How to convert an object into a sphere? I'm not sure if I understand it enough, but doesn't the Poincare conjecture prove any shape can be turned into a sphere? How would I go about transforming such an object? Like let's say I have a rectangular prism with length, width, and height 3,4,5. How would I find the characteristics of the sphere after I create it. This is topology correct? What would I study or look up to understand? Thanks for any help! Update: From the previous comment "A prism is the boundary of a convex subset of R3 R3 so just pick a point on the interior and project along a line from that point onto a sphere which is large enough to enclose the prism. This will define a homeomorphism from the prism to the sphere." Can someone explain this in more detail?

$\
ewcommand{\Reals}{\mathbf{R}}$The stereogram shows a rectangular prism being mapped to a larger sphere by radial projection from the center. (The action on the vertices is indicated by dashed lines.)

![Stereogram of mapping a rectangular prism to a sphere](

Analytically, center the prism $P$ at the origin, let $R$ denote the radius of the sphere $S$, and let $$ T(x, y, z) = \frac{R}{\sqrt{x^{2} + y^{2} + z^{2}}}(x, y, z) $$ denote radial projection from $\Reals^{3} \setminus\\{(0, 0, 0)\\}$, the complement of the origin, to the sphere $S$. The mapping $T$ is continuous on its domain, the prism $P$ is a closed, bounded (i.e., compact) subset of $\Reals^{3}$ that does not contain the origin and that intersects each ray from the origin exactly once. It follows that $T$ defines a continuous bijection from $P$ to $S$. Since $P$ is compact, this bijection is a homeomorphism.