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?
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
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).