Dancing fly formations explained

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Arno
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Post by Arno »

Dustin wrote:Cool! I can't wait to see the Big 5 in action, but I'm too lazy to try out all 4^5 possible combinations of initial directions of the butterflies - so I'll be patient until there's a video/article about them ;D
I can give you the initial directions, no problem! In the usual notation (with . as empty space):
.r.
u.d
l..
.l.

There will be an article about this formation, however, I first want to understand its working better and see if I can find a launch-method.
Next episode is on The 4 Elements; it is already written but I still need to publish one more video to refer to.
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Post by Dustin »

Cool, thanks! Actually I find the second phase, i.e. the one after the one you posted, most striking - here the butterflies form kind of a "W" which looks very similar to the star constellation "Cassiopeia"! Also in this phase, all butterflies face either up or down, so it's relatively easy to remember!
Boulder Dash X Rock, Paper, Scissors:
ROCKFORD collects DIAMOND, digs DIRT
DIAMOND outvalues DIRT & BOULDER
DIRT carries BOULDER, blocks FIREFLY
BOULDER kills FIREFLY & ROCKFORD
FIREFLY kills ROCKFORD, guards DIAMOND
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Post by Arno »

It is a very interesting formation indeed, in particular because no vortex is involved! I don't know yet how I'm going to tell a story behind it, but hopefully it will be somewhat more exiting and dramatic than just "5 butterflies partying together". ;)

And yes, I like to watch star constellations, and how they shift over the seasons. I don't have a telescope (quite expensive and too much choice), but for example, the Pleiades through binoculars looks very cool too! :)
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Post by Dustin »

Actually, when I think about it, I want to make an official name suggestion - renaming The Big 5 into Cassiopeia! The reasons-
1. The second phase - which could still be newly defined as first phase - looks very similar.
2. Cassiopeia has 5 stars.
3. The Big 5 are a quite outstanding formation - no vortex and no relations to any other formation - so I think an outstanding name would be very fitting!

I always like to look for the Big Dipper and/or Cassiopeia when I'm on my way home in the late evening! During the winter, I also like the Orion and the "Winter Hexagon" a lot! :)
Boulder Dash X Rock, Paper, Scissors:
ROCKFORD collects DIAMOND, digs DIRT
DIAMOND outvalues DIRT & BOULDER
DIRT carries BOULDER, blocks FIREFLY
BOULDER kills FIREFLY & ROCKFORD
FIREFLY kills ROCKFORD, guards DIAMOND
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Post by LogicDeLuxe »

Maybe a Boulder Dash wiki would be in order. So much research has to be well organized and easily accessible.
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Post by Arno »

Dustin wrote:Actually, when I think about it, I want to make an official name suggestion - renaming The Big 5 into Cassiopeia!
Good idea! To be honest, I wasn't really satisfied with the current name and also I'm not the world's most creative person when it comes to invent striking names. This suggestion fits very well; thanks Dustin! I will update the first post and the overview picture soon. :)
Dustin wrote:I always like to look for the Big Dipper and/or Cassiopeia when I'm on my way home in the late evening! During the winter, I also like the Orion and the "Winter Hexagon" a lot! :)
Indeed, Orion is one of my favourites, in particular after I watched a documentary on the controversional theory that the outline of the Giza pyramids match the 3 stars in Orion's Belt. :)
LogicDeLuxe wrote:Maybe a Boulder Dash wiki would be in order. So much research has to be well organized and easily accessible.
Sounds useful indeed, although I don't know yet whether I will do more BD research after these dancing fly articles. :)
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Post by Dustin »

