Is a hypersphere of non-integer dimension a fractal? 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?
 A: 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. 
A: As I remember works of Benua Mandelbrot, the fractal object must have not only the non-integer dimension but also the property of the self-similarity. For example, any part of a straight line segment also is a line segment, so in this case we have the self-similarity but do not have the non-integer dimension. The hypersphere in a space with the non-integer dimension can be a fractal if it is self-similar, but I doubt that is a fact.
