# Fractal "dimension"

I have been learning about the idea of "dimension" of a shape in terms of how its "mass" scales down when we cut it into self-similar parts.

For example,

However, the term "dimension" is closely linked with the idea of degree of freedom. So my question is, is there any sense in which the dimension of a fractal is linked with a "fractional" degree of freedom? Perhaps leading to some implications of a thermodynamic nature?

• YoungPhysicist

DaveC426913
Gold Member
The word fractal is, in fact, a short form of fractional dimension.

Simplistically, if I understand it correctly, if your 1-dimensional line is complex enough, you can ultimately reach virtually any point in the 2-dimensional plane. The degree to which you can do this is the fractional component.
i.e. if you can reach m points in the 2-dimensional plane, you have a 1-point-n dimensional structure.

• Swamp Thing
Mark44
Mentor
However, the term "dimension" is closely linked with the idea of degree of freedom. So my question is, is there any sense in which the dimension of a fractal is linked with a "fractional" degree of freedom?
As far as I know, there is no connection between fractional dimension and degrees of freedom.

DaveC426913
Gold Member
As far as I know, there is no connection between fractional dimension and degrees of freedom.
? Then I must be mistaken.
AFAIK, something is 1D if you have one degree of freedom (such as a line).
If that line is sufficiently curved, you can achieve an amount of freedom greater than one but less than two.

Mark44
Mentor
? Then I must be mistaken.
AFAIK, something is 1D if you have one degree of freedom (such as a line).
If that line is sufficiently curved, you can achieve an amount of freedom greater than one but less than two.
Normally "degrees of freedom" are integral values, but in non-standard regression (which is something new to me), non-integer values are permitted. I haven't read of the term "degrees of freedom" used in discussions of fractals, though, but I haven't done any recent reading about fractals, so maybe someone has made the connection.

The usual meaning of the term fractal is that a curve or surface is self-similar, meaning that it looks the same even when magnified over and over. The usual method of measuring curves assumes that for small enough intervals, the curve is approximately a straight line segment, so we can add up all the pieces to get the overall length of the curve. On the other hand a curve such as the Koch snowflake has little jaggies at all resolutions, so no matter how much we magnify the image, we can find any straight line segments. Mathematically, this curve is nowhere differentiable. Although the area of the Koch snowflake is 1.6 times as large as the triangle it is based on, the perimeter of the Koch snowflake grows large without bound (see https://en.wikipedia.org/wiki/Koch_snowflake). Normal geometric curves have a defined perimeter, so because the perimeter of this figure becomes infinite, but encompasses a finite area, it is considered to have a dimension somewhere between 1 and 2 (about 1.26 according to the wiki article).

I have been learning about the idea of "dimension" of a shape in terms of how its "mass" scales down when we cut it into self-similar parts.

For example,

However, the term "dimension" is closely linked with the idea of degree of freedom. So my question is, is there any sense in which the dimension of a fractal is linked with a "fractional" degree of freedom? Perhaps leading to some implications of a thermodynamic nature?

I think that the conection exists, look at this paper : "Effective degrees of freedom of a random walk on a fractal" (Alexander S. Balankin), if in the studied phenomena, space-time structure could be treated as fractal certain phenomena developed in it could respond with fractional degrees of freedom when looking at it in an adecuate scale range

There is an very interesting book from Laurent Nottale : Scale Relativity and Fractal Space-Time, about these kind of problems

• Swamp Thing