Planar and Spherical co-ordinates using just one number

Question

I'd like to specify positions on a plane or in 3D space using just a single number

I can see one way to do it in a plane, using a spiral. I simply start at the centre and travel along the spiral a given distance.

I can clearly specify any point on the plane this way, to any required degree of precision, with a single length measurement.

Question

(a) Is there a more intuitive* function to use whilst still using only a single co-ordinate?

(b) What 3D function can I use to generalise this spiral (or similar function) to a 3D space?

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*Note: A spiral is intuitive in that, the bigger the number, the further you are away radially from the origin. Of course, the actual distance from the origin and direction are not obvious without further manipulation of the measurement function.

Answer 1

The spiral does not work in the strict mathematical sense, i.e. there will always be gaps between successive loops of the spiral. In order to actually cover the entire plane (or even a small subset like a filled in square $[0,1]^2$ you need to use a relatively pathological object called a space-filling curve. But with a spiral, you do pass through say, every cube with integer coordinates if your spiral is tightly wound enough.

It would seem to me that this issue

Of course, the actual distance from the origin and direction are not obvious without further manipulation of the measurement function

disappears once you choose a certain fixed spiral, because the act of choosing the spiral also forces you to say what the angle and radius of each point is.

One choice, which does indeed pass through every box with integer coordinates (although some will be right on the border) is the spiral whose point at angle $\theta\ge 0$ is at distance $\theta/360º$. So after each round of 360º, the distance from the origin advances by $1$. (This sort of spiral that is linear in the angle is precisely what your picture is. There are other spirals that get more or less packed for larger angles..)

If you would like to have integer coordinates for your 'spiral', a reasonable choice is maybe something like the Ulam spiral.

As for a 3D generalisation of a densely packed spiral, I find it hard to visualise. Perhaps you can take some inspiration from yarn balls...?