## Thursday, December 4, 2008

One of the most unfair criticisms I hear about this project is that it somehow misuses or misunderstands the term "dimensions". In "What Would a Linelander Really See?", we talked about how the definition of dimensions that this project uses is aligned with the most basic definition of dimensions as found in wikipedia. What I want people to understand is that the extra dimensions physicists are describing are still spatial dimensions (or, as some physicists call them, "space-like dimensions"), and my way of visualizing the dimensions builds from that premise. Here's an easily-related concept from wikipedia: the point-line-plane postulate.

Point-line-plane postulate

The point-line-plane postulate in geometry is a collective of three assumptions (axioms) that are the basis for Euclidean geometry in three or more dimensions.

1. Unique Line Assumption
There is exactly one line passing through two distinct points.

2. Number Line Assumption
Every line is a set of points which can be put into a one-to-one correspondence with the real numbers. Any point can correspond with 0 (zero) and any other point can correspond with 1 (one).

3. Dimension Assumption
Given a line in a plane, there exists at least one point in the plane that is not on the line. Given a plane in space, there exists at least one point in space that is not in the plane.
I've added the italics on "three or more dimensions" for emphasis, as that sums up the game we're playing here: using what we know about the first three dimensions, we can continue to stack one dimension upon another using the same logic.

Here's one more idea for you to consider - in You are Me and We are All Together, we talked about renowned physicist Richard Feynman's proposal that the reason all electrons are absolutely identical is because there is really just one electron in the universe, whizzing back and forth within timelessness, and the trillions of electrons we see around us are just multiple copies of that same electron as it completes its journey back and forth from the beginning to the end of time. Why are all electrons identical? If you look up "point particle" you'll see that electrons are described as "point-like particles", which means they actually have no size and no dimension - just like the point with which we start our tenth dimension animation.

So this time around, let's think back to the original animation and review how this way of visualizing the extra dimensions relates to "points" that are moving within the dimensions, and how we can start from our first three dimensions with which we're so familiar to imagine the extra dimensions beyond spacetime.

Then, we can imagine a second point
and create a line segment with these two points at the end
We can imagine a line passing through these two points
extending to infinity in either direction.

Adding all of the possible values together
on either side of the point we started from
creates a perfect symmetry
which adds back up to where we started:
Zero, a point of indeterminate size.

This will be true
no matter where you start
or how many dimensions you're imagining
Because each new dimension adds two more directions
and they will always head towards infinity in either direction.
But what do we mean by infinity?

Infinity is a tricky word. Is there more than one infinity?
Or is it more correct to say that there are many ways to get to infinity?
If I start counting 0,1,2,3... and so on,
there's no end to the numbers that I could count.
If I start counting 0,2,4,6,8... and so on,
there's also no end to the numbers that I could count.
If I start dividing any number in half, and half again, and half again,
there's no end to the number of times I could keep dividing that number in half.
Each is a way to get to infinity.

Are each of the infinities we just imagined a different size?
We should always keep reminding ourselves - Infinity is Not a Number
So even though one infinite set can be a subset of another infinite set
(which means that saying one version of infinity is larger than another
does have a certain usefulness in helping to imagine all this)
ultimately all infinities are the same size,
because all infinities are of indeterminate size
Just like the point we started from.

Let's imagine that first and second point again.
They could have been anywhere, in any dimension
And there would be an infinite number of points
on the line that passes through those 2 points
But even though there are an infinite number of points on that line,
there could still be another point that we could imagine that's not on that line
No matter where we place that additional point, we'll now have to think of not a line
But a plane, and that plane will extend to infinity in both of the new directions we just added

Again, this new point could have been anywhere
As long as it isn't on the line we started from
But no matter where we place it the plane we're creating
is still just a subset of all possible planes
And no matter what plane we imagine, we can still add an additional point
That is not on that plane and requires us to add an additional dimension
Again, with two new directions that extend both ways to infinity

This cycle can be repeated endlessly -
define a system, add a point that isn't within that system
add a dimension for that new point to be within
which adds two new opposing directions that each extend to infinity
in either of those two new directions.

For the first few dimensions, this is easy to imagine with graphs and arrays -
we can imagine a two-dimensional data set, a two-dimensional array (x,y)
with values for two different co-ordinates:
X on one axis, Y on the other, simple to draw on a piece of paper.

But this gets harder to picture as the number of dimensions climb:
So while we can easily define a seven-dimensional array with seven different co-ordinates,
Visualizing the graph that could represent such an array
is not an easy thing for our 3D sensibilities to accomplish.

So...
If we're trying to visualize our reality as coming from extra dimensions
it's helpful for us to keep imagining what new degree of freedom
The "point-line-plane postulate's" idea of using a current dimension
to define a line,
the next dimension up to define a plane
and the dimension above that to define a space created
by those potential lines and potential planes
Is a way for us to keep visualizing
past what we as 3D creatures

We're going to continue talking about these ideas next time, with an entry called "You are a Point Within the Omniverse". In it, we're going to go back to the idea that the omniverse is an enfolded symmetry state, which we can think of as a perfectly balanced zero. But one of the ideas we haven't talked about much is how that symmetry state is always ready to fall out of balance and create a universe - it's like a pencil balanced on its tip, always ready to fall one way or another and create a new pattern in the information that becomes our reality or any other.

To finish, a song sung for me by Ron Scott, one of the 26 songs attached to this project. This one is about the mysterious spark of life and consciousness, a point moving within the omniverse. The song is called "Burn the Candle Brightly".