Models of fractal with rational valued dimension Most constructive calculable models of geometric objects with non integer dimensionality as Cantor set, Koch curve, Sierpinsky triangle, etc.;  give rise to values of dimensionality that are integer base logarithms  of integers as $\frac{\log(2)}{\log(3)}$, $\frac{\log(3)}{\log(4)}$, $\frac{\log(3)}{\log(2)}$, etc. ; and in consequence are trascendental numbers because of Lindemann results. 
Do you know some finite deterministic recipe for constructing geometric objects with rational values of dimensionality as $\frac{1}{2}$ or $\frac{3}{2}$? 
Conversely: Could it be shown that such recipes don't exist at all?  
 A: The "usual" ternary Cantor set is the fixed point (or attractor, or invariant set) of an iterated function system with two maps on $\mathbb{R}$:
$$\varphi_1(x) = \frac{1}{3}x \qquad\text{and}\qquad \varphi_2(x) = \frac{1}{3} x + \frac{2}{3}. $$
That is, the Cantor set is the unique nonempty compact set $\mathscr{C}$ such that
$$\mathscr{C} = \varphi_1(\mathscr{C}) \cup \varphi_2(\mathscr{C}). $$
As you note, the dimension of the Cantor set is $\log_{3}(2)$ (though this depends on what notion of dimension we are using—I'll assume that the Hausdorff dimension or the box-counting dimension is meant; in this case, they coincide).  It turns out that this dimension can be computed in terms of the iterated function system.
Suppose that $\{\varphi_j\}_{j=1}^{J}$ is a finite collection of maps on $\mathbb{R}^d$ of the form
$$ \varphi_j(x) = r_j O_j x + b_j, $$
where each $r_j \in (0,1)$ is the contraction ratio of the map $\varphi_j$, each $O_j$ is some unitary map (i.e. a reflection in $\mathbb{R}$, or a rotation or reflection in $\mathbb{R}^2$), and each $b_j \in \mathbb{R}^d$ is a translation (such a map is called a contracting similitude, and the collection of maps is a self-similar iterated function system).  A fairly powerful theorem (the Banach fixed-point theorem) guarantees that there will be a unique nonempty compact set $\mathscr{A}$ such that
$$ \mathscr{A} = \bigcup_{j=1}^{J} \varphi_j(\mathscr{A}). $$
Moreover, if the iterated function system is sufficiently "well-separated," i.e. if there is some open set $U$ such that


*

*$\varphi_j(U) \subseteq U$ for all $j$, and

*$\varphi_j(U) \cap \varphi_k(U) = \emptyset$ for all $j\ne k$,


then the Hausdorff dimension of $\mathscr{A}$ will be the unique real solution $s$ to the Moran equation, which is given by
$$ 1 = \sum_{j=1}^{J} r_j^s. $$
If all of the contraction ratios are the same, i.e. if $r_j = r$ for all $j$, then the Moran equation simplifies to
$$ 1 = \sum_{j=1}^{J} r^s = J r^s
\implies r^s = \frac{1}{J}
\implies s \log(r) = -\log(J)
\implies s = -\frac{\log(J)}{\log(r)}. $$
Note that $r < 1$, hence $\log(r) < 0$, which implies that $s$ is positive (which is what we would expect).

All of this gives us one possible way of engineering sets of arbitrary Hausdorff dimension.  For example, suppose that we wish to construct a set with Hausdorff dimension $\frac{1}{2}$.  In a fair and just world, we could find such a set as the attractor of an iterated function system $
\{\varphi_1, \varphi_2\}$ on $\mathbb{R}^2$, so suppose that we live in a fair and just world.  Since the dimension of our set $s = \frac{1}{2}$, and we have two maps, the Moran equation gives us
$$ \frac{1}{2} = -\frac{\log(2)}{\log(r)}
\implies \log(r) = -2\log(2)
\implies r = \mathrm{e}^{-2\log(2)} = \frac{1}{4}. $$
Therefore an iterated function system with two maps of contraction ratio $\frac{1}{4}$ will get the job done, assuming that we can find one which satisfies the separation condition above.  But this can be done by setting
$$ \varphi_1(x) = \frac{1}{4} x
\qquad\text{and}\qquad
\varphi_2(x) = \frac{1}{4}x + \frac{3}{4}. $$
Note that
$$ \varphi_1((0,1)) = (0,\tfrac{1}{4})
\qquad\text{and}\qquad
\varphi_2((0,1)) = (\tfrac{3}{4},1), $$
where both sets are disjoint and contained in $(0,1)$.  Thus this iterated function system does, in fact, satisfy the separation condition, and we do live in a fair and just world!  In other words, we have found a way of building a set with Hausdorff dimension $\frac{1}{2}$.

More generally, given any positive real number $s$, it is possible to build an iterated function system with attractor $\mathscr{A}$ such that the Hausdorff dimension of $\mathscr{A}$ is equal to $s$.  The general idea is the same as shown above, but the details might require a little more care.  For example, if $s > 1$, then more than two maps will be required to build such a set, and we'll have to work in a larger ambient space (i.e. $\mathbb{R}^d$ with $d > 1$).  Alternatively, we could construct Cantor-like sets such as the one above, then start working with Cartesian products (though there are some other technical details which need to be addressed).