Arno wrote: Good idea! To be honest, I wasn't really satisfied with the current name and also I'm not the world's most creative person when it comes to invent striking names. This suggestion fits very well; thanks Dustin! I will update the first post and the overview picture soon.
Oh, I really like finding striking names! You probably noted that when looking through the caves of Dustin's BD11 that I already sent you ;D In fact, I really like the big majority of the DFF names very much!
--> BnB and FnF are just perfect names in my opinion - the name implies that two FF/BF are involved and one can also guess from the names that the two DFF are fully related!
--> Fright - 3,4,5 I also like a lot! The "ingrediants" of these DFF and also their moving direction combined with a word game - what could describe such a frightening formation better?
--> The lifted vortices - here the names tell you that a vortex is just lifted upwards, but the vortex itself doesn't change - in contrast to most other vortex-based DFF, where the jealous rival/s successfully disturb the dancing pair! The only important thing that one can not see in the name is that these two DFF have "mixed genders", unlike most other DFF. This would be cool to have in the name, but how? "Mixed Lifted F/B vortex" would sound a bit odd to me. Perhaps "B-lifted F-vortex" and "F-lifted B-vortex" but this also feels to me more complicated than necessary because this would sound as if there should also be a "F-lifted F-vortex" and a "B-lifted B-vortex". So I think the actual names are quite good. Also the p-mirrored relationship can easily be guessed from the names.
--> Gear Wheels - that's a really funny and visually fitting name! Of course, there are also disadvantages - the name doesn't tell you anything about the relationship to the lifted vortices, the DFF direction or how many and which flies are involved. Here I do have a suggestion solving some of these disadvantages - you could name it "Lifted Gear Wheels". After all, usually gear wheels don't move, so this extra word would give the DFF their dynamics, while also telling you its direction and give someone who only reads the names at least a chance to guess that this DFF has sth to do with the lifted vortices. Then the three DFF "Lifted B/F-vortex" and "Lifted Gear Wheels" would have the only real disadvantage of not telling you about their outstanding charakter in terms of their mixed "genders", but on the other hand, you can't have every-everything in the names ;D
--> The 4 elements still have potential in my opinion. You only know that there are 4 flies involved and the name isn't as optically striking as "Gear Wheels" is. On the other hand, the name sounds kind of positively mysterious to me, and also the 4 elements had some mentions in your articles, so I would let the name as it is in order not to confuse anyone who has already read about "The 4 elements" under this name. Still, I have a suggestion - "The Butterfright Quartet", which sounds like four butterflies who have "desperately" tried to imitate a Fright formation and somehow "half-succeeded" - however, this would drastically change the "character" of this DFF - while "The 4 elements" sounds kind of outstanding and universal, my suggestion would make them sound rather ridiculous! ;D
--> About the other DFF in the list - I think their names are quite fitting. The sinking and crawling DFF are quite "standard" in the sense that they are built around vortices - they are, aren't they? - and have only one type of flies. So just "categorizing" them in terms of type and number of flies and moving directions looks like the best thing one can do - and at the same time, the relationships between these DFF become quite clear!
So my suggestion apart from the Big 5 / Cassiopeia is-
1. change "Gear Wheels" into "Lifted Gear Wheels" - I think it definitely gains sth and nobody will be confused by this little extra word.
2. Perhaaaaps rename "the 4 elements", but I think I'd rather let the name be as it is.
Boulder Dash X Rock, Paper, Scissors:
ROCKFORD collects DIAMOND, digs DIRT
DIAMOND outvalues DIRT & BOULDER
DIRT carries BOULDER, blocks FIREFLY
BOULDER kills FIREFLY & ROCKFORD
FIREFLY kills ROCKFORD, guards DIAMOND
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Post by Arno »

Thank you Dustin for this extensive review of the DFF names! :)
Indeed you have quite creative cave names in your Dustin's BD series; I've noticed that.

I'm convinced to add the word "Lifted" to Gear Wheels. This makes the name more complete as it suggests the connection with the Lifted F/B-vortex formations. Moreover, it is only a small change and the name "Gear Wheels" can still be used as shorthand, without confusing it with other DFF's.
I think I'll leave The 4 Elements as it is for now.

The "Crawling" and "Sinking" formations are indeed all built around vortices. Until now, FnF, BnB and Cassiopeia are the only vortex-free DFF's! :)
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Post by Arno »

Episode 9: The 4 Elements

This episode focuses on one particular dancing fly formation, The 4 Elements, which consists of 4 butterflies moving on a diagonal line, from top-right to bottom-left.

The properties of this formation are as follows:
- Name: The 4 Elements
- Composition: 4 BF
- Short loop: 6 phases
- Long loop: 12 phases
- Shift per short loop: 1 Down, 1 Left

This formations moves in quite a fast pace, and in a regularly looking pattern. It seems that every fly has its own role and the formation looks balanced, therefore, I found ‘The 4 Elements’ a suitable name for this formation.
This video shows a cave where The 4 Elements must be launched. Other funny examples are this video and this one, where this formation is used as a timer.

