The discussion centers on the transformation of a rectangle into a torus while excluding edge splicing. Participants assert that this transformation, when splicing is omitted, can be classified as a homomorphism. They propose the use of rotation matrices to establish a bijective function mapping between the two spaces. Additionally, the transformation is characterized as both a homeomorphism and a diffeomorphism, provided that no folds occur during the bending process.
PREREQUISITESMathematicians, topologists, and students studying geometric transformations, particularly those interested in the properties of shapes and their mappings.
If you don't clue the edges, then it is only a "bending", which is both, homeomorph and diffeomorph (as long as you don't fold it somewhere).aheight said:I understand the transformation in general is not homomorphic but what about the transformation minus the splices, that is, contort it all the way up to and not including splicing the edges? Isn't that a homomorphism? Can't we define a bijective function (rotation matrices) to map the two spaces? Seems also to be diffeomorphism as well or no?