Posted: Mon Aug 26, 2019 4:51 pm
Arno - big respect for your effort. You did massive documentation of something which will be useful for most extreme BD players. 
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Dancing fly formations explained
Page 2 of 8
Posted: Mon Aug 26, 2019 6:32 pm
Thanks V-12! More to come! Next article on Fright-3,4,5
Posted: Sat Aug 31, 2019 11:59 am
Episode 5: Fright-3, 4 and 5
This is the first episode about formations involving more than 2 flies. We will dive into the details of 3 related dancing fly formations, Fright-3, 4 and 5. These 3 formations all move to the right and consist of resp. 3, 4 and 5 Fireflies. Also, to me these formations look quite frightful, hence the name!
This video shows all 3 formations (in reversed order).
The properties of these formations are as follows:
- Name: Fright-3 / Fright-4 / Fright-5
- Composition: 3 / 4 / 5 FF
- Short loop: 7 / 8 / 3 phases
- Long loop: 21 / 32 / 15 phases
- Shift per short loop: 1 / 2 / 1 Right
How do these formations work?
Fright-3
We start with Fright-3. As you will see, this formation demonstrates a general idea on which many dancing fly formations are based. It is possible to explain the main plot of this formation in the form of a somewhat dramatic story!
This story goes as follows:
Two fireflies are intimately dancing together. A third firefly, an angry and jealous rival, is frustratingly running around the dancing couple. Suddenly he breaks in, and brutally takes over the dance with one of the two flies. The remaining fly is now kicked out, has lost his dancing partner, and unwillingly takes over the role of the jealous rival. And so, the story repeats: the jealous fly starts running around the new dancing pair, until he finds a moment to break in, etc…
I’m telling the story in this way because it turns out that many dancing fly formations can be explained in a variation of this story.
Below picture shows the 7 phases of the short loop of Fright-3, in the same format as before.
Phase 1-7 show a vortex of 2 FF, the blue and red one. This vortex represents the “dancing couple” from the above story. (Note that actually, the 2 FF in a vortex are spinning independently. However, since they are spinning in the same 2x2 square, it looks like a dancing pair of FF.)
Also in phase 1-7, the green FF moves counter-clockwise around the vortex, from left-bottom to right-top. This is the “jealous rival”. At the corners, the FF can freely move to its left. In all other phases (until phase 7), you can easily check that the cave scanning rules determine that the green FF moves along either the red or blue FF, so it moves around the dancing couple. Between phase 7 and 1’ the “break-in” happens. This is the moment when the cave scanning rules determine that the green FF moves into the 2x2 area of the vortex. Now the red FF is pulled to the right side, starting a new vortex/dance with the green FF. The poor blue FF just follows its default path to its left, but is effectively kicked out from the vortex area. The vortex has moved 1 step to the right (leaving the blue FF behind), and so the whole formation has moved 1 step to the right.
Interesting to note is what happens after phase 1’. The blue FF has taken over the role of the green FF and will thus run counter-clockwise around the vortex, until he breaks in again at the right-top corner to start another dance. With the story in mind, you would probably expect that the blue FF takes revenge on the green FF and choses his original dance partner, the red FF. However, this is not what happens. Instead, the blue FF starts dancing with the green FF (his former rival!) and now the innocent red FF is rudely kicked out and left with the role of the rival.
And so we see, after each short loop the 3 FF change roles, such that after 3 short loops each fly has fulfilled all 3 roles once and returns to its original position. This makes that the long loop takes 3*7 = 21 phases.
Fright-4
Fright-4 is a dancing formation similar to Fright-3. It just involves one additional FF. In terms of the story, there are now 2 jealous rivals. Both are taking over the dance (vortex) in turn.
Below picture shows the 8 phases of the short loop. It starts with the same configuration as Fright-3 phase 1 where basically 1 FF (the yellow one) has been added.
Again, the formation is based around a dancing couple (vortex) consisting of the red and blue FF. In phase 1-4, the green and yellow FF are the two rivals, and move counter-clockwise around the vortex.
In phase 4, the yellow FF breaks in and starts a new dance with the red FF. The blue FF is kicked out. The green FF is still walking around the dancing couple.
In phase 5-8, the green and blue FF are the two rivals, walking around the vortex, with the green FF in front.
In phase 1’, the green FF has reached the point of breaking in and starts a dance with the red FF. The Yellow FF is kicked out. Since the configuration is now similar to phase 1, the short loop ends here by definition.
Within the short loop, the group moves 2 steps to the right. This means that Fright-4 advances faster than Fright-3, since on average Fright-4 needs 4 phases to move 1 step (while Fright-3 needs 7 phases).
After the last phase the flies have changed roles. Each FF will fulfil each role one by one. This implies that it takes 4 short loops before all flies have returned to their original position within the group. Thus, the long loop takes 4*8 = 32 phases.
Fright-5
The reason why Fright-4 has more phases than Fright-3 (8 versus 7) and moves 2 steps during a short loop is that the “queue” of FF moving around the vortex contains a gap that could be filled by one additional FF. Fright-5 exactly fills this gap. Another FF is added, so there are now 3 rivals.
As below picture shows, there are only 3 phases of the short loop. Basically a fifth FF (the purple one) has been added to the Fright-4 configuration.
Since the queue around the vortex has been filled up, the short loop of this formation has only 3 phases. At the end of the short loop one FF takes over the dance, another is put back to the queue and the remaining two FF in the queue have moved one position forward in the queue.
As Fright-5 needs only 3 phases to move 1 step, this formation is again faster than Fright-4, which (on average) needs 4 phases to move 1 step.
Also here it is clear that each FF will take each role and show up at each position one by one. Hence, it takes 5 short loops before all flies have returned to their original position within the group, so that the long loop takes 5*3 = 15 phases.
