4 dimensions

[mathwonk]

i recall that some of the tightly rolled dimensions are those of a calabi - yau complex threefold, but this only accounts for 6 real dimensions. what is the 7th?

 

[Moo Of Doom]

i recall that some of the tightly rolled dimensions are those of a calabi - yau complex threefold, but this only accounts for 6 real dimensions. what is the 7th?

Here's a quote from The Elegant Universe that should answer that (from the notes at the back of the book):

With the discovery of M-theory and the recognition of an eleventh dimension, string theorists have begun studying ways of curling up all seven extra dimensions in a manner that puts them all on more or less equal footing. The possible choices for such seven-dimensional manifolds are known as Joyce manifolds, after Domenic Joyce of Oxford University, who is credited with finding the first techniques for their mathematical construction.

 

[mathwonk]

it answers it in the sense of giving it a name, but not in the sense of explaining what that name means.

(To me, a calabi yau threefold is a complex three dimensional manifold with trivial canonical line bundle, i.e. with a never vanishing alternating three - form on its tangent bundle.)

As such, it is a 3 dimensional analog of an elliptic curve. For instance, the hypersurface of complex projective 4 space defined by a general homogeneous polynomial of fifth degree is a Calabi - Yau.

 

[Moo Of Doom]

Sorry, but I don't know more about Joyce manifolds than you do. You can always Google it. Or maybe someone else on here happens to be able to explain them...

 

[selfAdjoint]

Sorry, but I don't know more about Joyce manifolds than you do. You can always Google it. Or maybe someone else on here happens to be able to explain them...

Try this: http://www.maths.ox.ac.uk/~joyce/mrrev.html. Joyce wrote a book.

 

[mathwonk]

thanks, that was very helpful, except i do not know which group G2 refers to.

i.e. apparently a joyce manifold is a compact riemannian manifold with holonomy group G2.

 

[meckano]

Hi, sorry for the lengthy post, but it is something I have been working towards for a few years.
I liked the 2D being trying to see 3D as time.
- there are many ideas I liked, so here I post. Again, sorry for it's length.
-- There is, at the end, a quantitative look at the 4th and well, all dimensions.

simplify calculations for Circular: Vector(1D), Area(2D), Volume(3D)...
In my search to find the the flip side of Pi, a never ending number,
I found the following:

Formulae for circles in any Dimension:

diameter ^ (dimension being worked in: 1D, 2D, 3D...) *

( π )
( _______________________________________ )
( 2 * (dimension being worked in: 1D, 2D, 3D...) )

The birth of the second part of the above equation is explained in
the following example.

2D, Area:
A circle with a diameter of 4 has an area of... π * r^2 = 12.5664
I like 1, so
I found how many circles of diameter 1 were needed to equal the area
of that 4d circle.
The answer is 16: 12.5664 / 0.7854
a) 16 is also the square of the diameter.
b) 0.7854 = area of circle with a diameter of 1.
c) 0.7854 happens to be π / 4 or
π / (2 * (dimension in question, in this case 2D)) =
π / 2 * 2 = π / 4

3D, Volume:
I did the same check with 3D: (4/3) * π * r^3 and
I found that it takes d^3 circles with a diameter of 1.
so again
4^3 = 64,
Volume = 64 * 0.5256(Volume for dia. of 1) = @33.5103
π / 2 * 3 = 0.5256

1D, Vector:
Please note this is supposed to be simple, it is 1D.
- and it relates to circles.
If you travel straight for 4 feet, how many circular feet have you traveled?
4^1 * (π / 2 * 1) = 4 * (π / 2) = 2π
What does this mean?
well we know that the circumference is π * d,
and we don't need to go in a complete circle,
we only need half the circle, hence C = 4π, 4π / 2 = 2π.

The 1D example is used to prove the validity of the formula.

so
1D gives us the circular or arc vector in feet
2D gives us the area in square feet
3D gives us the volume in cubic feet
4D gives us the ?(warp of)? spacetime in ?(quad feet)?
- is it the radius of, or area of, or volume of... who knows.
- but 4D may give radius of,
- then 5D area of,
- and 6d the volume of, warping. (still thinking about what the 4D, 5D... is really telling me.)

If the 4D is tested on a circle with a d of 4, against a circle with a d of 1,
it would take 256 for the circle with a d of 1 to fill or do the same as
the circle with a d of 4.
- assuming the 4th dimension is spacetime,
as it is accepted that space and time are together the 4th dimension,
then what is going on that it takes 256 of the circle with a d of 1
to do the same as the circle with a d of 4?

