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2019-11-22 10:10 nesdori  
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this cave is demoing a new formation called "the 6 elements" which previously was thought to be non-existent and it thus is a little surprise. It's related to previously known "the 4 elements" formation and each of its phases are extensions from some phase of "4-elements". first i was going to demo this new formation by colliding "4 elements" with two circling flies to form "6 elements". although it's possible in theory, it's not easy to do. it's not possible with just free circling flies without some interaction from rockford. because i wanted to keep this demo cave simple i decided to demo the formation as ready-made. the original "The 4 elements" formation has been added to side of it to make comparison easier.  
  
again, before you read on, i recommend opening  Arno 's article about "the 4 elements" because i'll use the same phase numbering:  
  
<a href="https://www.boulder-dash.nl/forum/viewtopic.php?t=636&postdays=0&postorder=asc&start=53" target="external">https://www.boulder-dash.nl/forum/viewtopic.php?t=636&postdays=0&postorder=asc&start=53</a>  
  
from this on, "the 6 elements" is written "6-e" and "the 4 elements" is "4-e".  
  
Each phase of 6-e fully contains some phase of 4-e enhanced with two additional bflies.  the cycle length of 6-e is only 4 frames as opposed to 6 frames of 4-e. the behaviour of flies is similar to 4-e; each fly moves around a certain path clockwise. for some flies, it takes 2 cycles or 8 frames to get into a initial position and for some it takes 4 cycles = 16 frames. as this implies, the long cycle loop length of the entire formation is 16 frames.  
  
like 4-e, 6-e moves one step down and one left during a cycle. because the cycle is only 4 frames, 6-e moves faster than 4-e. like with fright-3 - 5, adding some flies seems to speed up the formation.  
  
because each phase of 6-e is an extension of some phase of 4-e, how can its cycle be only 4 frames without it getting out of sync with 4-e phases? well this is the interesting part and 6-e has a clever solution for it :)  
  
Here are all phases of 6-e (l=left, r=right, u=up, d=down):  
  
1:  
..R.  
.R..  
L.D.  
.L.D  
  
2:  
U.D.  
.D..  
L.D.  
...D  
  
3:  
..R.  
.L.D  
L.L.  
...L  
  
4:  
.U..  
U.D.  
.L.D  
..L.  
  
let's compare these with 4-e phases.  
  
6-e phase 1 is clearly an extension from 4-e phase 1. so is 2, 3 and 4. they all are extensions from 4-e phase of same number. so it looks like that 6-e is skipping two last states of 4-e but how can it then get back on track and start with extended 4-e phase 1 again? let's see..  
  
4-e phase 5 clearly is an unique state not contained in any 6-e phase. but 4-e phase 6 looks familiar.. i have seen this somewhere. aha! it's fully contained in 6-e phase 4! three leftmost flies of 4-e phase 4 are same as three rightmost flies of phase 6, 6-e phase 4 takes advantage of this and merges them both perfectly into a formation of just 6 flies and so 6-e is able to represent 5 phases of 4-e with just 4 phases skipping just phase 5 of 4-e. because phase 6 of 4-e is now present, it's easy to get back on sync and start over with phase 1.  
i was truly amazed when i realized this trick. 6-e phase 4 is just a beautiful composition :)  
  
so now that 6-e has been discovered, is there 8-e or 10-e? after some tries and analysis i think that there isn't 8-e or 10-e. 6-e is already moving so fast that adding flies to it causes it to fall apart. so i think the only way to create 8-e is to fake it by putting two 4-e:s close to each other.

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2019-11-22 12:42 dustin  
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wow, fascinating!  
i did some analysis myself and have some additions and even corrections.  
  
4-e phase 5 is actually present in 6-e phase 3! and there's still more to it. i number the butterflies of 6-e phase 1 as follows  
..1.  
.2..  
3.4.  
.5.6  
  
now let's closely regard the bflies 1,2,3,4. these four bflies form the following 4-e phases.  
4-e phase 1 -- 4-e phase 2 -- 4-e phase 3 -- 4-e phase 4  
then, betwenn 6-e phase 4 and 6-e phase 1, they get drifted apart and don't form a new formation.  
  
now let's look at bflies 2,3,4,5! starting from the above diagram, they behave as follows  
4-e phase 3 -- 4-e phase 4 -- 4-e phase 5 -- 4-e phase 6 -- 4-e phase 1  
so after a 6-e short loop, the bflies 2,3,4,5 take over the role of the bflies 1,2,3,4!  
  
so the question arises - who takes over the role of 2,3,4,5 after a 6-e short loop? and what on earth is bfly 6 doing? well, you might guess that the answers to both questions are related! ;d  
the role of bflies 1,2,3,4 is taken over by the bflies 5,3,4,6 between 6-e phases 4 and 1.   
in  nesdori 's phase 4, the topmost bfly is number 2 and the rightmost bfly is number 1. deleting those gives  
....  
u.d.  
.l..  
..l.  
this would be 4-e phase 2 if the left-facing bflies were facing down instead. but even though it's "not perfect", these four bflies still form 4-e phase 3 next frame and thereby take over the role of bflies 2,3,4,5!  
  
so here's a summary of what happens between 6-e phase 4 and phase 1.  
1. bflies 1,2,3,4 are drifted apart and don't form any formation during the new short loop.  
2. bflies 2,3,4,5 take over the role of 1,2,3,4  
3. bflies 5,3,4,6, which hadn't formed anything meaningful before, now happen to take over the role of 2,3,4,5.  
voila - cycle closed! ;d

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2019-11-22 12:51 dustin  
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i should probably add some more diagrams which show which bfly is which in each 6-e phase.  
  
6-e phase 1 see below  
  
6-e phase 2  
3.1.  
.2..  
5.4.  
...6  
  
6-e phase 3  
..3.  
.2.1  
5.4.  
...6  
  
6-e phase 4  
.2..  
5.3.  
.4.1  
..6.  
  
6-e phase 1 after 1 short loop  
..2.  
.5..  
4.3.  
.6.1

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2019-11-22 12:54 nesdori  
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 dustin , you are right! i missed the phase 5 being also present. and wow, what a great in-depth analysis by you! i hadn't got so deep myself yet. and you did that right off after i released this demo. another amazement for me.. :)

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2019-11-22 12:59 nesdori  
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and btw, i'm away for a few days. no playing and probably no commenting during the weekend. and no new formations.. :D

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2019-11-22 13:09 dustin  
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hi-5  nesdori  for that teamwork! i wonder how much  Arno  should offer us for this... ;d

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2019-11-22 16:12 shredder  
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great!

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2019-11-22 19:52 Arno  
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wow, amazing new formation  nesdori , great discovery! :-)  
  
indeed a beautiful turbo version of 4-elements!