size of dimension

Higher-dimensional geometry (previously "Polyshapes").

size of dimension

Postby thigle » Sun Jan 29, 2006 11:58 pm

what does it mean that a dimension is small ? or to the contrary.
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Postby Marek14 » Mon Jan 30, 2006 9:23 am

Where? In a string theory, it means that the dimension in question is curled in a circle which is, for all practical definitions, small (on the order of Planck length).
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Postby thigle » Mon Jan 30, 2006 12:09 pm

in considering the 4th spatial dimension, it can be flat, or curled under Planck.

what about the macro, or global case (like for exemple the Rieman's hypersphere S3/V4 ), what is the 4th dimension like ? a circle with the centre at infinity ? or a circle at infinity ? or some loop or twist ? :?
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Postby wendy » Tue Jan 31, 2006 9:35 am

A small dimension means any number of dimensions less than or equal to eight.
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Postby thigle » Tue Jan 31, 2006 4:28 pm

small in that sense would be related to the 'valence' of the coordinate base ? i might call it then 'dimensionality'.

but why is it as you state ? what does it mean that all dimensionalities less or equal to eight are small ? does it have anything to do with those (for me 'mysterious') octonions ? what is the essential reason for this '8 limit of smallness' ?

or is this derived from your personal (supra)empiric experience/workplay with dimensions ?
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Postby thigle » Thu Feb 09, 2006 12:25 pm

i try again: why is it that [smaller =< 8] ?
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Postby wendy » Fri Feb 10, 2006 8:13 am

Eight dimensions is where the strange things that we use filter out, and new strange things start to appear. I could have picked any number, like 124 or 15.

It's just that at eight, is the limit of my unaided visualisation.

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Postby thigle » Fri Feb 10, 2006 5:04 pm

thank you. :D if you mind...:

hmm, so 8 as the empirically testable Wild (that you call strange) threshold, beyond which lies the Other, that you call 'new strange'.

i don't think you could have picked ANY number, could you even pick ? or isn't it that it's just so ? it seems from your last sentence. so why 124 or 15 invalid exemple ?

apart from you, Coxeter, Boole's daugter, and few others are not many that i came across, that mentioned their (rather unusual) ability to visualise higher than 4 dimensions. however, (although i cannot find the coxeter reference), i believe, that (if my lousy memory doesn't trick me) the maximum number of dimensions anyone of these exceptionally skilled imaginers was always less than 8.

for me it means another communicated EMPIRIC data, that support the finding that normal rationality (as we usually consider/know it) breaks down like division algebras, into genuine newness, suprarationality or existential awareness. as Palmer, Smith, Goertzel, Bateson, Young(the helicopter inventor), Dante, Boddhidharma and hundreds others have noticed.

so the question is then, if as you say,
...just that at eight, is the limit of my unaided visualisation.
what are the tools for aided visualisation ? some generalisation ? for ex., Tony Smith uses some formula for decomposing CL(>8) into some simpler handable thing.

bluntly, what concepts/tools help you handle the unhandable ?
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Postby wendy » Sat Feb 11, 2006 8:13 am

I mainly draw various lace-cities, if need be, or other abstractions. Really, an awful lot of what is in the polygloss gets used quite a lot. Lace-cities, deep holes (stations), modular number theory, all sorts of little tidbits that people tell you.

I am not sure that Coxeter could visualise more than three dimensions. He relies too heavily on number, and the word-style suggests that he is looking at numbers and equations, rather than polytope and model.

Six dimensions is the largest that i maintain static models for natively. In higher dimensions, one can sense threads that run higher, and can use a combination of visual calculations and some plain old common sense to gather what is happening in seven or eight dimensions. But these threads are exhausted there.

In higher dimensions, there are different threads, which i have felt, but do not completely understand to this time.

15 dimensions is where an interesting lattice of eff 1.5 q falls. This contains both the t-basic and q-quartercubic. One notes that there are much more efficient lattices, the maximun obsurbed is somewhere like 8 q.

In 124 dimensions, one should see a super-efficient quasi-lattice, based on the pentagonal quaterions. These should be assorted slices of a 248d lattice, but i am not completely sure. The numbers there indicate that it should work, but it amounts to something like putting a sqrt(8) diameter sphere into a cube.

In practice, though, i rarely stray over eight dimensions, save to forage in the lattices.

