some complex combination of the two. Is it even remotely possible that there are other shapes that we cannot conceive of, because they donít exist in our visible world, or perhaps we donít see them? Can it be mathematically proved, or disproved that any new kinds of shapes are impossible? Keep any mathematical explanations as simple as possible please; I have trouble adding up my expenses!

 
Ameena
599522.  Sat Aug 15, 2009 4:26 pm Reply with quote

I think that's getting into other dimensions - we can see and think in three (or four if Time is considered the fourth). Our brains are 3-D. We can't conceive of anything outside that - there might be a 10-dimensional object floating right in front of you but because we only see in three dimensions, you wouldn't be able to see it.
At least, that's sort of how I understand it. Vaguely.

 
Posital
599555.  Sat Aug 15, 2009 5:08 pm Reply with quote

You might want to consider the "shape" of an electron orbit...

 
Flash
599762.  Sun Aug 16, 2009 5:59 am Reply with quote

Welcome to the boards, ALanQIman. If you haven't already come across it, you might want to look into a book called Flatland, by Edwin Abbott, published in 1884. It envisages a two-dimensional world inhabited by polygons who are unable to conceive of a third dimension. When they do encounter a (three-dimensional) sphere they perceive it as a straight line.

There's a wikipedia article: http://en.wikipedia.org/wiki/Flatland

 
Jenny
599915.  Sun Aug 16, 2009 10:51 am Reply with quote

Actually, that does raise an interesting point - if there was something that existed in more dimensions than we can perceive, I wonder if we would see it but not as it 'really' is.

 
Moosh
599926.  Sun Aug 16, 2009 11:03 am Reply with quote

Just a niggle with what Flash said, wouldn't a two-dimensional being encountering a sphere perceive it as a circle, not a straight line?

Edit for grammar


Last edited by Moosh on Sun Aug 16, 2009 2:35 pm; edited 1 time in total

 
Posital
599975.  Sun Aug 16, 2009 2:20 pm Reply with quote

Moosh wrote:
Just a niggle with what Flash said, wouldn't a two-dimensional being encountering a sphere would perceive it as a circle, not a straight line?
Yes - Otherwise, we'd perceive spheres as circles which we don't - our perception is 3d rich - and I assume flatlanders would have similar 2d perceptual abilities.

Although a drawing/painting/photo would be a line, but with the perceptual cues (eg shading) to allow mr 2d to understand it as a circle.

 
bobwilson
600049.  Sun Aug 16, 2009 7:17 pm Reply with quote

Jenny wrote:
Actually, that does raise an interesting point - if there was something that existed in more dimensions than we can perceive, I wonder if we would see it but not as it 'really' is.


Which is exactly the stuff that physicists are wrestling with when they discuss the 10/11 dimensional existence. EG Why is Gravity so incredibly weak when compared to the other three fundamental forces? Possibly because the majority of gravitational force is experienced in the higher dimensions which we can't perceive.

 
Flash
600091.  Mon Aug 17, 2009 1:20 am Reply with quote

Moosh wrote:
wouldn't a two-dimensional being encountering a sphere perceive it as a circle, not a straight line?


No, they don't see things in "plan", as though looking at them from above - as they exist only on the surface of the paper, as it were, they see everything sideways-on. They see each other as straight lines as well (of varying lengths, depending on which way they are standing), except for the females which are in fact straight lines, so that they perceive an approaching female as a point.

The bit of the sphere that they can discern is the circle which intersects the plane on which they live - but it's seen sideways-on.

 
Posital
600096.  Mon Aug 17, 2009 2:18 am Reply with quote

I think you're missing the point here, Flash.

The key word is perceive.

We perceive a hyper-sphere as a sphere - in the same way a flatlander would perceive a sphere as a circle.

Even if they lack binocular vision, they could use focus (depth of vision) or movement to spot a circle.

Edwin Abbott is plain wrong if he expressed this view back in 1884 - it was intended more as a social commentary (at the time) than anything else.

 
Flash
600107.  Mon Aug 17, 2009 3:04 am Reply with quote

I'm not really making a point, just conveying what it says in the book. Here's the text:
Quote:
I dare say you will suppose that we could at least distinguish by sight the Triangles, Squares, and other figures, moving about as I have described them. On the contrary, we could see nothing of the kind, not at least so as to distinguish one figure from another. Nothing was visible, nor could be visible, to us, except Straight Lines; and the necessity of this I will speedily demonstrate.

Place a penny on the middle of one of your tables in Space; and leaning over it, look down upon it. It will appear a circle.

But now, drawing back to the edge of the table, gradually lower your eye (thus bringing yourself more and more into the condition of the inhabitants of Flatland), and you will find the penny becoming more and more oval to your view, and at last when you have placed your eye exactly on the edge of the table (so that you are, as it were, actually a Flatlander) the penny will then have ceased to appear oval at all, and will have become, so far as you can see, a straight line.

The same thing would happen if you were to treat in the same way a Triangle, or a Square, or any other figure cut out from pasteboard. As soon as you look at it with your eye on the edge of the table, you will find that it ceases to appear to you as a figure, and that it becomes in appearance a straight line.

 
Moosh
600338.  Mon Aug 17, 2009 9:50 am Reply with quote

But if you move around the penny, and closer to and away from it, then you'd see it was circular. I think Posital has something on the depth-perception thing as well.

If you look at a ball, you can only see a two-dimensional circle, but you can perceive it's three-dimensional by moving around it, and by the shading and so on. I can't see why it would be different for a circle in a two-dimensional world.

 
Jenny
600490.  Mon Aug 17, 2009 4:28 pm Reply with quote

I think we're taking an essentially light-hearted book a tad too literally here...

 
ColinM
600497.  Mon Aug 17, 2009 4:41 pm Reply with quote

I don't understand what you're disagreeing about. As far as I can tell you all agree that the visual information they get from a circle is a (coloured) line, and from this they infer that the object is a circle. Posital, Flash, Abbot, and Moosh all seem to be saying that very thing. Are you really just disagreeing about which bit should be called "perceiving"?

As to the original question, the answer is Quite Interesting. Which is to say that it depends on what the question is.

If I may regress a hundred years or so, the Stuff in the Real World is made up of atoms. These are little blobs of mass all joined together. So the shapes we can make in the Real World are, at best, the sort of shapes we can make by joining little blobs together. That pretty convincingly leaves us with lines and curves of various descriptions, however many dimensions they might happen to sit in.

Mathematics, on the other hand, has some other things to try. Not satisfied with what the Real World has to offer, nineteenth century mathematicians thought up some alternatives. Consider Peano's space filling curve - that is, a one dimensional line twisted up so it completely fills an area. Or Koch's snowflake, which is an infinitely long line which bounds a finite area.

These days such things rejoice under the collective (but not terribly specific) name "fractal". They really don't have to be made of curves or straight lines in any normal sense.

 
Posital
600542.  Mon Aug 17, 2009 6:52 pm Reply with quote

Jenny wrote:
I think we're taking an essentially light-hearted book a tad too literally here...
But it's (laughingly) cited in the case for the defence, m'lud.

 

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