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Could someone give and explicit homeomorphism between the tetrahedron with vertices $(0,0,0)$, $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ and the $2-$sphere?

Thanks

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1 Answer

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As with most problems about mapping things* continuously to spheres, this involves normalizing a vector.

All we need to do first is to move the origin inside our object. The function $\alpha : (x, y, z) \mapsto (x - 1/4, y - 1/4, z - 1/4)$ should do the trick.

If we let $\beta : (x, y, z) \mapsto \frac{(x, y, z)}{\sqrt{x^2 + y^2 + z^2}}$ be the normalization map, then $\beta \circ \alpha$ is a homeomorphism to the $2$-sphere.

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* convex, closed surfaces

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