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Postby papernuke » Wed Jul 12, 2006 7:10 pm

where did u read that? all dimensions are infinetly large, like our own, the universe
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Postby Nick » Wed Jul 12, 2006 9:58 pm

Icon wrote:where did u read that? all dimensions are infinetly large, like our own, the universe


The Universe's size is quite finite. It's just really, really big. And because space only exists inside the Universe, the three axes that we know of is finite as well.

Just imagine a 2d piece of paper. That's a 2d world. Now, imagine a pencil on top of it. That's a piece of 3d. Anything above the paper (3 dimensionally), but outside the pencil, is undefined. The pencil is the size of the 3d universe, and anything outside of it has no space (or time).

Hope that helps :)
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Postby Keiji » Wed Jul 12, 2006 11:08 pm

The universe may well be infinite. The fact that it's expanding is not proof that the universe is finite.
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Postby Nick » Wed Jul 12, 2006 11:13 pm

Rob wrote:The universe may well be infinite. The fact that it's expanding is not proof that the universe is finite.


If the Universe was infinitly large, there would no room to expand to. :roll:
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Postby Marek14 » Thu Jul 13, 2006 5:09 am

irockyou wrote:
Rob wrote:The universe may well be infinite. The fact that it's expanding is not proof that the universe is finite.


If the Universe was infinitly large, there would no room to expand to. :roll:


Would you care to explain the logic, please?
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Postby wendy » Thu Jul 13, 2006 8:52 am

It is clear to show that the stars are in some way uniformly distributed, even in the vaguest sense, because the sky at night is black. This means that there is only a finite abount of light being generated out there.

Specifically, the sum of

sum(I/d^2) over all stars, is finite, where

I = intensity of star
d = distance of star.

It is also possible for the universe to be expanding without the stars moving. The usual model is to think of dots on a balloon. These do not move, relative to the surface of the balloon, but as more air is put into the balloon, the distance becomes greater.

When we think of, eg 4d or 3d or 8d, these dimensions are taken to be freely extendable. That is, there is as much up as left in 3d, and as much t_1 as t_7 in 8d.

A model of the present world has 11 dimensions, of which 7 are wrapped in plankian dimensions, while one is rendered as complex (ie, time). This means that we can only access 3d.

The reason we judge this to be 3d, is the same reason we judge a sheet of paper as 2d, and a cotton thread as 1d, even though each of these actually are 3d objects. Some of the dimensions of both the page and the thread are so small that we discount their pressence.

If one posits that we live in 5d, but two of these are would tightly, then it really is 3d, rather like a chorous petid (3d-extensive 5d-solid). A sheet of paper on the same rule is a hedrous (2d extensive) chorid (3d solid).
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Postby papernuke » Sat Jul 22, 2006 4:00 am

Where did you read that? All dimensions are infinetly large. For example our dimension, the Universe is infinetly large.
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Postby Marek14 » Sat Jul 22, 2006 8:38 am

Icon wrote:Where did you read that? All dimensions are infinetly large. For example our dimension, the Universe is infinetly large.


How do you know that? :) I don't think there's a definite answer to the question "is the universe infinite?" yet.
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Postby wendy » Sat Jul 22, 2006 8:54 am

The universe, as it affects us, is bounded by the speed of light over the age of the universe. This means that the radius is of the order of 14e9 light years, and since a light year is of the order of 1e17 c/ms, it means it is about 14 e26 c/ms. This is about the size of avagadro's number, N_a = 273.16e24.

The number of particles in the universe is estimated to lie in the range of 1e80, which gives roughly a density of 1 particle per cc.

The geometry at a large scale (ie > galaxies) is zero-curvature, but the extremely large model is that the shape is a poincare dodecahedron, a finite torus-tiling cell from {5,3,3}, which means that 120 universes make the complete glome. The radius is 22e26 c/ms

While 1e27 c/ms is a fairly large distance, it is not infinity.
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Postby Nick » Sat Jul 22, 2006 12:25 pm

Marek14 wrote:
Icon wrote:Where did you read that? All dimensions are infinetly large. For example our dimension, the Universe is infinetly large.


How do you know that? :) I don't think there's a definite answer to the question "is the universe infinite?" yet.


Well, the Universe had to start somewhere. Assuming the Big Bang Theory, the Universe can only expand as fast as the speed of light; and it hasn't been an infinite number of years since the Universe was created.