What about Fright-6?
Given the series of extensions of Fright-3 with 4 and 5 FF, it is natural to ask if a Fright-6 formation exists. The answer is no. With Fright-5, the queue of “rivals” waiting to take over the dance, is simply full. The path from the position where a fly is kicked out from the dance to the position where the fly breaks in again is 7 steps long. Within these 7 steps there is room for at max. 3 flies, since moving flies also need a distance of at least 1 step.
So it is not possible to continue this series of dancing formations with a 6 FF variant. This does not mean that 6 FF dancing formations cannot exist at all. Such formations do exist, but will just show a different dancing pattern.
Why do the Fright-3, 4 and 5 formations move to the right?
These formations are all based around a vortex. When the vortex moves into a certain direction, this determines the direction of the whole group.
For Fright-3, the crucial step is phase 7. Here, the vortex (or dancing pair) moves to the right because the approaching fly takes one of the two dancing partners away, at the right side. In the other formations, similar shifts occur.
The moving direction of the vortex is determined by the location where the rival FF breaks in. During phase 1-6 this FF is running around the vortex. Only in the corners the FF can move to its left. In all other phases the FF is forced to move forward since the vortex is blocking its left side. This is due to the cave scanning order. Phase 7 is the first moment when the rival FF gets precedence and can move into the 2x2 square of the dancing couple (vortex). This location of breaking in determines the movement direction in the end.
How to launch a Fright-3, 4 or 5?
One way to launch a Fright-3 is by first creating a vortex of 2 FF, and then “add” a third FF such that one of the phases is triggered. This video illustrates two variants of this method.
Fright-4 and 5 can be launched in a similar way, just by connecting 2 or 3 fireflies to the vortex. See videos for Fright-4 and Fright-5.
Apart from these methods, I have tried out another approach starting with a Fright-3, add one FF to the formation (while it is moving) to create a Fright-4, and afterwards add another FF to create a Fright-5. The problem, however, is that the FF naturally run in the wrong direction. The added FF needs to run counter-clockwise around the vortex, and until now I didn’t find a way to trigger such configuration without distorting the running formation.
Nevertheless, there are more ways. In this video the opposite is done: Fright-5 is decomposed step by step. First, a Fright-4 is created by removing a FF from Fright-5. Next, a Fright-3 is created by removing a FF from the Fright-4.
Finally, in this video a Fright-4 is triggered seemingly by chance from a large group of FF. Although the method works, it seems more a lucky shot rather than a clean method.
Other related questions
In this episode we examined dancing FF formations with 3 or more flies. This analysis gives rise to various related questions, like:
- Does a similar series of formation exist for butterflies?
- Or mixed variants?
- How many formations of 3 flies exist overall? (or 4 or 5?)
These are nice questions to be tackled in next episodes!
This is the first episode about formations involving more than 2 flies. We will dive into the details of 3 related dancing fly formations, Fright-3, 4 and 5. These 3 formations all move to the right and consist of resp. 3, 4 and 5 Fireflies. Also, to me these formations look quite frightful, hence the name!
This video shows all 3 formations (in reversed order).
The properties of these formations are as follows:
- Name: Fright-3 / Fright-4 / Fright-5
- Composition: 3 / 4 / 5 FF
- Short loop: 7 / 8 / 3 phases
- Long loop: 21 / 32 / 15 phases
- Shift per short loop: 1 / 2 / 1 Right
How do these formations work?
Fright-3
We start with Fright-3. As you will see, this formation demonstrates a general idea on which many dancing fly formations are based. It is possible to explain the main plot of this formation in the form of a somewhat dramatic story!
This story goes as follows:
Two fireflies are intimately dancing together. A third firefly, an angry and jealous rival, is frustratingly running around the dancing couple. Suddenly he breaks in, and brutally takes over the dance with one of the two flies. The remaining fly is now kicked out, has lost his dancing partner, and unwillingly takes over the role of the jealous rival. And so, the story repeats: the jealous fly starts running around the new dancing pair, until he finds a moment to break in, etc…
I’m telling the story in this way because it turns out that many dancing fly formations can be explained in a variation of this story.
Below picture shows the 7 phases of the short loop of Fright-3, in the same format as before.
Phase 1-7 show a vortex of 2 FF, the blue and red one. This vortex represents the “dancing couple” from the above story. (Note that actually, the 2 FF in a vortex are spinning independently. However, since they are spinning in the same 2x2 square, it looks like a dancing pair of FF.)
Also in phase 1-7, the green FF moves counter-clockwise around the vortex, from left-bottom to right-top. This is the “jealous rival”. At the corners, the FF can freely move to its left. In all other phases (until phase 7), you can easily check that the cave scanning rules determine that the green FF moves along either the red or blue FF, so it moves around the dancing couple. Between phase 7 and 1’ the “break-in” happens. This is the moment when the cave scanning rules determine that the green FF moves into the 2x2 area of the vortex. Now the red FF is pulled to the right side, starting a new vortex/dance with the green FF. The poor blue FF just follows its default path to its left, but is effectively kicked out from the vortex area. The vortex has moved 1 step to the right (leaving the blue FF behind), and so the whole formation has moved 1 step to the right.
Interesting to note is what happens after phase 1’. The blue FF has taken over the role of the green FF and will thus run counter-clockwise around the vortex, until he breaks in again at the right-top corner to start another dance. With the story in mind, you would probably expect that the blue FF takes revenge on the green FF and choses his original dance partner, the red FF. However, this is not what happens. Instead, the blue FF starts dancing with the green FF (his former rival!) and now the innocent red FF is rudely kicked out and left with the role of the rival.
And so we see, after each short loop the 3 FF change roles, such that after 3 short loops each fly has fulfilled all 3 roles once and returns to its original position. This makes that the long loop takes 3*7 = 21 phases.
