[seqfan] Re: Anyone have a 3-D "Ulam" spiral?

[Allan Wechsler]

I think I might not understand what the real application of this project is.

If we are hoping to use visual inspection to discover prime-rich cubic
polynomials, then the most important thing is that we introduce as little
complication as possible. We would want a scheme where, at the very least,
lighting up the cells with index numbers of the form x^3 or x^3 + 7 would
produce clearly visible trajectories. I think Kevin Ryde's recent
suggestion is the first one to have a chance of doing that. Sorting cells
in order of distance from the origin, and then breaking ties in some way,
would scramble cubic forms as badly as the primes are scrambled.

If finding prime-rich cubic forms is the agenda, my suggestion would be to
start with a 3-d analogue of Klauber's triangle: make a pyramid whose apex
is numbered 0 (or 1), the next layer of nine cells is numbered in the
obvious left-to-right top-to-bottom way, the next layer of 25 cells the
same, and so on.



On Tue, Apr 27, 2021 at 9:32 PM Kevin Ryde via SeqFan <seqfan at list.seqfan.eu>
wrote:

> primeness at borve.org (Neil Fernandez) writes:
> >
> > 3D Ulam spiral
>
> Maybe similar thing expressed a different way?:  On a cube make a
> horizontal spiral outwards on the top face, horizontal loops around to
> traverse the sides, and a horizontal spiral inwards on the bottom face.
> Then next layer by step down and spiral outwards on the bottom face, etc.
>
> I think this needs at least 1 diagonal step in the loops so as to make
> the spiral-inwards begin at a corner.  A diagonal step is unlike the 2D
> spiral of course.  Could have every loop-around take a diagonal step at
> its end so each loop begins on a corner.
>
>
> Among self-similar curves, Peano gave a 3D form for filling a unit
> cube which can be done in integers to fill one octant x,y,z>=0.
> A couple of authors have thought about how Hilbert's curve might best
> go in 3D, again in one octant unless you invent some sort of additional
> reflections or something to go to negatives.
>
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