TITLE: How to show that $\{(x,y,z)\in\mathbb{R}^3:x^4+y^4+z^4=1\}$ is diffeomorphic to the $2$-sphere.
QUESTION [5 upvotes]: How to show that the "squared sphere"
  $$\tilde{S}^2=\{(x,y,z)\in\mathbb{R}^3:x^4+y^4+z^4=1\}$$
  is diffeomorphism to the standard $2$-sphere
  $$S^2=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2=1\}?$$

One obvious map between the two is
$$F:S^2\to\tilde{S}^2,\quad F(x,y,z)=(x^2,y^2,z^2),$$
but this is clearly not surjective since it maps only to non-negative numbers. What to do?
Here is how it looks:

REPLY [3 votes]: I would think of having a balloon inside a box.  You inflate the balloon until it completely fills the box.  Mathematically, you pretend like the origin is the center of the balloon and the box.  Each point in the balloon is going to expand directly away from the origin until it reaches the box.  
So, for a point $(x,y,z)\in S^2$, take the vector $<x,y,z>$ and scale it by some  constant $c> 0$ so that $(cx,cy,cz)\in \hat{S}^2$.  I think you can figure out what the actual map should be.