Hi,

I think no such solutions exists in the field of natural numbers ( we should use infinitely many numbersas as we go to the infinity) However this is another question if we takes into account real numbers or complex numbers. It would be interesting to see such a solution.

Dear Michal,

The prototype or the common-sense idea of a spiral is the 2d-version, and to define a spiral in a higher or lower dimensional space is a generalization. Now the question is, what exactly could be the basis for such a generalization. Ignoring your remark concerning prime numbers, I would think the following approach makes sense.

A (2d) spiral is generated by moving along a circle with a continuously shrinking radius. This motion can be described by a vector consisting of a cosine-function (x-axis) and a 90 degree phase shifted cosine function, i.e. a sine function (y-axis). A generalization might be to use another 90 degree phase shifted function that describes the elevation, i.e. angle between the vector and the x-y-plane. This would result in a motion on the surface of a sphere. Now let the radius of the sphere shrink again, and your 3d-spiral would look like a dumbbell.

In 1d you would just delete the y-component of the vector, resulting in a weakening pendulum motion rather than in the left-right jumping motion of your example b).

ps I just tried to imagine the form. Dumbbell? It depends on how fast the radius changes, but of course the shape will be always be asymmetric.