How does this formation work?

Below picture shows the 6 phases of the short loop of The 4 Elements:

Image

The story behind this formation seems, at first sight, similar to Fright-4. Two butterflies (red and blue) are dancing together in a vortex-shape, and two jealous rivals (green and yellow) are running around the dancing couple. At a certain point (see phase 6 and 1’), the green rival-BF breaks into the dancing couple. But unlike Fright-x, where the rival takes over the dance with one of the partners, this time he drifts both partners apart and starts a dance with the yellow butterfly behind him. Now the red and blue butterflies take over the role of the rivals.

In this formation, two BF couples can be distinguished (red-blue and green-yellow), so each BF will always dance with the same partner! This formation looks similar to Lifted Gear Wheels, in the sense that two couples are competing for the same space to dance and thereby move the whole group to a certain direction.
Because each fly will fulfil only two different roles over time, it takes just two short loops before all flies have returned to their original position. This makes that the long loop takes 2*6 = 12 phases.

Why does this formation move down-left?

As can be seen from phase 6 and 1’, the natural point to break into the BF-vortex is at the left/bottom. Consequently, the new vortex is created at one step to the left and one step below the original vortex. Since the vortex is the central item around which the formation is built, effectively the whole formation moves down-left.

Relations with other dancing formations?

Actually we already know a relation from the previous episode. However, to be sure that we don’t miss any others, let’s do the analysis for The 4 Elements from scratch, by answering the following questions.

Does this formation form a mirrored pair?
Remember the recent conversation between Dustin and me about chessboard patterns in dancing fly formations? In the light of this phenomenon, I'd say it is unlikely that a mirrored counterpart exists for The 4 Elements. It is quickly observed that all 4 BF in this formation are configured in a chessboard pattern, where all 4 BF are placed on the same “square color” on the imaginary chessboard. This makes it quite likely that at some point two BF are interacting in a way which breaks the condition for a mirrored pair (see previous episode).
Indeed, in phase 4, the red and green BF compete for the same position to move to. Due to the row-by-row cave scanning, the red BF “wins”. If column-by-column cave scanning would be applied, then the green BF would “win”. This proves that the condition is broken, i.e. The 4 Elements is not part of a mirrored pair.

Perhaps a p-mirrored pair?
To see whether The 4 Elements has a p-mirrored counterpart formation, you have to build the mirrored image of all 6 phases in the Construction Kit and test whether they lead to a dancing FF formation. It turns out that mirroring phase 1-4 gives a Fright-4 (to be precise, phase 5-8 of Fright-4). So The 4 Elements and Fright-4 are a p-mirrored pair (which we already knew from previous episode).
As an example, below pictures shows phase 1 and its mirror image, which is phase 5 of Fright-4.

Image

If we create the mirror image of phase 5 and 6 in the Construction Kit, it turns out that these setups lead indirectly to a phase of Fright-4.
Thus, Fright-4 is the only p-mirrored counterpart of The 4 Elements.

How to launch The 4 Elements?

One way to launch The 4 Elements is by first creating a vortex of 2 BF and connect 2 more BF at the right place and with the right timing (this is similar to how Fright-4 can be launched). This video illustrates this method. Probably, there exist several other methods.

Can this formation be extended with more butterflies?

This formation works because at the moment of breaking into the vortex, 2 BF arrive closely after one another, such that they can create a new vortex together. If we try to extend this to 5 BF in total (a potential p-mirror of Fright-5), the queue of rival-BF would consist of 3 BF. However, the path from the first position where a BF is kicked out from the vortex until the position where a BF breaks in again is 8 steps long. With 3 BF present on this path, at least one BF won’t have a partner BF closely behind him (i.e. with one step in between). When this BF breaks in, the vortex is broken without creating a new one, which ultimately leads to all 5 BF spinning around individually.

But what about 6 butterflies? Well, this sounds promising at first, since 4 BF (an even number again!) will be in the queue of rival-BF. Unfortunately, the 8-step path around the vortex is just too short since we need one “empty” position more than there are BF on this path. I’ve tried it out in the Construction Kit, and with 6 BF in total, one of them is quickly pushed aside and the group continues with 5 BF, which gives the same result as described above, so in the end all 6 BF are spinning around.

