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

Jan Ritsema van Eck j.ritsemavaneck at planet.nl
Wed Apr 28 08:01:07 CEST 2021

If a double spiral is allowed, you can do without the diagonal step. The first branche goes to the top face, the second one to the bottom face, and then they spiral towards each other in the way described by Kevin. They meet in the middle, at the centres of two opposing side faces. They move out one step and fill the next shell in similar fashion, continuing the same movement and "spiralling in" to the centres of the top and bottom faces.


> Op 28 apr. 2021, om 02:11 heeft Kevin Ryde via SeqFan <seqfan at list.seqfan.eu> het volgende geschreven:
> 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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