Fright-4
Fright-4 is a dancing formation similar to Fright-3. It just involves one additional FF. In terms of the story, there are now 2 jealous rivals. Both are taking over the dance (vortex) in turn.
Below picture shows the 8 phases of the short loop. It starts with the same configuration as Fright-3 phase 1 where basically 1 FF (the yellow one) has been added.
Again, the formation is based around a dancing couple (vortex) consisting of the red and blue FF. In phase 1-4, the green and yellow FF are the two rivals, and move counter-clockwise around the vortex.
In phase 4, the yellow FF breaks in and starts a new dance with the red FF. The blue FF is kicked out. The green FF is still walking around the dancing couple.
In phase 5-8, the green and blue FF are the two rivals, walking around the vortex, with the green FF in front.
In phase 1’, the green FF has reached the point of breaking in and starts a dance with the red FF. The Yellow FF is kicked out. Since the configuration is now similar to phase 1, the short loop ends here by definition.
Within the short loop, the group moves 2 steps to the right. This means that Fright-4 advances faster than Fright-3, since on average Fright-4 needs 4 phases to move 1 step (while Fright-3 needs 7 phases).
After the last phase the flies have changed roles. Each FF will fulfil each role one by one. This implies that it takes 4 short loops before all flies have returned to their original position within the group. Thus, the long loop takes 4*8 = 32 phases.
Fright-5
The reason why Fright-4 has more phases than Fright-3 (8 versus 7) and moves 2 steps during a short loop is that the “queue” of FF moving around the vortex contains a gap that could be filled by one additional FF. Fright-5 exactly fills this gap. Another FF is added, so there are now 3 rivals.
As below picture shows, there are only 3 phases of the short loop. Basically a fifth FF (the purple one) has been added to the Fright-4 configuration.
Since the queue around the vortex has been filled up, the short loop of this formation has only 3 phases. At the end of the short loop one FF takes over the dance, another is put back to the queue and the remaining two FF in the queue have moved one position forward in the queue.
As Fright-5 needs only 3 phases to move 1 step, this formation is again faster than Fright-4, which (on average) needs 4 phases to move 1 step.
Also here it is clear that each FF will take each role and show up at each position one by one. Hence, it takes 5 short loops before all flies have returned to their original position within the group, so that the long loop takes 5*3 = 15 phases.
What about Fright-6?
Given the series of extensions of Fright-3 with 4 and 5 FF, it is natural to ask if a Fright-6 formation exists. The answer is no. With Fright-5, the queue of “rivals” waiting to take over the dance, is simply full. The path from the position where a fly is kicked out from the dance to the position where the fly breaks in again is 7 steps long. Within these 7 steps there is room for at max. 3 flies, since moving flies also need a distance of at least 1 step.
So it is not possible to continue this series of dancing formations with a 6 FF variant. This does not mean that 6 FF dancing formations cannot exist at all. Such formations do exist, but will just show a different dancing pattern.
Why do the Fright-3, 4 and 5 formations move to the right?
These formations are all based around a vortex. When the vortex moves into a certain direction, this determines the direction of the whole group.
For Fright-3, the crucial step is phase 7. Here, the vortex (or dancing pair) moves to the right because the approaching fly takes one of the two dancing partners away, at the right side. In the other formations, similar shifts occur.
The moving direction of the vortex is determined by the location where the rival FF breaks in. During phase 1-6 this FF is running around the vortex. Only in the corners the FF can move to its left. In all other phases the FF is forced to move forward since the vortex is blocking its left side. This is due to the cave scanning order. Phase 7 is the first moment when the rival FF gets precedence and can move into the 2x2 square of the dancing couple (vortex). This location of breaking in determines the movement direction in the end.
How to launch a Fright-3, 4 or 5?
One way to launch a Fright-3 is by first creating a vortex of 2 FF, and then “add” a third FF such that one of the phases is triggered. This video illustrates two variants of this method.
Fright-4 and 5 can be launched in a similar way, just by connecting 2 or 3 fireflies to the vortex. See videos for Fright-4 and Fright-5.
Apart from these methods, I have tried out another approach starting with a Fright-3, add one FF to the formation (while it is moving) to create a Fright-4, and afterwards add another FF to create a Fright-5. The problem, however, is that the FF naturally run in the wrong direction. The added FF needs to run counter-clockwise around the vortex, and until now I didn’t find a way to trigger such configuration without distorting the running formation.
Nevertheless, there are more ways. In this video the opposite is done: Fright-5 is decomposed step by step. First, a Fright-4 is created by removing a FF from Fright-5. Next, a Fright-3 is created by removing a FF from the Fright-4.
Finally, in this video a Fright-4 is triggered seemingly by chance from a large group of FF. Although the method works, it seems more a lucky shot rather than a clean method.
Other related questions
In this episode we examined dancing FF formations with 3 or more flies. This analysis gives rise to various related questions, like:
- Does a similar series of formation exist for butterflies?
- Or mixed variants?
- How many formations of 3 flies exist overall? (or 4 or 5?)
These are nice questions to be tackled in next episodes!
Posted: Sun Sep 01, 2019 6:38 pm
Wow - having read all this, now I want to implement some formations in Dustin's BD 11!
Posted: Sun Sep 01, 2019 7:31 pm
Thanks and excellent idea!Dustin wrote:Wow - having read all this, now I want to implement some formations in Dustin's BD 11!
Posted: Sun Sep 01, 2019 7:42 pm
By the way, I think it's funny that these formations could be used in "hard-core" caves where the player must find a launch method, as well as in fun caves where the formation is running from the start.