So we conclude that, while the Fright-x series works with 3, 4 and 5 FF, The 4 Elements is a unique formation which only works with the exact amount of 4 BF.
Although an extension with more BF is not possible, there exists a surprising other type of extension.

Extension with free riding firefly

As we have seen previously, the Fright-x formations can be extended with 1 or 2 free riding butterflies. In a similar way, it is possible to extend The 4 Elements with 1 free riding firefly.
This video shows this formation and that it can be launched by first launching The 4 Elements and let it collide with a spinning FF.
Below picture shows the 6 phases of The 4 Elements, with an additional light blue FF.

Image

Clearly, the butterflies influence the movement of the firefly. In phase 1 it is pushed down and in phase 6 it is pushed left, so eventually the FF makes the same down-left movement of the whole formation. Also note that basically the FF spins around once (which takes 4 frames) and is 2 times pushed into a certain direction by a butterfly, which perfectly fits into the 6 phases of The 4 Elements.
Conversely, the firefly does not influence the movement of the butterflies. In any frame where a BF tends to move to a location occupied by the FF, the FF is scanned first and thus has been moved before the BF is scanned. So indeed this formation falls into the category of formations with free riders.
Also, this formation, The 4 Elements + 1FF, is a p-mirrored counterpart of Fright-4 + 1BF.
An extension with 2 fireflies is not possible. I’ve tried to add another FF in a similar way as Fright-4 + 2BF, but this breaks the formation.

Final note

Several times, I’ve been surprised when I discovering that something funny can be done with a dancing fly formation. In this video, you can see that The 4 Elements can be easily converted into another dancing formation. This formation, Crawling B4, will be discussed next time, together with its mirrored counterpart, Crawling F4.
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Post by Dustin »

Oh, I'm mentioned in a scientific artice! Cool! :D
Regarding the nice, regular moving pattern, I think that "The 4 elements" is a quite fitting name as well! They clearly first move down and then left, and in phases 3 and 6, they form a beautiful "o", so with regard to their beautiful optics, I think now that it is a much better name than my suggestion!
The fact that they can be forced tu "turn back" as Crawling B4 is really funny! :D Is it even possible, with some obstacles at the right positions, to make Crawling B4 reconvert into the 4 elements, so that the BF move back and forth from the top-right to the bottom-left and vice versa?
Boulder Dash X Rock, Paper, Scissors:
ROCKFORD collects DIAMOND, digs DIRT
DIAMOND outvalues DIRT & BOULDER
DIRT carries BOULDER, blocks FIREFLY
BOULDER kills FIREFLY & ROCKFORD
FIREFLY kills ROCKFORD, guards DIAMOND
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Post by Arno »

Dustin wrote:Is it even possible, with some obstacles at the right positions, to make Crawling B4 reconvert into the 4 elements, so that the BF move back and forth from the top-right to the bottom-left and vice versa?
I did some small tests in the editor, and yes, this is indeed possible!
Also, it is possible to convert Fright-4 into Crawling-F4 and back to Fright-4. This way, the formation could travel throught the cave in a zig-zag path by alternating direction to the east and south-west.
I have some very cool cave ideas now...! :D
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Post by Dustin »

Arno wrote: I have some very cool cave ideas now...! :D
I'm sure you do! Looking forward to see them :D
Boulder Dash X Rock, Paper, Scissors:
ROCKFORD collects DIAMOND, digs DIRT
DIAMOND outvalues DIRT & BOULDER
DIRT carries BOULDER, blocks FIREFLY
BOULDER kills FIREFLY & ROCKFORD
FIREFLY kills ROCKFORD, guards DIAMOND
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Post by Arno »

Episode 10: Crawling-F4 and -B4

This is possibly the last episode…… well, the last one about dancing fly formations with 4 or less flies. ;)
Yes, all 4-fly formations that have been discovered until now already have a place in this thread, except Crawling-F4 and –B4, which will be covered by this episode.

In episode 8 we introduced the concepts of mirrored pair and p-mirrored pair. A mirrored pair of dancing fly formations has the same number of phases, opposite fly genders, and each phase of one member can be mirrored in the diagonal line to result in a phase of the other member.
Until now, one mirrored pair has been discussed: FnF & BnB. As we will see in this episode, Crawling-F4 and –B4 make up the second example of a mirrored pair.

