Can you provide examples of fractal coordinates for points?

Question

On a 1D line there is one real number needed to uniquely locate a point by coordinates.

On a 2D plane, a pair of real numbers are needed.

But what if a fractal has dimension d, between 1 and 2 ($1<d<2$), how can points on the fractal be specified with more than one, but less than 2 coordinates?

Can you provide examples of 1.xD coordinates for points?

Edit: I'm asking for a numerical example, so this is not the same question tan this one.

A point in the plane can be specified by a single number in base 4 if each one of the symbols represents one quadrant

O=0
→=1
↑=i
↗=1+i
. is a decimal separator

a point with coordinates 1.5+2.5i can be written ↑→.↗ meaning (the dot is a decimal separator)

$\uparrow \rightarrow . \nearrow = i*2^1+1*2^0+(1+i)*2^{-1} $

That representation system can be rewritten as pair of numbers in decimal system.

On the same way, a point on sierpinski triangle can be written in base 3 using a symbol for each of the 3 subtriangles Any point on the fractal can be specified with arbitrary precision with a string like

←→↑←↑↑←←→.←↑←→←←↑

where each position is multiplied by a power of 2 and

←=0
→=1
↑=0.5+(3/4)^½i
. is a decimal separator

So, I expect that number to be decomposed into 1.D coordinates as a real number, and something else.

On information theory, a character may represent a non integer number of bits, and that is solved by combining multiple characters, so I expected a similar solution for expressing coordinates.

Note that on the real line, if base N is used, the characters are multiplied by power of 4, but on other dimensions, it looks like the dimension depend on the difference between the base N and the power used as positional weight.

Answer 1

You're confusing the topological dimension of an embedding with the intrinsic dimension. You need as many (integer) coordinates as the topological dimension.

Answer 2

When we talk of "dimensions" of fractals we mean something entirely different that the number of independant spatial coordinates to determine a point.

So the question of how to indicate a point doesn't apply or even make sense.

So what does the dimensions of a fractal mean?

In euclidean geometry, if we increase the radius of an $n$dimensional object in $\mathbb R^n$ by a factor of $k$, then the hyper-volume of the object increase by a factor of $k^n$. And it is the linear increase causes an exponential growth in volume that we mean by "dimension" and that has nothing to do with coordinates.

So the area of a fraction increases in comparison to an increase in linear size by a non-integer factor.

So in that definition it is fractional dimensional. But it has nothing to do with coordinates.

https://medium.com/@cosinekitty/understanding-fractional-dimensions-f2ed2e4e1600

(Sometimes we teach by analogy. But then analogy breaks. The classic example if this is $b^{n} = \underbrace{b\times b\times b......\times b}_{n\text{ times}}$. And as just by counting we must have $b^{n+m} = b^nb^m$. But what if $x\not \in \mathbb Z$? What does $b^x$ mean? Well, knowing that $b^{x+y} = b^xb^y$ we come up with another definition based on... other stuff. So $2^\pi = 8.824977827076287623856429604208$...... Punch it into a calculator! That's what you get. But does that mean $2^\pi = \underbrace{2\times2\times 2\times ....}_{\pi \text{ times}} = 8.824977827076287623856429604208$???? How the heck do you multiply a number by itself $\pi$ times? Does that make any sense? The answer is "no" it doesn't make sense but that is not what $b^x$ means any more. It means something else.)

Answer 3

Such a coordinate system won't exist. I draw an analogy with polynomials. A linear function can be specified by two reals and a quadratic can be specified by three. Yet when we attempt to find a polynomial to represent a function like $ x^{1.5} $, you do not get an expression with 1.5 coefficients, but infinitely many. The series expansion is infinite.

Similarly, it would take infinite information to specify a point on the Sierpinski triangle. So there is something fundamentally different about fractional-dimensional spaces versus integral dimensional spaces, just as polynomials are different from functions involving fractional powers of $ x $.