Thanks to the gamma function the formula for the surface of a unit http://mathworld.wolfram.com/Hypersphere.html $$ S(n) = \frac{2 \pi^{n/2}}{\Gamma(n/2)} $$ allows to calculate the surface of a hypersphere of non-integer dimensions. I wanted to know, what is the number of dimensions I need, so that the surface of a n-sphere (with radius 1) equals the area of a square (with "radius" 1), which means solving the equation $$ 4 = \frac{2 \pi^{n/2}}{\Gamma(n/2)} $$ for $n$. Since $S(n)$ has a maximum at $n=7.256...$ one get't two positive solutions: $$ n=1.534...\\ n=15.86... $$ (see https://www.wolframalpha.com/input/?i=2Pi%5E(n%2F2)%2FGamma(n%2F2)+%3D%3D+4).
Now my questions is: Since the number of dimensions of the n-sphere from this equation is a non-integer, does that mean such a sphere would be a fractal? If so, is it possible to construct a n-sphere with 1.534 dimensions somehow and draw it?
Since no one has posted anything yet, I'll at least give you something to think about.
The d-dimensional generalization of the Euclidean length formula is,
$$(1) \quad L=\sqrt{\sum_{n=1}^d x^2_n}$$
For fractional dimensions, we need to be able to evaluate the quantities inside the square root in a consistent manner.
In the pleasant case that $x_i=x$, we have,
$$(2) \quad \sum_{n=1}^d x^2_n=d \cdot x^2$$
If we assume $(2)$ holds for non-integer $d$ then we can substitute $(2)$ into $(1)$ and obtain,
$$(3) \quad L=|x| \cdot \sqrt{d}$$
Now, for a d-dimensional sphere, the points that make up the object are found by finding the set of all spatial points such that $(1)$ holds. Here, we are solving a subset of that problem. Namely, we are looking for spatial points, such that the coordinates can be permuted and still satisfy $(1)$.
Solving for $x$, we obtain,
$$(4) \quad |x|=\cfrac{L}{\sqrt{d}}$$ $$\Rightarrow x=\pm \cfrac{L}{\sqrt{d}}$$
For a unit sphere, $L=1$, and $d=1.534$, we have,
$$x=\pm 0.80739...$$
We also have the scenario where $x_i=L$ and $x_j=0$ for $j \not =i$. In this case also we have the conditions of $(1)$ satisfied still without having fully defined what coordinates are in fractional dimensions.
What does any of that mean? IDK...but with appropriate axioms there's nothing that prevents interpretation.
Extra Bit: I should mention that if a proper formalism was developed, we'd be seeing these "fractals" in their natural habitat. There's no reason to assume that they'd look like typical fractals, if viewed from this vantage point.