The properties of both formation are as follows:
- Name: Crawling-F4 / Crawling-B4
- Composition: 4 FF / 4 BF
- Short loop: 7 / 7 phases
- Long loop: 28 / 28 phases
- Shift per short loop: 1 Down-Left / 1 Up-Right

I use the word “crawling” in the name because these formations move in a somewhat slow, irregular pattern. Compared to other formations (e.g. Fright-x and The 4 Elements) the movements of the Crawling-formations look slightly more chaotic, like a fluttering insect that’s flying along a winding rather than straight line.

How do these formations work?

Below pictures show the 7 phases of the short loop of both formations. It is easy to see that each phase of Crawling-F4 can be mirrored in the diagonal line to give the corresponding phase of Crawling-B4 (and vice versa):

Crawling-F4:
Image

Crawling-B4:
Image

The story behind these formations is basically that of The 4 Elements. Two flies (red and blue) are dancing together in a vortex-shape, and two rivals (green and yellow) are running around the dancing couple. At a certain point (see phase 7), the green rival breaks into the dancing couple, starts a dance with the yellow fly behind him, and drifts the red and blue partners apart. Again, two couples can be distinguished (red-blue and green-yellow), so each fly will always dance with the same partner.

So far the story is very similar to The 4 Elements. Perhaps this sounds a bit boring, right? Isn’t there anything more exciting that could be said about the Crawling formations?
Well, certainly there is. First of all, both couples of flies seem “balanced” in the sense that we cannot speak of a leader and a follower. In phase 1 the green fly moves at front and the yellow fly follows him. After 2 short loops, these roles will be reversed: the yellow fly moves at front and will ultimately break into the vortex and start a dance with the green fly. In this sense, the Crawling formations can be compared with Lifted Gear Wheels, which also involves two balanced couples. In contrast, The 4 Elements could be compared with Lifted F- and B-vortex, since in these formations each couple has a clear leader and follower. That is, the same partner will move at front every time when the couple is drifted apart.
For the Crawling formations, the above story implies that each fly fulfils all 4 different roles over time, so it takes 4 short loops before all flies have returned to their original position. This makes that the long loop takes 4*7 = 28 phases.

Although we have touched upon a difference, still the main plot of the Crawling formations can be told in a similar way as for The 4 Elements. A good question that may come to mind is the following. If Crawling-B4 and The 4 Elements are so much similar, why are they two different formations anyway?
Below picture shows a phase from The 4 Elements and from Crawling-B4.

Image

Note that in both snapshots the red/blue vortex is in the same state, while the green/yellow rivals have moved one step further in Crawling-B4 than in The 4 Elements. In a similar way, Fright-4 and Crawling-F4 could be compared:

Image

Here, the green/yellow rivals are walking at the same position, while the red/blue vortex is in the reversed state. Thus, both sets of formations are comparable in the sense that they all involve 2 rivals walking around 2 dancers, but with a 1-frame timing difference of the rivals relative to the dancers.
This 1-frame difference has several consequences.

Visual difference

The first consequence is mainly about the appearance of the formations to a viewer. The Fright-x formations and The 4 Elements are always shaped in a chessboard pattern, so that the rivals never touch the dancers. In the Crawling formations, on the other hand, the rivals walk around by sliding along one of the dancers.
This makes (in my experience) that the Crawling formations look slightly more chaotic.
In fact, after FnF and BnB, this is the first set of unique formations where flies are touching each other within a phase. All other formations discussed earlier are in a chessboard pattern (except when free riding flies are involved).

Difference in relations

Secondly, the 1-frame timing difference also “explains” (at least it makes it more likely) that Fright-4 and The 4 Elements are only partially (p-)mirrored, while the Crawling formations are a fully mirrored pair:
- Fright-4 and The 4 Elements are shaped in a chessboard pattern. This increases chances that two diagonally placed flies intend to move to the same position, such that row-by-row or column-by-column cave scanning would give different “traffic rules”, and thereby different outcomes. This breaks the condition for a mirrored pair (see episode 8 ).
- Crawling-F4/B4 do not follow such chessboard pattern. In all situations where two flies interact, both flies are placed next to each other, on the same row or column. This implies that row-by-row or column-by-column scanning would not make a difference, hence the condition for a mirrored pair is perfectly satisfied.

