On Tue, Sep 24, 2019 at 5:38 PM rwg <rwg@ma.sdf.org> wrote:

-------- Original Message -------- Subject: Re: [math-fun] Hilbert's ≠ Peano's Date: 2019-09-24 08:24 From: Brad Klee <bradklee@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Reply-To: math-fun <math-fun@mailman.xmission.com>

Subj:Re: [math-fun] Hilbert's ≠ Peano's Isn't the valency of a multi-point limited by the valency of vertices in the unit cell tiling? Square unit cells seem to imply that quadruple points have the most possible degeneracy. Or am I wrong?

That sounds right. It's true of Julian's sextuple pointer: That sounds right. It's true of Julian's *http://howwords.com/triangles/?a_=sexttri&radiusPct=90&nRotors=6251&dtPct=3&... <http://howwords.com/triangles/?a_=sexttri&radiusPct=90&nRotors=6251&dtPct=3&rotorSort=1&cmn_=32&cmt_=highcontrast&cmabs_=false&stepsPerCycle_=20&delayMS_=0>* (In a fleeting moment of mental clarity, I managed to "invert" this fractal. Sure enough, piecewiserecursivefractal returned preimage sextuples.) But when he sent it he said he can go higher than 6. Maybe with a wild tiling? —rwg

Considering only tiling topology, it would also be possible to have triple points, but it looks like the symmetry of the unit pattern forbids this. The curve snakes across the unit cell between opposite corners, so divides any Z^2/3^n lattice into subsets 2*Z^2/3^n and 2*Z^2/3^n + 1. One associates to quadruple points and the other to double points. This is already visible in the nice "tetraskelion" pic.

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From the perspective of multipoints, Wunderlich "Figur 5" is the most interesting. It looks to have double, triple, and quadruple points.

--Brad

On Tue, Sep 24, 2019 at 7:38 AM Bill Gosper <billgosper@gmail.com> wrote:

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Here's a raw sandbox with Peano and Wunderlich samplings: https://www.wolframcloud.com/obj/1fa6d29d-7451-4e68-ae7b-dded77390040 The "tetraskelion"s are dense with quadruple points. I need to figure out how, with Julian's tools, to make inversepeano and see if there are quintuple or sextuple points (which would be surprising). —rwg

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