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 Posted: Mon Apr 15th, 2013 03:31 am
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kurtwaters
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From Jee-Host:

On a side note - let's have a test of imagination. Try and make a verbal description of how would you imagine a place with fractional value of number of dimensions. Let's say, instead of mainstream 3 dimensions let it be 3.14 dimensions.
kurtwaters:
My first thought was do you mean the three familiar dimensions plus another .14 dimension or a small bit added to the 3 familiar dimensions. I chose the latter.

 So I imagine a world of 3.14/3 or three 1.046.... dimensions.

     My mind then approaches the problem in this manner:
a point (no dimensions ) would lengthen a little tiny bit.
a line ( one dimension) widen a little tiny bit a plane (two dimensions) would be raised a little tiny bit
a cube ( three dimensions) would cast a funny shadow.
so the 3.14 dimensional world would be a weird, blurry world as though I had the wrong glasses prescription. I would be able to feel the edge of a finite plane. And any point I tried to make would expand. I will continue to imagine.

Jee-Host:
If I may, can I suggest a couple of known conceptions that may add some more perspective to this task of imagining you decided to undertake? Feel free not to read them if you want to do this completely on your own. I'll make the text white so you won't accidentally read it if you don't want to. You're probably familiar with 4-dimensional imaginary hypercube (aka tesseract). It's kind of a common conception, there are even videos of 2D and 3D 'shadows' of tesseract as a computer model. I think observing the way it behaves both mathematically and in terms of theoretical physics can be a decent starting point for this imagination challenge. Another good conception is geometry. Most common Euclid's geometry bases on the fact that sum on triangle angles is always 180 degrees. But there are approaches that challenge this premise. Riemann and Lobachevsky went opposite ways on this and ended up having planes looking like sphere and distinct saddle-like form respectively. I get the impression that you think that you can imagine 3.14 dimensions more accurately than you already did. Thus I'm not going to give my take on it yet to not influence your effort. Please tell me if(when) you'd like me to do it.
Mike:
I came upon your puzzle late at night, wrote my initial reaction in this e-mail and then went to bed. As I lie awake waiting for sleep to come I began to realize problems. How does one actually define a dimension? A line can only be 2, right? A point always 0, no? Then I fell asleep.

     In the morning I decided I needed help and before I opened this e-mail I discovered Hausdorff dimensions. Now I have read your e-mail and the cleverly hidden clues and I am imagining a fractional dimension space that would curve and twist. From the point of view of an observer inside this space he would believe he was walking a straight line, but someone outside would observe him walking in curves and would the curves be continuous? And on and on it goes, my attempting to understand.

     So,now, I am ready to hear your take on it. My major in college was Math, but I had to abandon that study because I had to work to make money to pay the bills because I had a wife and two kids by then. I couldn't do both at the same time. My kids are now grown, I divorced the first wife and married another and I am, so to speak, going back to school.


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 Posted: Mon Apr 15th, 2013 03:32 am
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kurtwaters
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Jee-Host wrote

"I think you've chosen the right course of thought in your analysis. Especially the part you mentioned about outside observer. This in itself suggests change of dimensional value, so to speak, throughout the space. It logical that such a change would be gradual rather than abrupt. However, in this case it is important to find out exact significance of using fractional number of dimensions as opposed to integer number. overall, any use of relative numbering such as this is a result of a convenience effort. For any systematic purpose we can use any other numbering, taking any significant (by our account) change as additional integer step. All this doesn't answer about the nature of such a measurement we use to think of as dimension, what it signifies.

In my worldview model I separate two conceptions - space and matter. Difference between them lies not only in their nature (matter being substance and space - not), but also in their relative quantifyiability. Space, unlike matter, we can't precisely quantify. Thus I suppose space to be infinite and matter - finite. But both of them interact with each other, thus they all have mutual effect. Space is anisotropic, and whatever anisotropy it has in any given part of it results in consequential distribution of matter. Now matter we can quantify by types or forms, whichever term you like. Common point being matter's features in interactions.

We consist of matter. Any change in macro mutual affection between space and matter (for simplicity reasons let's narrow it to just matter changes for now) directly affects us, not 'balances' in us as in separate system. What I'm saying is that quantifiable result of matter types affecting the space is what seems to us like space dimensionality. By certain account this measurement for 'our' part of space is 3,00017. Now should 'our' part of space have more or less matter types - that value would change and so would our vision of conception of dimensions. Now obviously this suggests that places with different dimension value do exist since space is anisotropic. But I think I've strayed too far from the point about fractional dimensions."

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 Posted: Mon Apr 15th, 2013 03:47 am
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kurtwaters
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Mana: 
did you mean 3.00017 instead of 3,00017

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 Posted: Mon Apr 15th, 2013 08:57 am
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Jee-Host[gm]
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Yes. Here in written notation we use comma instead of dot. I sometimes forget that over there things are not the same in terms of syntax, even when it comes to mathematics.

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 Posted: Mon Apr 15th, 2013 12:38 pm
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kurtwaters
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Mana: 
I know.

I thought, perhaps, the comma had some other significance besides being a syntax replacement for a decimal point, hence the question.

My thought process is now along these lines:

what is a dimension? A parameter to describe a space?

In the 3 dimension system I am familiar with if I want to describe a cube I can do so by taking a point and copying it until I have a line, copy a line until I have a plane, copy a plane until I have a cube. The cube has 3 dimensions.

An integer dimension and fractional dimension differ in the method I choose to copy in order to describe a space. I can choose different copy methods for convenience.

Copy may be a poor choice of word for what I mean.

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 Posted: Mon Apr 15th, 2013 02:43 pm
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Jee-Host[gm]
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Whenever I approach writing about a certain conception I tend to formulate answers to 3 basic questions about it:

- What (it is)?
- How (it works)?
- Why (it functions)?

All three shoud be view in succession, because essentially they are just extensions.

So in this instance we have a concept of dimensions.

1. What (is a dimension)? Dimension is as simple as it gets - is technically our way to measure depth (one of them, to be exact). In onther words - it is a way to measure how space manifests itself in regard of our organs of sense. But that explanation is completely trivial and would only satisfy a brainwashed bloke. Because measure we gain that way is just a convenience, it has no tangible physical follow-up. So the proper question would be - what is the reason behind us making such a measurement as dimension. What process accommodates for a change that provides us with a reason to make such a measurement. Looking at it as on a certain interaction between matter and space I make a hypothesis that the event in question is growing influence of matter in this calculation. If we are to imagine dimensional value changing (increasing or decreasing), we can view it as according change in matter influence over space due to whatever reason. No matter the reason we now have a certain grasp on how to formulate the second question.

2. How that aforementioned interaction occurs? Let's picture most efficient configuration as plane or straight line. And anything, that influences this configuration - bring about a curve. If space is like a plane with pits of different shapes and size here and there, then matter would be like watter filling those up until surface is a plane. Effecively matter is what negates those curves as much as it required to balance them out. Curves being a condition based on space's anisotropical imperfection in any given part of itself. Thus the bigger the curve is - bigger dimension value would be.

3. Why all this would happen? Now that is one excellent question. Something I don't yet know how to crack on the larger scale. Even though I have some theories flying about, they are in a too raw state yet to share.

Last edited on Tue Apr 16th, 2013 07:29 am by Jee-Host[gm]

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