Posted: Sun Sep 01, 2019 8:15 pm
That's true! As for Dustin's BD 11, it will be hard to find a way to force the player to create a formation with only classical elements, i.e. no dolls.
Posted: Sun Sep 01, 2019 10:08 pm
It could for instance be necessary to launch such formations to send flies to an island (or several islands) in the middle of the cave.Dustin wrote:That's true! As for Dustin's BD 11, it will be hard to find a way to force the player to create a formation with only classical elements, i.e. no dolls.
Posted: Mon Sep 02, 2019 8:55 am
This is so funnyArno wrote:Two fireflies are intimately dancing together. A third firefly, an angry and jealous rival, is frustratingly running around the dancing couple. Suddenly he breaks in, and brutally takes over the dance with one of the two flies. The remaining fly is now kicked out, has lost his dancing partner, and unwillingly takes over the role of the jealous rival. And so, the story repeats: the jealous fly starts running around the new dancing pair, until he finds a moment to break in, etc…
And still I'm wonder why they are called fireflies since they doesn;t even look similar to fly Posted: Mon Sep 02, 2019 7:17 pm
Perhaps just because of the blinking animationV-12/Tropyx wrote:This is so funny
Anyway, I'm aware of at least one BD remake where they actually used a flying insect-type of character for the firefly: http://www.zxgames.com/en/boulderdash.shtml Posted: Tue Sep 03, 2019 9:48 pm
Thanx for the idea @Dr. Guru!
Posted: Sun Sep 08, 2019 12:05 pm
Episode 6: Lifted F-vortex, Lifted B-vortex and Gear Wheels
In this episode we will investigate (for the first time) a set of mixed formations, which involve a combination of fireflies and butterflies. All 3 formations move upwards and share a similar method. This video shows these 3 formations.
The properties of these formations are as follows:
- Name: Lifted F-vortex / Lifted B-vortex / Gear Wheels
- Composition: 2FF+1BF / 1FF+2BF / 2FF+2BF
- Short loop: 5 / 5 / 3 phases
- Long loop: 5 / 5 / 6 phases
- Shift per short loop: 1 / 1 / 1 Up
How do these formations work?
Lifted F-vortex
Do you remember the story from previous article, about the dancing couple and the jealous rival? Well, here the Lifted F-vortex can be explained by similar, yet slightly different story. Again, the story involves a dancing couple of fireflies. And again, a jealous rival is involved, this time in the guise of a butterfly. At a certain point, the butterfly breaks into the dancing FF couple, drifting both partners apart. This time, however, both FF partners stay loyal to each other. None of them start a dance with the butterfly. Instead, they both move upwards and start another dance together! Too bad for the butterfly… but he doesn’t give up! He starts walking around the dancing couple again until he finds a point to break in again, etc… But no matter how hard he tries, the butterfly will never dance with a firefly. What a painful sight…
Below picture shows the 5 phases of the short loop of the Lifted F-vortex.
In phase 2-5, both FF are “dancing” and the BF is running clockwise around the dancing couple (FF vortex) at the bottom and left side. At phase 4 the BF cannot yet break in, because the cave scanning order (from top to bottom) determines that the dark blue FF has precedence over the BF to take the conflicting position. At phase 5, the BF will finally break in, since at this point the cave scanning order (from left to right) determines that the BF has precedence over the light blue FF to take the conflicting position. In phase 1’, the BF has taken this position and has drifted both FF apart. As can be seen in phase 1 and 2, the BF next makes a usual move to its right (=down), which helps the two FF to occupy the space above the BF to start another dance. Now the BF has ended up again at the bottom of the vortex and starts moving clockwise around the vortex again.
After the last phase each fly has returned to its original position. Even the two FF have not swapped their role. In phase 1’ both FF move upward, with the light blue FF in front. This was already the case in phase 1. So it seems that in this partnership of fireflies, the light blue FF is a directive “leader”, while the dark blue FF just follows him!
In conclusion, this formation has a long loop of 5 phases, the same as the short loop.
Lifted B-vortex
The Lifted B-vortex could be explained by a similar story. This time 2 BF are dancing, while a FF is running around, until he breaks in at a certain point, drifting both BF apart. Again, both BF stay loyal to each other, move up together, and start another dance.
Below picture shows the 5 phases of the short loop of the Lifted B-vortex.
All phases are very similar to the Lifted F-vortex. The main difference is that the FF now runs counter-clockwise at the right side of the dancing pair (instead of left) and breaks in at the top-right (instead of the left-top). This is all due to cave scanning order. In phase 3, the dark green BF gets precedence over the FF since the cave scan runs from left to right. In phase 5, the FF gets precedence over the light green BF since the cave scan runs from top to bottom. Now the FF breaks in and the two BF are drifted apart. In phase 2 the FF moves to the left such that the two BF have free space to start another dance.
Also here, after the last phase each fly has returned to its original position. The two BF have kept their role: the light green BF is the “leader” moving in front in phase 1 and 1’, and the dark green BF is the “follower”.
Thus, also this formation has a long loop of 5 phases, the same as the short loop.
Gear Wheels
This formation combines the principles of Lifted F-vortex and Lifted B-vortex, resulting in a formation that rises faster than the former two.
In terms of the story, this formation looks like two couples, 2FF and 2BF, competing for the same space to dance. Yet, to me, when watching this formation moving up, I imagined a pair of gear wheels, spinning in opposite direction while pushing each other in the same (upward) direction. Hence the name.
Below picture shows the 3 phases of the short loop.
In phase 1 both FF are dancing while the leading BF (light green) is ready to break in.
In phase 2 the light green BF has broken in. The FF are drifted apart, making free space for the BF to dance.
In phase 3 both BF are dancing, while the leading FF (light blue) is ready to break in.
In phase 1’ the light blue FF has broken in and the BF are drifted apart. At the same time the FF have started another dance.