π = Pi (to be clear :)
If comparing a circle with a d of 1 with that of one with a d of 4 then:
in 1D we need 4 to go the same distance: π/2 vs. 2π ... (4^1=4)
in 2D we need 16 to cover the same area: π/4 vs. 4π ... (4^2=16)
in 3D we need 64 to fill the same volume: π/6 vs. 64π/6 ... (4^3=64)
in 4D we need 256 to xxxx? the same xxxx?: π/8 vs. 256π/8 ... (4^4=256)

Teachers will not like this but:
2D, area:
Take any diameter, square it, multiply by π/4 = area
3D, volume:
Take any diameter, cube it, multiply by π/6 = volume

 

[meckano]

There is a link from another thread that explains the beach sand going into the sand box.
It is under the title of topology. http://en.wikipedia.org/wiki/Topology

I'm refering to the line that reads:
However, it is not possible to deform a sphere into a circle by a bicontinuous one-to-one transformation.

If you take a square foot and stand in the middle of one side, looking toward the opposite side, there is an infinite number of steps. But,
if you set a step-size, to keep things 'real', set by the smallest known particle/wave/energy-level, there is now a finite number of steps.
It would then be possible to transform a 2D representation into a bunch of lines.
- good for getting a reference frame from which to 'see' what is going on. (something I do alot of.)

Now back to the sand. To start:
1 layer of sand(2D + 3D estimate based on 2D diameter) 2 ft^2 into a 3D box volume of 2ft^3.
- I won't use 1 for dimension lengths. Based on my formula for circles, a diameter of 1 is the only one that gets smaller as we prgress into the next dimension.
-- re: 1D gives circular vector π/2 ft, 2D gives area π/4 ft^2, 3D gives volume π/6 ft^3...
(makes me think of water, being the only one to expand below 4C.)
CHANGE: We know there is a guesstimated volume fill limit based on the dimensions of the 2D sand, so:
2 ft^1 * 2 ft^2 = 2(ft^1 * ft^2) = volume of 2 ft^3
ADDITION: volume / volume of sand grains = how many grains.

Now in the 3D to 4D example,
there must be a limit, but what?
Mathematically it is, if dealing with circles and trusting my formula:
2 ft^1, so:
2 ft^1 * 2 ft^3 = ?spacetime? of 2 ft^4 of sand

But that number will only hold true if I can get a relation between all dimensions.
CHANGE: - hmm found something here to help get the 4th dimension wording. It's in the wording.

This is here to help in understanding why I no longer deal with the size of the grains of sand in 4D:
0D: Can not divide by zero. Not yet part of the 'real' world. See quantum physics?
We first had to calculate the physical dimensions of the mass-x grains of sand we would work with.
- quantum physics I presume.(based on strong, weak, whatever forces.)
-- I will think of that as quantum dimensions for now.(Amplitude, Spin...)

1D: Used to calculate the boundary lines/vectors of pysical space to fit siz-x grains of sand for diameter y ft.
CHANGE: ---- not yet a tangiable dimension. remember sand has Second dimension.
2D: Used to calculate the boundary area of pysical space to fit size-x grains of sand for diameter y ft^2.
---- Not yet a tangiable dimension. remember sand has Third dimension.
3D: Used to calculate the boundary volume of pysical space to fit size-x grains of sand for diameter y ft^3.
---- Now a tangiable dimension. remember sand occupies 4D spacetime, or
------ is spacetime the start of something larger? One that covers 2, 3, or more dimensions?

a reach here?
4D: Used to calculate the boundary lines/vectors of spacetime that is affected by mass-x sand * grains in diameter y ft^3.
5D: Used to calculate the boundary area of spacetime that is affected by mass-x sand * grains in diameter y ft^3.
6D: Used to calculate the boundary volume of spacetime that is affected by mass-x sand * grains in diameter y ft^3.

The following is babbling, but I left it in.
after writing all that,
I'm not sure now if the 4th dimension is looking at a larger or smaller scale of the sand.
- it may be looking at the quantum properties of circle with diameter y.
-- so much to think about.
is spacetime the 4th dimension? it can't be the 1st? or just another?
OR
is the 4th dimension, like I say, just on a smaller physical/energy scale but felt farther out, like gravity?
Can't see it, but it's effects are surely seen in real time on large pysical items.

 

[Doodle Bob]

thanks, that was very helpful, except i do not know which group G2 refers to.

i.e. apparently a joyce manifold is a compact riemannian manifold with holonomy group G2.