Also, consider this: we see a bunch of stars at night. These stars are millions of miles away. If the Universe was infinitely large, then there are an infinite number of stars; we would see every single star's light (because it would have all reached the earth after an infinite time period), and the night would be as bright as day.
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Postby houserichichi » Sat Jul 22, 2006 1:42 pm

I suspect that the CMB is what leads us to believe the universe is of finite size. At any rate the debate between whether space is continuous or discrete at the sub-Planck length scales is still unresolved. If it's continuous then there are infinitely many points between here and the edge of the universe but the universe may or may not still be of finite length. If it's discrete then there are finitely many points between here and the edge of the universe which still may or may not be of finite length.

Short and sweet there's no "proof" of either, just educated speculation...and as far as I understand it the majority believe it to be of finite length and locally continuous.

And back to the original question of the thread, small dimension (I assume in the KK-theory or strings, or any of that) refers to sub-Planck scale dimensions that are far removed from our every day. Topologicall you can think of said dimension as compact.
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Postby Marek14 » Sun Jul 23, 2006 5:44 am

irockyou wrote:
Marek14 wrote:
Icon wrote:Where did you read that? All dimensions are infinetly large. For example our dimension, the Universe is infinetly large.


How do you know that? :) I don't think there's a definite answer to the question "is the universe infinite?" yet.


Well, the Universe had to start somewhere. Assuming the Big Bang Theory, the Universe can only expand as fast as the speed of light; and it hasn't been an infinite number of years since the Universe was created.


Except that if the curvature of universe is not positive, then it would already start to exist at infinite size. Which is kind of contraintuitive, but not actually impossible.


Also, consider this: we see a bunch of stars at night. These stars are millions of miles away. If the Universe was infinitely large, then there are an infinite number of stars; we would see every single star's light (because it would have all reached the earth after an infinite time period), and the night would be as bright as day.


What you talk about is Olbert's paradox. Today it's resolved by the notion that universe is only of finite age, so there's no infinite time period involved (also, stars only live for finite time). Still not an argument against finitely old and infinitely large universe, I'm afraid.

http://en.wikipedia.org/wiki/Infinity states:

"Infinity in cosmology

An intriguing question is whether actual infinity exists in our physical universe: Are there infinitely many stars? Does the universe have infinite volume? Does space "go on forever"? This is an important open question of cosmology. Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By walking/sailing/driving straight long enough, you'll return to the exact spot you started from. The universe, at least in principle, might have a similar topology; if you fly your space ship straight ahead long enough, perhaps you would eventually revisit your starting point.

If the universe is indeed ever expanding as science suggests then you could never get back to your starting point even on an infinite time scale."

Since it's an important open question, I have serious doubts that a Big Bang that creates infinitely large space is considered paradoxical enough to eliminate the infinite-universe option.

Wendy: I think that the Poincare dodecahedron theory is not currently accepted - weren't there some inconsistent data later?

From http://en.wikipedia.org/wiki/Universe:

Secondly, whether the universe is multiply connected, is unknown. The universe has no spatial boundary according to the standard Big Bang model, but nevertheless may be spatially finite (compact). This can be understood using a two-dimensional analogy: the surface of a sphere has no edge, but nonetheless has a finite area. It is a two-dimensional surface with constant curvature in a third dimension. The 3-sphere is a three-dimensional equivalent in which all three dimensions are constantly curved in a fourth.

Since this article talks about WMAP, it would appear that the question is far from settled.
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Postby wendy » Sun Jul 23, 2006 7:07 am

Whether the poincare dodecahedron is or is not accepted, the relevance is that one can largely deal with space as if of various curvature.

Still, must get around to modifying the "encyclopedae galaxia" one day.

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Postby bo198214 » Sun Jul 23, 2006 10:00 am

houserichichi wrote:If it's discrete then there are finitely many points between here and the edge of the universe which still may or may not be of finite length.

Ahm, how can there be finitely many points if it is of infinite length?! :o

Topologicall you can think of said dimension as compact.

Am not really sure whether this helps Icon .... or if they are at all any people except us two on this board which know what "compact" means.
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Postby houserichichi » Sun Jul 23, 2006 1:50 pm

Oops, very much my mistake. It could be of infinite size and made up of infinitely many points but locally the universe could be discrete. Tis nothing more than a fart of the brain on my part. Excuse. Pass the match.

As per the idea behind a compact space, I suggest this link. I only added the word for you and I...it's like our secret handshake, bo!
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Postby bo198214 » Sun Jul 23, 2006 8:40 pm

*ggg*
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