If you mean a spiral then in polar coordinates the function r(theta)=k theta (where k is some constant) will work. If you want a vector field then something like F(x,y)=(-y,x) will work.
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No shape is mathematical really unless it has been created by a mathematical formula, but is certainly a geometric shape. But anything which is a 2D or 3D shape is geometric. My improvement: A catenary curve from a mathematical equation such as cosh x, is a mathematical and natural shape. Maby each other arch can be approximated by a mathematical formula.
Any shape is mathematical.
It is a cube.cube
a washer
sphere