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 referring 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 a lot 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 physical 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 physical 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 physical 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 physical items.