In contrast to the Lifted F- and B-vortex formations, at the end of a short loop of Gear Wheels the FF and BF partners have swapped role. Over the repeating cycles, each partner will in turn move at front at the times when their dance is interrupted. This is because the break-in happens earlier, before the leading partner has completed a full round of spinning.
Thus, it takes 2 short loops to complete a long loop, hence the long loop takes 2*3 = 6 phases.
Why do these formations move upward?
Like the Fright-x formations from previous episode, also this set of formations is based around a vortex. When the vortex moves into a certain direction, this determines the direction of the whole group. The moving direction of the vortex is determined by the moment of the break-in. For both the Lifted F- and B-vortex this happens just after phase 5. In both cases, the dancing partners are drifted aside, either to the left or right side. Since FF and BF are running in opposite direction, in both formations they move upward, and start another dance at the location just one step above their original location.
In the Gear Wheels formation, both scenarios occur within the same formation. Since both vortices move up, the whole formation moves up.
How to launch these formations?
As usual, each formation can be launch by triggering any of its phases.
For the Lifted F-Vortex, one way is to first create a FF vortex, and then connect a BF at the bottom-left with the right timing, such that the BF will move along the vortex until it breaks in.
The Lifted B-Vortex can be launched similarly, by creating a BF vortex and connecting a FF at the bottom.
Gear Wheels is more tricky to launch. Basically, two vortices have to be created at the same time. One with 2 FF and one with 2 BF. With the help of a boulder, it is possible to first create a pair of spinning flies, 1 FF and 1 BF, sharing one position within the 2x2 squares in which they are spinning. Next, a FF and BF should both approach from the bottom at the same time, such that both will form a vortex with their already spinning partner.
Note that the method for Gear Wheels can also be used to create a Lifted F-Vortex or Lifted B-vortex. Simply leave one approaching FF or BF out to create either formation.
This video shows the launch methods described above.
Free riders
We end this episode with a funny variant of the Lifted F-vortex. It is possible to extend this formation with one “free riding” butterfly. This effect was discovered recently by Shredder.
This additional BF does not influence the movements of the other flies. So these flies are not “aware” that someone is traveling with them; he is basically traveling as stowaway.
The overall formation still moves upwards, with the same speed.
Below picture shows the 5 phases of this formation, almost identical to the Lifted F-vortex, but with the light green BF added at the left side.
In phase 5, the light green BF is lifted up by the dark green BF. In all other phases, the light green BF is just spinning around.
In fact, it is possible to extend this formation with even more butterflies. The light green BF could lift up another extra BF at its left side, and so on. This gives theoretically an unlimited set of dancing fly formations! This video shows a cave where this idea is applied. To me these formations look like flying kites, waving in the wind.
Another video shows a huge variant of this formation, using 100 flies in total, limited by the maximum cave size (100x100) at Krissz’ Boulder Dash site. A world record for now!
Note, however, that all these additional BF are free riders, and have no role in keeping the group together or determining the speed and direction of the group. For this reason, this set of formations is not included in the list of dancing fly formations in the first post of this thread. I deliberately limit this list to unique formations, where every fly interacts with another fly at least once.
In the next episode we will move away from these huge formations, and step back to smaller formations by trying to answer a relevant question regarding 3-fly formations.
In this episode we will investigate (for the first time) a set of mixed formations, which involve a combination of fireflies and butterflies. All 3 formations move upwards and share a similar method. This video shows these 3 formations.
The properties of these formations are as follows:
- Name: Lifted F-vortex / Lifted B-vortex / Gear Wheels
- Composition: 2FF+1BF / 1FF+2BF / 2FF+2BF
- Short loop: 5 / 5 / 3 phases
- Long loop: 5 / 5 / 6 phases
- Shift per short loop: 1 / 1 / 1 Up
How do these formations work?
Lifted F-vortex
Do you remember the story from previous article, about the dancing couple and the jealous rival? Well, here the Lifted F-vortex can be explained by similar, yet slightly different story. Again, the story involves a dancing couple of fireflies. And again, a jealous rival is involved, this time in the guise of a butterfly. At a certain point, the butterfly breaks into the dancing FF couple, drifting both partners apart. This time, however, both FF partners stay loyal to each other. None of them start a dance with the butterfly. Instead, they both move upwards and start another dance together! Too bad for the butterfly… but he doesn’t give up! He starts walking around the dancing couple again until he finds a point to break in again, etc… But no matter how hard he tries, the butterfly will never dance with a firefly. What a painful sight…
Below picture shows the 5 phases of the short loop of the Lifted F-vortex.
In phase 2-5, both FF are “dancing” and the BF is running clockwise around the dancing couple (FF vortex) at the bottom and left side. At phase 4 the BF cannot yet break in, because the cave scanning order (from top to bottom) determines that the dark blue FF has precedence over the BF to take the conflicting position. At phase 5, the BF will finally break in, since at this point the cave scanning order (from left to right) determines that the BF has precedence over the light blue FF to take the conflicting position. In phase 1’, the BF has taken this position and has drifted both FF apart. As can be seen in phase 1 and 2, the BF next makes a usual move to its right (=down), which helps the two FF to occupy the space above the BF to start another dance. Now the BF has ended up again at the bottom of the vortex and starts moving clockwise around the vortex again.
After the last phase each fly has returned to its original position. Even the two FF have not swapped their role. In phase 1’ both FF move upward, with the light blue FF in front. This was already the case in phase 1. So it seems that in this partnership of fireflies, the light blue FF is a directive “leader”, while the dark blue FF just follows him!
In conclusion, this formation has a long loop of 5 phases, the same as the short loop.
