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 2021-04-09, 21:03 #4269 Xyzzy     "Mike" Aug 2002 25·3·5·17 Posts
 2021-04-15, 11:15 #4270 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 137748 Posts The eyes have it When: ⅈ = √-1 Then: (((ⅈ^ⅈ)^ⅈ)^ⅈ)^ⅈ = ⅈ And ⅈ^ⅈ^ⅈ^ⅈ^...^ⅈ^ⅈ converges to ~ 0.438282... + 0.360592...ⅈ
2021-04-15, 13:13   #4271
Dr Sardonicus

Feb 2017
Nowhere

23·34·7 Posts

Quote:
 Originally Posted by retina When: ⅈ = √-1 Then: (((ⅈ^ⅈ)^ⅈ)^ⅈ)^ⅈ = ⅈ And ⅈ^ⅈ^ⅈ^ⅈ^...^ⅈ^ⅈ converges to ~ 0.438282... + 0.360592...ⅈ
Good one! There are two different sequences here. The first one repeats five four terms cyclically, the second is the one that converges to the value indicated:
Code:
? z=I;for(i=1,20,print(i" "z);z=z^I)
1 I
2 0.20787957635076190854695561983497877003 + 0.E-39*I
3 0.E-38 - 1.0000000000000000000000000000000000000*I
4 4.8104773809653516554730356667038331263 + 0.E-57*I
5 0.E-57 + 1.0000000000000000000000000000000000000*I
6 0.20787957635076190854695561983497877003 + 0.E-58*I
7 3.186183822264904554 E-58 - 1.0000000000000000000000000000000000000*I
8 4.8104773809653516554730356667038331263 + 0.E-57*I
9 3.186183822264904554 E-58 + 1.0000000000000000000000000000000000000*I
10 0.20787957635076190854695561983497877003 + 0.E-58*I
11 3.186183822264904554 E-58 - 1.0000000000000000000000000000000000000*I
12 4.8104773809653516554730356667038331263 + 0.E-57*I
13 3.186183822264904554 E-58 + 1.0000000000000000000000000000000000000*I
14 0.20787957635076190854695561983497877003 + 0.E-58*I
15 3.186183822264904554 E-58 - 1.0000000000000000000000000000000000000*I
16 4.8104773809653516554730356667038331263 + 0.E-57*I
17 3.186183822264904554 E-58 + 1.0000000000000000000000000000000000000*I
18 0.20787957635076190854695561983497877003 + 0.E-58*I
19 3.186183822264904554 E-58 - 1.0000000000000000000000000000000000000*I
20 4.8104773809653516554730356667038331263 + 0.E-57*I
? z=I;for(i=1,20,print(i" "z);z=I^z)
1 I
2 0.20787957635076190854695561983497877003 + 0.E-39*I
3 0.94715899807237838065347535201819333350 + 0.32076444997930853466011684587486314010*I
4 0.050092236109321075325057857667404946465 + 0.60211652703600378785675053054092402757*I
5 0.38716618108611499414398349627185887944 + 0.030527081605484415140640453871444887915*I
6 0.78227568243395340528654343812104529543 + 0.54460655765798973477763897335697905935*I
7 0.14256182316366677621483372940966688336 + 0.40046652533708730698492897511841708916*I
8 0.51978639640785427894557902248600215208 + 0.11838419641581427825928144766679494853*I
9 0.56858861727189698753818358443166667583 + 0.60507840679780382839423641672697902899*I
10 0.24236524682521112389090823201045302323 + 0.30115059207131774722222823190168820179*I
11 0.57848868337710091563881022955895887525 + 0.23152973530683357064541382292107588487*I
12 0.42734013269162773482179644068826711761 + 0.54823121734356120368611146605201176309*I
13 0.33096710435764648494764410889019260596 + 0.26289184279459072161601256135202806100*I
14 0.57427101539042836742244120057133428331 + 0.32871562363049941282674521086088084983*I
15 0.36994804215669755616736899867123840508 + 0.46817319337186955733969232674886127136*I
16 0.40063349486673663827824023315392161379 + 0.26312021335073463213299399846450098581*I
17 0.53474389203394446098262230758571465046 + 0.38932777320075335892804042289024294020*I
18 0.36211373777488820944581031376300750280 + 0.40396445699053973513867483335988567832*I
19 0.44669687505662957512158203846002241697 + 0.28556732839726256985973488224519152118*I
20 0.48769614755247653922390493148645597016 + 0.41217486803024428049635565128868599910*I

