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):
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.
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:
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.
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.
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:
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
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!