Different walking path and break-in point

The fact that Fright-4 and Crawling-F4 differ basically by a 1-frame timing difference of the rivals versus the dancers also has as a consequence that the natural walking path of the rivals and the natural point to break into the vortex are different. In Fright-4 the rivals walk at the left, bottom and right side of the vortex before breaking in. In any situation of interaction between a rival and a dancer, the dancer is scanned before the rival and takes the position where the rival intended to go. Within the same frame, the rival must move forward. This gives the typical chessboard pattern. As soon as the rival is scanned before the dancer (the other way around), the break-in happens. This happens at the top-right corner for Fright-4.
In Crawling-F4, the rivals walk at the top and left side before breaking in at the bottom-left. In any case of interaction with a dancer, the rival is initially placed next to the dancer, but the rival is scanned first, so that he will move forward. This way, the rival will slide along the dancer. (Except in the corners; there the rival loses touch with the dancer, but one frame later, after changing direction, he touches the other dancing partner.) This pattern, where rivals slide along a dancer, continues until the dancer is scanned before the rival. Then, the dancer makes room free for the rival so that the rival breaks in. This happens at the bottom-left for Crawling-F4.

In the same way, The 4 Elements and Crawling-B4 can be compared. In The 4 Elements the rivals walk at the left and bottom side (the dancers are scanned before the rivals) after which a rival breaks in at the left-bottom. In Crawling-B4, the rivals walk at the left and top side (the rivals are scanned before the dancers), after which a rival breaks in at the right-top.

Thus, we have seen that the cave scanning order determines that the 1-frame timing difference between two formations gives a different path for the rivals and a different break-in point, which ultimately gives two different formations, moving in different directions.

Different #phases

Another notable difference between Crawling-B4 and The 4 Elements is that in The 4 Elements, at the phase of breaking in, the green fly starts the dance with the yellow fly while at the same time drifting the red and blue flies apart. In Crawling-B4 these events happen in two steps. In phase 7 the break-in happens, without immediately separating the dancing couple (so in this phase two vortices exist). In the next phase 1’ the red and blue flies are separated. This explains why the Crawling-B4 has 7 phases while The 4 Elements has just 6 phases.
This also explains why, as discussed earlier, Crawling-B4 has balanced couples, while the couples in The 4 Elements always have the same leader (walking at front) and follower (walking behind him). Crawling-B4 has 1 phase more, so that the vortex is running for 1 frame longer. This is enough to ensure that the fly which is leading at the time when the vortex is created will become the follower after the vortex is split.

In conclusion, even though the basics of the Crawling-formations are similar to The 4 Elements, there are some remarkable differences as well, which result in unique formations, each with their own characteristics.

Why do these formation move down-left and up-right?

Phases 7 and 1’ show that the natural point of breaking in is at the bottom-left for Crawling-F4 and at the right-top for Crawling-B4. The vortex therefore moves 1 down and 1 left for Crawling-F4, resp. 1 right and 1 up for Crawling-B4. As the formations are built around the vortex, the formations as a whole move in the same directions.

Funny to note is that mirroring the direction-arrows of these formations in the diagonal line gives two opposite directions, i.e. they make an angle of 180 degrees. The other mirrored pair that we have seen, FnF and BnB, move to the left resp. upwards, which gives a different angle: 90 degrees.
If we will ever discover a mirrored pair with one member moving along the diagonal line (so either up-left or down-right), we already know that the other member will move in exactly the same direction!

Also interesting to note is that Crawling-F4 is the second formation so far which moves down-left. The other formation moving in this direction is The 4 Elements. This video shows a race between both formations, which (if we ignore Rockford for now) is not surprisingly won by The 4 Elements since this formation needs 6 frames to advance 1 step while Crawling-F4 needs 7.

How to launch these formations?

Launching the Crawling-formations works similar to all other formations that are built around a vortex. That is:
Step 1: Create a vortex
Step 2: Connect two more flies at a certain side of the vortex, and with the correct timing.
The following videos show the solutions of two “mirrored caves”, each illustrating the launch of a Crawling-formation:
- Crawling-F4
- Crawling-B4

There are other ways to launch these formations. For instance, this video shows the first cave ever in which I’ve spotted a Crawling-B4. Note however that for this launch method 5 butterflies are needed. Nevertheless, to me it was quite an amazing invention by Nesdori at that time.

Relations with other dancing formations?

