What is the math from turning a circle into a sphere and beyond ?

Although hypercube can refer to a four dimensional cube, the term also refers to any n dimensional cube. The proper term for 4 dimensional cube is teseract. Hypersphere also refers to spheres in more than three dimensions. I did a google search and found this picture of a hypersphere, which I think does a pretty good job of representing it.

Calc 1, solids of revolution.

You can’t square a square into a cube. (x^{2) defines a square. (x}3) defines a cube.

Squaring the square (x²⁾2 is x^4, a hyper cube, not a cube.

@all Thanks

new question What is the square of a triangle?

The question has been answered. A tetrahedron is the 3 dimensional form of a triangle and the general term in k dimensions is k-simplex.

Good YouTube video Strange Spheres in Higher Dimensions – Numberphile

Also perfect shapes in higher dimensions – numberphile YouTube