# Does a fractal exist for any given dimension?

For example a Sierpinski Triangle has a dimension of $log_23 \approx 1.58$, so if I named a dimension such as $log_2301$ is there definitely a fractal that exists in that dimension?

Theorem: For any given $r > 0$, there are uncountable fractals with Hausdorff dimension $r$ in $n$-dimensional Euclidean space $\mathbb{R}^n$ ($n \geq - \lfloor -r \rfloor$).
The parenthetical condition just means we don't try to cram a $>n$ dimensional fractal into a $n$ dimensional space.