G_2 is the automorphism group of the octonian algebra. it is also one of the exceptional Lie groups that Calabi (or was it Cartan?) found could be the holonomy group of a compact, simply-connected Riemannian manifold.


For specifics, including the Dynkin diagram of its Lie algebra, go here:
http://en.wikipedia.org/wiki/G2_(mathematics)

 

[meckano]

I realize I have taken Pi*r^2 and 4/3*Pi*r^3 and found them to be dimension based.
1D: (2^1*Pi*r)/(2*1)
2D: (2^2*Pi*r^2)/(2*2)
3D: (2^3*Pi*r^3)/(2*3)

and started working on
4D: (2^4*Pi*r^4)/(2*4)
- glad most of the work is done for me.

 

[Werg22]

Well I don't know very much but I've seen a table where there was temporal dimensions on an axis and spacial dimensions on the other. I prefer to think it is better to seperate time and space into different kind of dimensions and assume that most things are studied in 3rd spacial dimension and 1st temporal dimension.

 

[meckano]

Thank you :)
I've just found a site that is dedicated to a 4th spacial dimension.
I will post some there and see what I get back.
- I have always seen a cube used to explain it. Will ask if a sphere would be easier to explain it.

I'm having a hard time seeing the universe as anything other than 3 dimensional.
- time slowing at fast speeds and this new spatial dimension site keep me going.
-- although spacial #4 may be non existent and time slowing may be an effect of what's going on in 3D.
- ofcourse, if I had the answer, I'd be onto something else that's half proven. :)

 

[meckano]

Wow, glad I did some reading here:
http://tetraspace.alkaline.org/

I now understand. :)
ok, so there are infinite 2D planes that make up a 3D object.
- I now see it as an irrelevant # of 2D planes.
-- 2D, to 3D, is more of a concept that can't exist alone.
There are also infinite 1D lines that make up a 2D object.
- again, irrelevant amount of.
So # of 3D planes in 4D must also be irrelevantly large and
- it also could not exist without the 4th Dimension.

Although,
taking a 4ftx4ft(a constraint) 2D picture and putting it in a 2ftx2ftx2ft(a constraint) 3D box,
one would have to cut and stack the 2D picture.
As it has no depth, there would be no filling whatsoever of the 3D box,
the picture would also loose all meaning, because
the line segments of each of the 4 2ftx2ft 2D cut-ups would overlap.
- seemingly impossible to extract and decipher.
We could keep stacking even after all area of the 2ftx2ft 2D plane was 'represented' by a line segment.
- but if the stacking is kept in mind, then if x number of 2D planes are full,
and the x+1 plane is empty, the first 2ftx2ft cut-up into the x+1 plane is kept intact when viewed from somewhere within that one 2ftx2ft cut-up.

Has me thinking of blackholes.

oh,
anyone know of this formula?
n D S = 2pi * C(n-2), C = S / (n)
i've heard it is the general formula for what I came up with.

 

[meckano]

after thought.
are computer hard drives storing information in 2D or 3D?

i know it is using 3D, but... ya know?

is one point in 3D the intersection of only 3 planes?
(thinking of point Zero on orthogonal axis)
if only 3 we'd need only the x, y, and z at Zero
or
the intesection of all those with angles about the x-y plus
those around the x-z plus
those around the y-z?

I understand getting from (0,0,0) to (0,0,1) would require one step in the z axis,
yet that is only direction and distance, not the make-up of (0,0,1)

Ok, I missed a few axis'.
After realizing that: about the x-y axis is the same as about the z axis,
I can now state that there are as many axis' through (0,0,0)
about which there are infinite 2D planes
as there are points on an infinitely large half sphere
centered on one axis, and resting on the 2 others.
- I understand many points would be in the same plane.
Formula for that? oh, many many many, ok. lol

 

[Werg22]

I think we have to start with this question in order to classify the dimensions;

Does time exists if there is no movement, meaning that everything is stationnary relatively to everything?

 

[meckano]

I think I know what you mean. I started on this whole path due to gyroscopes.
After looking at the pictures in the 4th dim. 'explanations',
and having seen a precessing gyro, imagined it forced to break-neck speeds of precession, and then moving in/through spacetime,
things pointed me towards quantum physics.

My original question, after some time with gyro's, was:
ok, can they cause propulsion, but
now it is:
ok, what are gyro's best suited for?
Better to have a plane than a bathing suit at 10 000 feet, and vice versa at sea level.
:)

 

[hypermorphism]

Does time exists if there is no movement, meaning that everything is stationnary relatively to everything?