Last fiddled with by Dr Sardonicus on 2021-04-15 at 15:38 Reason: I can't count

2021-04-15, 16:12   #4272
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

22·5·307 Posts

Quote:
 Originally Posted by retina When: ⅈ = √-1 Then: (((ⅈ^ⅈ)^ⅈ)^ⅈ)^ⅈ = ⅈ
Works for any x

(((x^ⅈ)^ⅈ)^ⅈ)^ⅈ = x

Last fiddled with by retina on 2021-04-15 at 16:15 Reason: Except for x == 0, of course

2021-04-16, 01:36   #4273
Dr Sardonicus

Feb 2017
Nowhere

10001101110002 Posts

Quote:
 Originally Posted by retina Works for any x (((x^ⅈ)^ⅈ)^ⅈ)^ⅈ = x
Blindly following the laws of exponents, yes, repeatedly raising to the i power you get x, xi, x-1, x-i, x, ...

Unfortunately, the "laws of exponents" can go haywire when the exponents aren't real numbers.

Example:

Code:
? z=100;for(i=0,4,print(i" "z);z=z^I)
0 100
1 -0.10701348355877020772086517528518239458 - 0.99425756941378968736161937190915602113*I
2 5.3549165552476473650304932958904718146 - 5.365580600568365703 E-41*I
3 -0.10701348355877020772086517528518239458 + 0.99425756941378968736161937190915602113*I
4 0.18674427317079888144302129348270303935 + 1.8711616494015361548219666197978930722 E-42*I

2021-04-16, 03:46   #4274
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

22·5·307 Posts

Quote:
 Originally Posted by Dr Sardonicus Blindly following the laws of exponents, yes, repeatedly raising to the i power you get x, xi, x-1, x-i, x, ... Unfortunately, the "laws of exponents" can go haywire when the exponents aren't real numbers.
Yes, you are correct. I was blind.

-e^pi < ||x|| <= e^pi

2021-04-16, 13:34   #4275
Xyzzy

"Mike"
Aug 2002

25×3×5×17 Posts

https://www.bbc.com/news/science-environment-56749105
Quote:
 Scientists in the US have developed a paint significantly "whiter than the whitest paint currently available".

2021-04-16, 14:13   #4276
Uncwilly
6809 > 6502

"""""""""""""""""""
Aug 2003
101×103 Posts

3·3,187 Posts

Quote:
Originally Posted by Xyzzy
Quote:
 Scientists in the US have developed a paint significantly "whiter than the whitest paint currently available".

2021-04-16, 18:26   #4277
PhilF

Feb 2005

617 Posts

Quote:
 Originally Posted by Xyzzy Scientists in the US have developed a paint significantly "whiter than the whitest paint currently available".
That makes it too politically incorrect to use.

2021-04-16, 20:30   #4278
kriesel

"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

2×5×7×73 Posts

Quote:
 Originally Posted by Xyzzy https://www.bbc.com/news/science-environment-56749105
Outstanding! For WHAM or WUPPE development, best NASA-approved paint we could use was solar spectrum absorptance ~23.%. More recently, there's 9% http://www.aztechnology.com/products...azw-la-ii.html.

2021-04-19, 00:06   #4279
firejuggler

Apr 2010
Over the rainbow

22×641 Posts

Quoting the video description
Quote:
 Originally Posted by Youtube We are very excited to share the first-ever camera collar footage from a wild wolf to our knowledge. We hope you enjoy seeing the world from a wolf's point of view! What is particularly fascinating is that this wolf (V089, a lone wolf) knew how to hunt and catch fish. He can be seen eating 3 different fish, which were all killed and consumed at the same spot along the Ash River. Based on the amount of time this wolf spent in this spot, it is clear this wolf killed more than 3 fish. However, the collar only took videos for 30 seconds at the beginning of every hour of daylight meaning we only got 7 minutes of video footage each day (14 hr of daylight x 30 second per hour). 7 minutes of footage a day is not that much. Luckily, we still captured some really neat stuff! Up to this point, we had only documented wolves from a single pack (the Bowman Bay Pack) hunting and killing fish at the same small creek. However, this footage clearly demonstrates that other wolves in our area know how to hunt fish and they do so in different areas. This revelation—in addition to some other info we learned in 2020 (i.e., we had another wolf from the Paradise Pack that went fishing…more about this soon!)—provides insight into the genesis and persistence of unique predation behaviors in wolf populations! We used a Vectronic-Aerospace camera collar for this footage. The collar worked great and we are excited to deploy more of these camera collars in the future!

Last fiddled with by firejuggler on 2021-04-19 at 00:15 Reason: formatting the quote

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