As Crawling-F4 and –B4 are a mirrored pair, they cannot have a p-mirrored relation with another dancing formation.
However, there are some other interesting effects to note. The 4 Elements can be converted to Crawling-B4 by putting some obstacle on the path of the formation. Here’s what happens. When the formation touches the obstacle, the 2 rival flies are fooled: while they expect to enjoy a nice dance soon, instead they are leaded around the obstacle object. The vortex stands still right next to the obstacle (this is no surprise: no rivals, no move). When the rivals return at the other side of the obstacle they touch the vortex again, but still have bad luck; although they move through the vortex, they first need to complete a path around the vortex before starting a new dance; this time the Crawling-B4 path, which gives a Crawling-B4 formation.

Conversely, it is also possible to convert Crawling-B4 back to The 4 Elements! For this effect, however, an obstacle of a different shape is necessary. This is to assure that the rivals return on the path along the vortex at the correct location and with the correct timing.
Because The 4 Elements and Crawling-B4 move in opposite direction, it is possible to create a scenario where 4 butterflies make an alternating dance in both formations. This video illustrates a cave using this effect.

Also, Fright-4 can be converted to Crawling-F4, and back to Fright-4. However, since these formations do not move in opposite direction, an alternating dance is not possible. Nevertheless, it is possible to let the formation travel through the cave in a zig-zag pattern, as shown in this video.

Final note

Although it is hard to prove, I have a strong feeling that we currently know all 4-fly dancing formations. Nevertheless, a surprise may be awaiting. (Note that I was recently surprised when Dustin discovered Fright-3 plus free-riding BF, which is in fact a 4-fly formation! :))
At least for now, we will continue with formations of 5 flies or more. Questions are plenty. For instance, given that Fright-4 can be converted into Crawling-F4, does Fright-5 have a such a relation with another formation as well? Good stuff for another episode!
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Dustin
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Post by Dustin »

Wow, a new mirrored pair! At first I thought that FnF and BnB would stay the only one, until I had a closer look at the very first post... ;D If I compare the number of known DFF with n flies - 2 for n=2, 3 for n=3 and 7 for n=4 including free-rider-formations - there's a clear ascending tendency. Now we have "only" 8 DFF with 5 flies - Fright3+2BF, Fright4+1BF, Fright5, The 4 el+1FF, Cassiopeia, Sinking-F5, and Crawling F/B5 - which is just one more than with 4 flies. This could make us think that there should still be a lot more DFF with 5 flies. But there's also a counter-argument. As so many DFF are built around a vortex, it makes sense that the number of DFF with n flies will reach its maximum for some small n and then decrease again - after all, there are only 12 "seats" free around a vortex! Now two questions arise to me. First, is Cassiopeia really just a casually popping-up DFF without a scheme that continues to pop up in other undiscovered DFF? Second, does the vortex-scheme give us more high-n DFF built around two or more vortices? I think the vortex-issue could be just tried out by trail-and-error.
Boulder Dash X Rock, Paper, Scissors:
ROCKFORD collects DIAMOND, digs DIRT
DIAMOND outvalues DIRT & BOULDER
DIRT carries BOULDER, blocks FIREFLY
BOULDER kills FIREFLY & ROCKFORD
FIREFLY kills ROCKFORD, guards DIAMOND
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Arno
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Post by Arno »

Dustin wrote:Now we have "only" 8 DFF with 5 flies - Fright3+2BF, Fright4+1BF, Fright5, The 4 el+1FF, Cassiopeia, Sinking-F5, and Crawling F/B5 - which is just one more than with 4 flies.
Actually, there is a 9th one: Lifted F-vortex + 2BF; this formation is theoretically extentable to any larger number (and so far the only one for which that is possible). Still 9 is relatively low and makes me wonder as well if there are more 5-fly formations. I still have a few ideas to try to perhaps find new formations. To be continued.... :)
Dustin wrote:First, is Cassiopeia really just a casually popping-up DFF without a scheme that continues to pop up in other undiscovered DFF?
Good question, I still have to analyze that formation more thoroughly to see if there is some reusable principle.
Dustin wrote:Second, does the vortex-scheme give us more high-n DFF built around two or more vortices? I think the vortex-issue could be just tried out by trail-and-error.
Well, Sinking-F6 contains 2 vortices in some of it's phases. It's a very complex formation (with its 18 short loop phases). So it seems not the end of the vortex-based series... :D
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