Peculiar nature of $x^{22}+y^{22}=1$

Question

We know that the equation of circle is $$x^2+y^2=1$$ Let $a=x^{11}$ and let $b=y^{11}$ so $$a^2+b^2=1$$ should also be the equation of a circle. But you see, desmos says that the equation is of a rounded square. Can anyone please explain me why is this happening$?$ If I'm wrong ( which should be absolutely true) why is this equation drawing a square. In my opinion, drawing a square needs four straight lines or two pairs of lines. Correct me if I'm wrong$?$

Answer 1

Consider the family of curves defined by the relation $$(x^n)^2 + (y^n)^2 = 1 \tag{1} $$ for positive integers $n$. The case of the circle obviously corresponds to $n = 1$. Your case that you plotted is $n = 11$.

You claim that the two cases should appear the same. However, a simple argument shows why this cannot be true. If we look at the intersection of such a curve with the line $y = x$, we find that the relation in Equation $(1)$ above becomes

$$(x^n)^2 + (x^n)^2 = 1, \tag{2}$$

which in turn implies that $$x^{2n} = 1/2,$$ or $$x = \pm \frac{1}{2^{1/(2n)}}. \tag{3}$$ Therefore, the points of intersection correspond to $$(x,y) \in \left\{\left(-\frac{1}{\sqrt[2n]{2}}, - \frac{1}{\sqrt[2n]{2}}\right), \left(\frac{1}{\sqrt[2n]{2}}, \frac{1}{\sqrt[2n]{2}}\right)\right\}. \tag{4}$$ And for $n = 1$, this is just $$(x,y) \in \left\{ \left(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right), \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \right\} \tag{5}$$ but for $n = 11$, we get something obviously different. Numerically, $1/\sqrt{2} \approx 0.707107$, but $1/\sqrt[22]{2} \approx 0.968984$. The fact that Equation $(3)$ still depends on the value of $n$ means that the curves corresponding to different values of $n$ cannot intersect the line $y = x$ in the same locations. Therefore, the curves cannot remain the same.

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On a more fundmanetal level, I think your question indicates a misunderstanding of which variables you are plotting in the coordinate plane, and how transformations between coordinate systems do not in general preserve the geometric properties of curves plotted in those systems.

For example, suppose I asked you to plot $y = x^2$. This is a standard parabola. But if I were to use your same reasoning, I could claim that it should be a line, because if I let $a = x^2$ and $b = y$, then we get $b = a$, and the plot of $b = a$ is a line.

But the flaw in that argument is that such a transformation of variables, namely $$(a,b) = (x^2,y)$$ does not need to preserve the geometry of the curves in question. If I plot $y = x^2$ with $x$ on the horizontal axis and $y$ on the vertical axis, then we get a parabola. And if I plot $b = a$ with $a$ on the horizontal axis and $b$ on the vertical axis, I get a line. But the $xy$-plot and the $ab$-plot have different shapes precisely because the relationship between them is nontrivial. Otherwise, you could claim every curve has the same shape through a suitable variable transformation. That is of course untrue.