Right, I remember now you mentioned that visualization tool before. It
turns out that Skip function actually was,
f=. 8 + (2 ^ ]) - (2 ^ 2) ^ ]
instead of,
g=. 8 + (2 ^ ]) - 2 ^ 2 ^ ]
which I was entertaining earlier last night.
Coincidentally, today I was exploring some solutions for f...
u New
- (u %. u D.1)
f New^:(22) 1
1.75372489
f New^:(22) 0j_0.5
1.75372489j9.06472028
f New^:(22) 1j_4
1.24627511j_4.53236014
which lead to the form,
,. f 1.7537248941553196 + ( i:3) * 0j9.0647202836543883
0j_1.42345214e_14
0j2.49224035e_14
0j1.24612017e_14
0
0j_1.24612017e_14
0j_2.49224035e_14
0j1.42345214e_14
and also to the form,
,. f 1.2462751058446806 + (1 + 2 * i:3) * 0j_4.5323601418271942
0j_1.58631202e_14
0j5.00674262e_15
0j_4.38300861e_15
0j4.38300861e_15
0j_5.00674262e_15
0j1.58631202e_14
0j3.67329831e_14
Your visualization verb,
viewrgb ccEnhPh f 10*sq 128
confirms that they are in the zigzag pattern that Louis mentioned.
Moreover, it not only depicts the relationship between f and g,
viewrgb ccEnhPh f 10*sq 128
but it also shows, as you pointed out, that the function g is quite more
interesting than f.
Thanks a lot for reminding me about viewrgb and its siblings and for making
them available in the first place. They are wonderful!
On Mon, Jan 29, 2018 at 10:08 AM, Andrew Nikitin <[email protected]> wrote:
>
> Jose Mario Quintana wrote:
>
> > but, it is not the only one,
> >
> > (X=. (8 + (2 ^ ]) - (2 ^ 2) ^ ])New (^:22) 0.5j_0.5)
> > 1.24627511j4.53236014
>
> There is way more than that.
>
> If you use "phase portrait visualization tool" (e.g. from here
> http://code.jsoftware.com/wiki/User:Andrew_Nikitin/Phase_portraits ) on
> a, say, 10x10 square:
>
> viewrgb ccEnhPh f 5*sq 128
>
> you will see a root near 3j4.53, near 3.43j1.72 and much more all
> intertwined with poles and getting denser and denser along 2 curves
> parallel to the real axis.
> And since exp(z) is periodic, same picture is repeated with period 0 j.
> 2p1%^.2
>
> Here is interesting piece of trivia: image of almost any straight line
> under exp(exp(z)) is dense in C so the behavior of anything with
> exp(exp(z)) in it
> (and 2^2^z is pretty much same thing as exp(exp(z))) becomes weird fast.
>
>
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