Lifted B-vortex
The Lifted B-vortex could be explained by a similar story. This time 2 BF are dancing, while a FF is running around, until he breaks in at a certain point, drifting both BF apart. Again, both BF stay loyal to each other, move up together, and start another dance.
Below picture shows the 5 phases of the short loop of the Lifted B-vortex.
All phases are very similar to the Lifted F-vortex. The main difference is that the FF now runs counter-clockwise at the right side of the dancing pair (instead of left) and breaks in at the top-right (instead of the left-top). This is all due to cave scanning order. In phase 3, the dark green BF gets precedence over the FF since the cave scan runs from left to right. In phase 5, the FF gets precedence over the light green BF since the cave scan runs from top to bottom. Now the FF breaks in and the two BF are drifted apart. In phase 2 the FF moves to the left such that the two BF have free space to start another dance.
Also here, after the last phase each fly has returned to its original position. The two BF have kept their role: the light green BF is the “leader” moving in front in phase 1 and 1’, and the dark green BF is the “follower”.
Thus, also this formation has a long loop of 5 phases, the same as the short loop.
Gear Wheels
This formation combines the principles of Lifted F-vortex and Lifted B-vortex, resulting in a formation that rises faster than the former two.
In terms of the story, this formation looks like two couples, 2FF and 2BF, competing for the same space to dance. Yet, to me, when watching this formation moving up, I imagined a pair of gear wheels, spinning in opposite direction while pushing each other in the same (upward) direction. Hence the name.
Below picture shows the 3 phases of the short loop.
In phase 1 both FF are dancing while the leading BF (light green) is ready to break in.
In phase 2 the light green BF has broken in. The FF are drifted apart, making free space for the BF to dance.
In phase 3 both BF are dancing, while the leading FF (light blue) is ready to break in.
In phase 1’ the light blue FF has broken in and the BF are drifted apart. At the same time the FF have started another dance.
In contrast to the Lifted F- and B-vortex formations, at the end of a short loop of Gear Wheels the FF and BF partners have swapped role. Over the repeating cycles, each partner will in turn move at front at the times when their dance is interrupted. This is because the break-in happens earlier, before the leading partner has completed a full round of spinning.
Thus, it takes 2 short loops to complete a long loop, hence the long loop takes 2*3 = 6 phases.
Why do these formations move upward?
Like the Fright-x formations from previous episode, also this set of formations is based around a vortex. When the vortex moves into a certain direction, this determines the direction of the whole group. The moving direction of the vortex is determined by the moment of the break-in. For both the Lifted F- and B-vortex this happens just after phase 5. In both cases, the dancing partners are drifted aside, either to the left or right side. Since FF and BF are running in opposite direction, in both formations they move upward, and start another dance at the location just one step above their original location.
In the Gear Wheels formation, both scenarios occur within the same formation. Since both vortices move up, the whole formation moves up.
How to launch these formations?
As usual, each formation can be launch by triggering any of its phases.
For the Lifted F-Vortex, one way is to first create a FF vortex, and then connect a BF at the bottom-left with the right timing, such that the BF will move along the vortex until it breaks in.
The Lifted B-Vortex can be launched similarly, by creating a BF vortex and connecting a FF at the bottom.
Gear Wheels is more tricky to launch. Basically, two vortices have to be created at the same time. One with 2 FF and one with 2 BF. With the help of a boulder, it is possible to first create a pair of spinning flies, 1 FF and 1 BF, sharing one position within the 2x2 squares in which they are spinning. Next, a FF and BF should both approach from the bottom at the same time, such that both will form a vortex with their already spinning partner.
Note that the method for Gear Wheels can also be used to create a Lifted F-Vortex or Lifted B-vortex. Simply leave one approaching FF or BF out to create either formation.
This video shows the launch methods described above.
Free riders
We end this episode with a funny variant of the Lifted F-vortex. It is possible to extend this formation with one “free riding” butterfly. This effect was discovered recently by Shredder.
This additional BF does not influence the movements of the other flies. So these flies are not “aware” that someone is traveling with them; he is basically traveling as stowaway.
The overall formation still moves upwards, with the same speed.
Below picture shows the 5 phases of this formation, almost identical to the Lifted F-vortex, but with the light green BF added at the left side.
In phase 5, the light green BF is lifted up by the dark green BF. In all other phases, the light green BF is just spinning around.
In fact, it is possible to extend this formation with even more butterflies. The light green BF could lift up another extra BF at its left side, and so on. This gives theoretically an unlimited set of dancing fly formations! This video shows a cave where this idea is applied. To me these formations look like flying kites, waving in the wind.
Another video shows a huge variant of this formation, using 100 flies in total, limited by the maximum cave size (100x100) at Krissz’ Boulder Dash site. A world record for now!
Note, however, that all these additional BF are free riders, and have no role in keeping the group together or determining the speed and direction of the group. For this reason, this set of formations is not included in the list of dancing fly formations in the first post of this thread. I deliberately limit this list to unique formations, where every fly interacts with another fly at least once.
In the next episode we will move away from these huge formations, and step back to smaller formations by trying to answer a relevant question regarding 3-fly formations.
Posted: Sat Sep 14, 2019 11:01 am
Episode 7: How many 3-fly formations exist?
This episode is all about dancing formations involving exactly 3 flies.
Until now we have discovered and discussed the following 3-fly formations:
- Fright-3: 3FF going right
- Lifted F-vortex: 2FF + 1BF going up
- Lifted B-vortex: 1FF + 2BF going up
A remarkable fact is that there isn’t any formation with 3 butterflies in this list. In addition, each of the other possible compositions occurs exactly once. For instance, Fright-3 is the only formation consisting of 3 fireflies.
The main question of this episode is:
How many dancing formations with 3 flies exist in total? Are there any other than those three listed above?
I am almost 100% sure that this is not the case. I’ll provide my argumentation below.