If that is true, how does one quantify "stationary" ? This was actually a lively debate during the creation of general relativity.
Note that the ascription of labels to coordinates is an application of mathematics to this particular physical universe, and has no deeper mathematical meaning. Saying that "the" fourth dimension is time is irrelevant to mathematics, which treats dimensions in far more general language.

 

[Werg22]

True, this is a metaphysical subject so mathematical relationships are not to be built. Mathematics seperate time and space, while this "theory" deals with their union. The fourth dimension could be called "spacetime". Also tell me if you agree, if we base ourselves on that theory, light would be solely dimension less.

 

[meckano]

I've been warned. oops.

 

[Werg22]

Yup, that is how I understand it.
Once you reach the speed of light:
1) time stops for you. The beginning of your light speed journey is the end.
2) you must be energy to attain that speed, no living there.
obviously there is more to decode. :)

about my bringing maths into this:
1) oops, sorry
2) I'm not a wizz, I use math to find comparisons, visualize, and then theorize.
like:
I understand in 3D why I need:
64 1"dia. spheres to equal 1 4"dia. sphere when dealing with volume,
and only 8 2"dia. spheres
and only 2.370repeating 3"dia. spheres

So to visualize 4D I use:
256 1"dia. abc's to equal 1 4"dia. abc,
vs. only 16 2"dia. abc's
vs. only 3.16... 3"dia. abc's

and unlike cubes, the diameter is a friendly 'known' place to start.
sometimes i think cubes to visualize the ^2 ness or ^3 edness.

Addit:
oops, stationary question was not for me.
well, light and gamma rays and whatnot are pretty good at defying gravity and being imparted with circular motion about something.
so I again get more appreciation for something, black holes.
- seem! to suck light back towards the living, but spits out some of it as dead gamma rays and whatnot.

Addit 2:
I don't see math as splitting time and space.
It lets me calculate what we call spacial difference.
As I understand space, time, and spacetime, they too are people made words to try and understand what we see.
If time is tied to space, and we calculate the space, are we not calculating the time as well?
That to me is the real question about spacetime. But I'm new and unbound by higher education which invariably imposes thought constraints too.
- Not just words, I always fought the math teacher, he loved me. I did well.
-- In higher maths, there were too many unknowns for my questions to be answered in time to graduate. Now That made/makes no sense to me.
So here we all are, some great thinkers without math skills, and some great mathematicians with their thought process in 'the box'.
- I'm working my free mind towards higher education. Join me, free your mind.
oops, that's the Mars movie. :)

Well, I do not quite agree on your "Addit 2" for the moment. Mathematics let you calculate spacial information indeed, but if spacetime is a dimension, then mathematics cannot represent it. That's why this dimension is split into two information; themporal and spatial. How to express spacetime in terms of mathematics? Also is spacetime absolute?

 

[meckano]

I've been warned. oops.

 

[meckano]

I've been warned. oops.

 

[MBakke]

Thougt of a thing, reading a topic about dimnesions...

if we think like this...

a point had 0 dimensions...
there are infinite many points in a line (which has 1 dimension)
there are infinte many lines in a square (which has 2 dimensions)
there are infinte many squares in a cube (which has 3 dimensions)

following this patterns, I see it reasonable to suggest:

that there are infinte many "cubes" (objects with 3 dimensions) in a 4 dimensional "object"...

so in an "field" (or what to call it) with 4 dimesnions we can fit EVERYTHING that we can see.. like the sun.... or maybe ven the whole univerese....

it might then be that our universe is 4 dimensional, an therfore has infinte "space" (3 dimensions).....

hope I wasnt confusing...

I just want you guys to know that this very old post isn't completely correct.

If the 3 dimensions in the 4D "box" are 5cmx5cmx5cm, you can not fit a 6cmx6cmx6cm cube in it. You can't fit a 10x10 paper (that is hard, "bendable" is not in this "theory") in a 5x5x5 cube. But if the volume of the 4D box was 5x5x5x5, then you can put 5 (quantity) of 5 cm x 5 cm x 5 cm x 1cm (all shapes in our universe are 4D - and more) in there. If the box was 0 size in the fourth dimension, it wouldn't exist in our universe. but if it was, you can fit infinite 0s in 10 cm of the fourth dimension, yes.


Just in case anyone happens to open this from Google like me, looking for answers.