Square with sides curving inwards

Question

Whilst plotting some charts on Desmos, it was noticed that the following formula $$x^{m}+y^{m}=k\\ \text{where}\begin{cases} 0<m=0.08n<1\\n\in \mathbb N, 0<n<12.5\\ k>0\end{cases}$$ plots a square with sides curving inwards.

Questions:
1. Why is this so, i.e. why does the curvy square appear only when $m$ is a multiple of $0.08$ and not for other values?
2. What is the name of the shape of the figure plotted?

The diagram below shows the plot for $x^{0.8}+y^{0.8}=10$, i.e. $m=10, k=10$.

$\qquad\qquad\qquad$

Edit

Some additional information has been found. The wiki reference here refers to this as a superellipse. However it does not address the question $1$ above, i.e. why does this figure appear only for powers which are multiples of $0.08$?

Answer 1

Why it doesn't work:

A super ellipse (in the form $|\frac{x}{a}|^n + |\frac{y}{b}|^n = 1$) only takes the form of the 4-star you noticed when $0<n<1$. So all values of $n \in (0,1)$ should produce a star. So why wasn't yours?

Notice the difference between the super ellipse equation and the equation you used ... $$x^n + y^n \neq |x|^n + |y|^n \quad\forall n \in \mathbb{Q}$$
(you can disregard $a$ and $b$, because they are only scaling factors)

So try putting the absolute value on $x$ and $y$ and you'll notice that your equation produces the desired output when $0<n<1$.

Why did $0.08$ work?

Note that the exponent $n$ can be expressed as $\frac{a}{b},\, a,b \in \mathbb{Z}$. Henceforth, when $x$ or $y$ is negative, $b$ (the denominator of $n$) is even, and $a$ (the numerator of $n$) is odd, you are taking an even root of negative number, which is a complex number. This is why the 2nd, 3rd, and 4th quadrants won't "show up" when you plot the equation.

Now $0.08 = \frac{2}{25}$, so when $x$ and $y$ are raised to that 2nd power they become positive and the 25th root allows for negatives without complex results. Try $n=0.4$ or $n=0.8$ without the absolute value and this should plot fine as well.