In order to reduce complexity when investigating the possible 3-fly formations, I’m going to make an important assumption:
Every 3-fly dancing formation involves a vortex of 2 flies and a third fly breaking in.
In other words, the story from previous episodes, with a dancing couple and a jealous rival, always applies. (At least, for 3-fly formations.)
This statement is hard to prove formally. Yet, I’m quite convinced that it must be true, and I have the following reasons.
First, the group must stay together while keep interacting. It is not allowed that one fly isolates from the group and just starts spinning around independently. I could not find a way to achieve this with 3 flies other than by using a vortex of 2 flies and let the third fly move around it.
Second, it is important that the group moves away from the current position. Otherwise, it is not a dancing fly formation. With a vortex, this can only be achieved when the vortex is interrupted at some point. If not, the vortex will just stay at the same position.
So, although there is no hard evidence currently, I’m almost sure a that the above assumption is valid. Let’s assume for now that it is indeed valid. So, there must be at least one phase consisting of a vortex and a third fly ready to break in.
Luckily, this assumption limits the set of all candidate setups considerably. Under this constraint, it is doable to just check them all. This is exactly what we will do below, for each possible composition.
3 Fireflies
Since a vortex consists of two flies spinning within the same 2x2 square, there are 8 locations where a third fly could break in. Below picture shows these 8 locations, numbered 1-8 clockwise.
With a vortex of 2 FF (dark blue) and 1 rival-FF (light blue), the following setups show all possibilities to let the rival-FF break in at the positions as numbered above:
Interesting to note is that location 8 has no possibility to break in. At the moment when the BD engine scans the rival-FF, always one of the two “dancing partners” is present at the left-bottom position of the vortex, blocking the way for the rival-FF. Either it is already blocked within the current frame, or the position is initially free but will be occupied in the same frame by the FF at the top-left (which is scanned before the rival-FF and moves down, blocking the way for the rival-FF).
This effect is interesting by itself, and is used, for instance, in this cave to block a growing wall.
Also remarkable is that location 4 has two different possibilities to break in. In each of the two stages of the vortex, a FF at that location will break in. This is because the light blue FF is scanned after the dark blue FF on the same line, but before the dark blue FF on the next line. So either the top-right position gets free before the light blue FF moves, or the position is already free and the light blue FF can move there unhampered.
I’ve tested these 8 scenarios in the Construction Kit, and the above picture indicates for each scenario what happens. One of the setups where the break-in happens at location 4 is exactly one of the phases of Fright-3 (phase 7, to be precise). So this setup gives a Fright-3 dancing formation. Two other setups lead indirectly to a Fright-3 as well. The remaining setups ultimately lead to all 3 FF spinning around. Hence, we did not find a new dancing fly formation.
In other words: Fright-3 is the only dancing fly formation with 3 fireflies.
2 Fireflies + 1 Butterfly
Because we assume that a vortex is involved, this vortex must consist of the 2 fireflies (since a vortex of 1 FF + 1 BF is not possible). Below picture shows all possible setups, basically the same as previous picture, but with the rival-FF replaced by a rival-BF.
For each setup it is indicated what happens after testing it in the Construction Kit. The setup with break-in at location 1 is exactly phase 5 of a Lifted F-vortex. Three other setups lead indirectly to a Lifted F-vortex. All remaining setups lead to a situation where all flies are spinning around.
Thus, Lifted F-vortex is the only dancing fly formation with 2 fireflies and 1 butterfly.
2 Butterflies + 1 Firefly
We will now look at the opposite setup: a vortex of 2 butterflies and a rival FF breaking in. Below picture shows all possible scenarios, where the FF breaks in at locations 1-8 as defined before.
Notice that a BF vortex doesn’t have the peculiarities of a FF vortex. For all 8 locations there is exactly one way to break in. With a BF vortex it is not possible to block any object (like a growing wall or another fly).
Again, the above picture shows the results of testing these scenarios. Not surprisingly, one setup (with break-in at location 3) is equal to phase 5 of a Lifted B-vortex. Two others lead indirectly to a Lifted B-vortex. All others lead to spinning flies.
Ergo, Lifted B-vortex is the only dancing fly formation with 2 butterflies and 1 firefly.
3 Butterflies
Finally let’s look at the situation where 3 butterflies are involved. Below picture shows all possible setups of a BF vortex and a rival-BF breaking in.
After testing these setups it appears that all of them ultimately lead to spinning butterflies. We haven’t found a dancing formation. Unless someone disproves the assumption I made at the beginning, we must conclude for now that a dancing fly formation with 3 butterflies does not exist.
Wrapping up
The three formations listed at the top of this episode are apparently the only 3-fly dancing fly formations. This is, of course, under an assumption for which a hard proof is not available yet. Still, I would be very surprised if this assumption would not be true, so there are other types of 3-fly formations, not involving a vortex and a third fly breaking in.
The most shocking finding for me is that a 3-BF formation does not exist, while a 3-FF formation does exist (Fright-3). We have seen in the earlier episodes that FnF (a 2-firefly formation) has a counterpart with 2 butterflies: the BnB. Also, the two Lifted F/B-vortex formations are counterparts with opposite composition. So why do these formations have such a counterpart, while Fright-3 has not?
It is time for the next episode on relations between formations.
This episode is all about dancing formations involving exactly 3 flies.
Until now we have discovered and discussed the following 3-fly formations:
- Fright-3: 3FF going right
- Lifted F-vortex: 2FF + 1BF going up
- Lifted B-vortex: 1FF + 2BF going up
A remarkable fact is that there isn’t any formation with 3 butterflies in this list. In addition, each of the other possible compositions occurs exactly once. For instance, Fright-3 is the only formation consisting of 3 fireflies.
