Is there a thing named a "spiral plane" which is a plane but it's spiral?

Question

Hello, I'm wondering if there is such thing like this. Is there a plane which is not flat but spiral and extending for infinity? I have drawn a representation for what I mean but it's not thorough it's getting narrower inside it and getting further apart from the outside and it extends for infinity and extends from either sides? do you understand me?

Answer 1

I was taught to call such an object a "cylinder." Begin with a spiral in the plane, for example http://en.wikipedia.org/wiki/Logarithmic_spiral , then put that in $\mathbb R^3$ by doing the same thing for every $z$ value.

Answer 2

If I understand you correctly, the shape you mean is given (up to similarity transformations) by the parametrisation $$(u,v) \mapsto (\exp(\alpha u)\cos(u),\exp(\alpha u)\sin(u),v)$$ where $\alpha$ is some nonzero real number. It is the Cartesian product of a logarithmic spiral with a straight line. I don't know if this shape has a name.