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Features - June 25, 2008
What is a Dimension Anyway?

This story is a supplement to the feature "Using Causality to Solve the
Puzzle of Quantum Spacetime" which was printed in the July 2008 issue of
Scientific American.

A Whole New Dimension to Space
In everyday life the number of dimensions refers to the minimum number
of measurements required to specify the position of an object, such as
latitude, longitude and altitude. Implicit in this definition is that
space is smooth and obeys the laws of classical physics.

But what if space is not so well behaved? What if its shape is
determined by quantum processes in which everyday notions cannot be
taken for granted? For these cases, physicists and mathematicians must
develop more sophisticated notions of dimensionality. The number of
dimensions need not even be an integer, as in the case of
fractals—patterns that look the same on all scales.

Cantor Set : Take a line, chop out the middle third and repeat ad
infinitum. The resulting fractal is larger than a solitary point but
smaller than a continuous line. Its Hausdorff dimension [see next page]
is 0.6309.

Sierpinski Gasket: A triangle from which ever smaller subtriangles have
been cut, this figure is intermediate between a one-dimensional line and
a 2-D surface. Its Hausdorff dimension is 1.5850.

Menger Sponge: A cube from which subcubes have been cut, this fractal is
a surface that partially spans a volume. Its Hausdorff dimension is
2.7268, similar to that of the human brain.

Generalized Definitions Of Dimensions

Hausdorff Dimension
Formulated by the early 20th-century German mathematician Felix
Hausdorff, this definition is based on how the volume, V, of a region
depends on its linear size, r. For ordinary three-dimensional space, V
is proportional to r3. The exponent gives the number of dimensions.
“Volume” can also refer to other measures of total size, such as area..
For the Sierpi´nski gasket, V is proportional to r1.5850, reflecting the
fact that this figure does not even fully cover an area.

Spectral Dimension
This definition describes how things spread through a medium over time,
be it an ink drop in a tank of water or a disease in a population. Each
molecule of water or individual in the population has a certain number
of closest neighbors, which determines the rate at which the ink or
disease diffuses. In a three-dimensional medium, a cloud of ink grows in
size as time to the 3/2 power. In the Sierpi´nski gasket, ink must ooze
through a twisty shape, so it spreads more slowly—as time to the 0.6826
power, corresponding to a spectral dimension of 1.3652.

Applying the Definitions
In general, different ways to calculate the number of dimensions give
different numbers, because they probe different aspects of the geometry.
For some geometric figures, the number of dimensions is not fixed. For
instance, diffusion may be a more complicated function than time to a
certain power.

Quantum-gravity simulations focus on the spectral dimension. They
imagine dropping a tiny being into one building block in the quantum
spacetime. From there the being walks around at random. The total number
of space­time building blocks it touches over a given period reveals the
spectral dimension.

The trouble with these interpretations is already exposed in this
brief. Within the description of spectral dimension we see the
statement
"In a three-dimensional medium, a cloud of ink grows in size as
time to the 3/2 power."
Thus the word dimension is being used to mean different things, some
of which are more fundamental than others. Even in standard usage if
we ask for the dimensions of a box we are not likely going to get the
answers two or three back. Instead we are likely to get back a series
of continuum based values. So our usage of this word is pretty badly
flawed since we are using it to mean two complementary things, where
the discrete and continuous properties within our space description
are being treated with the same term.

Here is a construction which stays within the traditional discrete
dimensional context yet allows an interpretation of more or less
space:
http://bandtechnology.com/ConicalStudy/conic.html
To state the level of consumption of a space is close, but this
construction poses that we can have as much of an n dimensional space
as we would like.
- Tim


Your conical diagrams / picture remind me of a scientific American

Timothy Golden BandTechnology.com wrote:

The trouble with these interpretations is already exposed in this
brief. Within the description of spectral dimension we see the
statement
"In a three-dimensional medium, a cloud of ink grows in size as
time to the 3/2 power."
Thus the word dimension is being used to mean different things, some
of which are more fundamental than others. Even in standard usage if
we ask for the dimensions of a box we are not likely going to get the
answers two or three back. Instead we are likely to get back a series
of continuum based values. So our usage of this word is pretty badly
flawed since we are using it to mean two complementary things, where
the discrete and continuous properties within our space description
are being treated with the same term.

Here is a construction which stays within the traditional discrete
dimensional context yet allows an interpretation of more or less
space:
http://bandtechnology.com/ConicalStudy/conic.html
To state the level of consumption of a space is close, but this
construction poses that we can have as much of an n dimensional space
as we would like.
- Tim

Your conical diagrams / picture remind me of a scientific American
article a friend sent me several years back on Thom's Catastrophe theory
( Sci Am Arr 1976):http://en.wikipedia.org/wiki/Catastrophe_theory

"In mathematics, catastrophe theory is a branch of bifurcation theory in
the study of dynamical systems."

Specifically the Swallow tail:
F(x)=x^5/5 -c*x^3/2-b*x^2/2-a*x

Swallowtail catastrophe

V = x^5 + ax^3 + bx^2 + cx \,

The control parameter space is three dimensional. The bifurcation set in
parameter space is made up of three surfaces of fold bifurcations, which
meet in two lines of cusp bifurcations, which in turn meet at a single
swallowtail bifurcation point.

As the parameters go through the surface of fold bifurcations, one
minimum and one maximum of the potential function disappear. At the cusp
bifurcations, two minima and one maximum are replaced by one minimum;
beyond them the fold bifurcations disappear. At the swallowtail point,
two minima and two maxima all meet at a single value of x. For values of
a>0, beyond the swallowtail, there is either one maximum-minimum pair,
or none at all, depending on the values of b and c. Two of the surfaces
of fold bifurcations, and the two lines of cusp bifurcations where they
meet for a<0, therefore disappear at the swallowtail point, to be
replaced with only a single surface of fold bifurcations remaining.
Salvador Dalí's last painting, The Swallow's Tail, was based on this
catastrophe.