I’m wondering if there’s a name for something I’ve noticed. I’ve defined it below in italics.
Sequences are comprised of elements. For example, 1-2-3-4 has four elements.
If a sequence is looped such that there is one more or one less element per cycle, the number of cycles it will take to return to the first element is equal to the number of elements (the length of the sequence).
For example, using the sequence 1-2-3-4:
Looping with one more element:
1-2-3-4-1
2-3-4-1-2
3-4-1-2-3
4-1-2-3-4
(back to 1 in four cycles)
Looping with one less element:
1-2-3
4-1-2
3-4-1
2-3-4
(back to 1 in four cycles)
The same results are obtained with sequences of five and six elements and, I’m guessing, all others.
Is there a name for this concept (defined above in italics)? Is it of any interest in mathematics or engineering?
I don’t know that there’s a specific name, but I’ll point out that it works for any two numbers that are relatively prime. N and N-1 are always relatively prime, so that works. But it works in other cases, too, like 3 and 5:
12312
31231
23123
12312
Sequence length 3, looping every 5, repeats after 3 cycles. But consider a case where the numbers are not relatively prime, like 4 and 6:
123412
341234
123412
The cycle repeats after only two cycles since 4 and 6 are not relatively prime.
If you have a word, say abccba, you can consider its cyclic permutations bccbaa, ccbaab, cbaabc, … Define a primitive word to be a word that is not a power of a shorter word, i.e., w=z^n implies n=1. Then a word is primitive if and only if it is distinct from all its cyclic rotations, so this seems to be the concept you are looking for.
I’m not entirely sure what you are going for, but it sounds a lot like modular arithmetic.
Seems like you are adding elements that are congruent to 1 or -1 modulo n (in your example n = 4). Sure enough, after n iterations you are back to the start.
@Typo_Knig:
Yes, I think what I’m looking for is an elementary part of cyclic permutation, which gets into… I’ll call it “repeating subsets of sequences.” I’m talking about something much more simple, but it seems to be closely related. I’ll give it some study and thought.
@Dr.Strangelove:
Those examples are different from what I’m looking for. You’re looping with two more elements whereas I’m talking about looping with one more or one less element. Applying my idea to the sequence 1-2-3 would look like this:
Looping with one more element:
1-2-3-1
2-3-1-2
3-1-2-3
(back to 1 in three cycles)
Looping with one less element:
1-2
3-1
2-3
(back to 1 in three cycles)
@DPRK:
You lost me there with the change from numbers to letters and words, and I’m not smart enough to understand the formula/equation you’ve provided. As I said, what I’m looking for is the name of the concept that I’ve defined in italics.
@Lance_Turbo:
Least common multiple. That might be it, and I thought it might be something basic that I’m just coming at in a weird way, as you say (which is why I asked if it’s of interest). But the cycles in your post, like Dr.Strangelove’s, loop with two more elements, and I’m talking about looping with one more or one less element.
@kk_fusion:
Modular arithmetic. Yeah, that clearly appears to be related to or maybe exactly what I’m talking about. I’ll look into it.
I understand that, but my point is that your original examples are just special cases of a more general principle. You are correct that if the loop length is one more or one less than the sequence length, then it will take a number of cycles equal to the sequence length to repeat.
But in addition to these cases, any time the two numbers are relatively prime, we can also expect the same behavior. N and N-1 are always relatively prime, so that fits. But there are other numbers as well.
But on the other hand, if the numbers are not relatively prime, then we will not hit the maximum sequence length. It will repeat after some shorter number of cycles, and miss some of the possible rotations.
Ah, okay. Thanks (ETA: thanks to both of you), I appreciate your efforts. I’ll say it again: I’m not smart enough to understand this at a higher level than simple observation. If you want to play around with the concept here, I’ll stand back and let you smart people have at it. (I say that without sarcasm.)
You should give yourself more credit! This is a non-obvious observation for someone not already familiar with it. Math is driven by just these sorts of explorations, as well as trying to generalize further. If you enjoyed making this discovery in the first place, you’ll probably enjoy exploring what happens with other kinds of numbers.
Thank you; you’ve very kind. I’m thrilled to see that smart people are giving it some thought!
The idea occurred to me in my childhood (not a long story but kind of weird and embarrassing), and I was later able to apply it to music. What some people call banjo rolls and other guitar techniques.
Ninjaed… I was also going to add that when ringing actual bells, there is some limitation on what permutations can be applied at any given moment. I do not know precisely what can be pulled off, but you can only slow down or speed up each bell by a maximum amount, so it would not be possible to ring 123456234561 without inserting intermediate permutations.
In general musical compositions you are not limited. Tone rows are pretty standard
and if you are good at combinatorics and have some musical ideas then many techniques are possible.
I don’t follow this. A traditional bell is held upside down until the bellringer triggers it. The bell will ring once and the ringer has applied enough energy to make it rotate back to its starting position.
Timing is not a case of speeding or slowing the bell, but holding it until it is time for it to strike. It takes skill and teamwork to get it right.
So you are saying one could ring 123456
immediately followed by 234561
by holding off bell #1? In that case bells 2, 3, 4, and 5 each ring after 5 beats but 1 must not ring until 11 places later.
In change ringing, each bell in a ring is numbered from 1 (the highest) to 𝑛 (the lowest). The bells sound in a sequence of rows. The movement between one row and the next is called a change.
There are two rules.
Each bell sounds exactly once in each row.
In any change, each bell can move at most one position.
There is a Lord Peter Wimsey quote (W Sayers) in which he says something like: “Foreigners might play tunes with their bells, but the English prefer mathematics.”
There are many books and papers written about the mathematical sequences involved for different numbers of bells.
Also related is the concept of beat frequencies. If you have one wave at one frequency, and superimpose it with another wave at a slightly different frequency, you’ll get a wave that increases and decreases its amplitude, with a frequency equal to the difference of the two source frequencies.
