# Stitch edge of disk to edge of square in 3D space (with deformations) [closed]

Topologically each point on the edge of a square can be mapped uniquelly (including the corners?) to a point on the edge of a circle.

It seems it might be possible to deform in 3D space the square and the disc so that they can be joined at the edge without any gaps. A colleague managed to do that in the real world with paper objects. Obviously this is not a precise result, so it may be wrong.

As a commenter said, the deformation should be isometric.

Can you make Mathematica solve this problem and generate a 3D surface showing a isometrically deformed square and disc joined at the edge?

Pictures taken from http://www.unitaryflow.com/2015/03/round-squares-exist.html

• You need more constraints on your problem. I can just generate you e.g. a cube with a face cut off, and assert that "this is a deformed circle." – djp Mar 14 '15 at 22:12
• @djp: Given the example of paper cutouts, I assume that the deformation is meant to be isometric. See also: dForm – Rahul Mar 15 '15 at 4:53
• @Rahul interesting. I wish the OP had a little more of "this is what I've tried" etc. – djp Mar 15 '15 at 10:47

circ := {Cos[#], Sin[#]} &