The main question of this episode is:
How many dancing formations with 3 flies exist in total? Are there any other than those three listed above?
I am almost 100% sure that this is not the case. I’ll provide my argumentation below.
In order to reduce complexity when investigating the possible 3-fly formations, I’m going to make an important assumption:
Every 3-fly dancing formation involves a vortex of 2 flies and a third fly breaking in.
In other words, the story from previous episodes, with a dancing couple and a jealous rival, always applies. (At least, for 3-fly formations.)
This statement is hard to prove formally. Yet, I’m quite convinced that it must be true, and I have the following reasons.
First, the group must stay together while keep interacting. It is not allowed that one fly isolates from the group and just starts spinning around independently. I could not find a way to achieve this with 3 flies other than by using a vortex of 2 flies and let the third fly move around it.
Second, it is important that the group moves away from the current position. Otherwise, it is not a dancing fly formation. With a vortex, this can only be achieved when the vortex is interrupted at some point. If not, the vortex will just stay at the same position.
So, although there is no hard evidence currently, I’m almost sure a that the above assumption is valid. Let’s assume for now that it is indeed valid. So, there must be at least one phase consisting of a vortex and a third fly ready to break in.
Luckily, this assumption limits the set of all candidate setups considerably. Under this constraint, it is doable to just check them all. This is exactly what we will do below, for each possible composition.
3 Fireflies
Since a vortex consists of two flies spinning within the same 2x2 square, there are 8 locations where a third fly could break in. Below picture shows these 8 locations, numbered 1-8 clockwise.
With a vortex of 2 FF (dark blue) and 1 rival-FF (light blue), the following setups show all possibilities to let the rival-FF break in at the positions as numbered above:
Interesting to note is that location 8 has no possibility to break in. At the moment when the BD engine scans the rival-FF, always one of the two “dancing partners” is present at the left-bottom position of the vortex, blocking the way for the rival-FF. Either it is already blocked within the current frame, or the position is initially free but will be occupied in the same frame by the FF at the top-left (which is scanned before the rival-FF and moves down, blocking the way for the rival-FF).
This effect is interesting by itself, and is used, for instance, in this cave to block a growing wall.
Also remarkable is that location 4 has two different possibilities to break in. In each of the two stages of the vortex, a FF at that location will break in. This is because the light blue FF is scanned after the dark blue FF on the same line, but before the dark blue FF on the next line. So either the top-right position gets free before the light blue FF moves, or the position is already free and the light blue FF can move there unhampered.
I’ve tested these 8 scenarios in the Construction Kit, and the above picture indicates for each scenario what happens. One of the setups where the break-in happens at location 4 is exactly one of the phases of Fright-3 (phase 7, to be precise). So this setup gives a Fright-3 dancing formation. Two other setups lead indirectly to a Fright-3 as well. The remaining setups ultimately lead to all 3 FF spinning around. Hence, we did not find a new dancing fly formation.
In other words: Fright-3 is the only dancing fly formation with 3 fireflies.
2 Fireflies + 1 Butterfly
Because we assume that a vortex is involved, this vortex must consist of the 2 fireflies (since a vortex of 1 FF + 1 BF is not possible). Below picture shows all possible setups, basically the same as previous picture, but with the rival-FF replaced by a rival-BF.
For each setup it is indicated what happens after testing it in the Construction Kit. The setup with break-in at location 1 is exactly phase 5 of a Lifted F-vortex. Three other setups lead indirectly to a Lifted F-vortex. All remaining setups lead to a situation where all flies are spinning around.
Thus, Lifted F-vortex is the only dancing fly formation with 2 fireflies and 1 butterfly.
2 Butterflies + 1 Firefly
We will now look at the opposite setup: a vortex of 2 butterflies and a rival FF breaking in. Below picture shows all possible scenarios, where the FF breaks in at locations 1-8 as defined before.
Notice that a BF vortex doesn’t have the peculiarities of a FF vortex. For all 8 locations there is exactly one way to break in. With a BF vortex it is not possible to block any object (like a growing wall or another fly).
Again, the above picture shows the results of testing these scenarios. Not surprisingly, one setup (with break-in at location 3) is equal to phase 5 of a Lifted B-vortex. Two others lead indirectly to a Lifted B-vortex. All others lead to spinning flies.
Ergo, Lifted B-vortex is the only dancing fly formation with 2 butterflies and 1 firefly.
3 Butterflies
Finally let’s look at the situation where 3 butterflies are involved. Below picture shows all possible setups of a BF vortex and a rival-BF breaking in.
After testing these setups it appears that all of them ultimately lead to spinning butterflies. We haven’t found a dancing formation. Unless someone disproves the assumption I made at the beginning, we must conclude for now that a dancing fly formation with 3 butterflies does not exist.
Wrapping up
The three formations listed at the top of this episode are apparently the only 3-fly dancing fly formations. This is, of course, under an assumption for which a hard proof is not available yet. Still, I would be very surprised if this assumption would not be true, so there are other types of 3-fly formations, not involving a vortex and a third fly breaking in.
The most shocking finding for me is that a 3-BF formation does not exist, while a 3-FF formation does exist (Fright-3). We have seen in the earlier episodes that FnF (a 2-firefly formation) has a counterpart with 2 butterflies: the BnB. Also, the two Lifted F/B-vortex formations are counterparts with opposite composition. So why do these formations have such a counterpart, while Fright-3 has not?
It is time for the next episode on relations between formations.
Posted: Sat Sep 14, 2019 6:17 pm
You are really a TRUE Boulder Dash fan...
Posted: Sat Sep 14, 2019 6:42 pm
Yeah I love to be surprised when I or someone else discovers new possibilities with the BD engine. And that, after 35 years, for a simple platform game with a few different elements....!