Every number in the sequence which equals 1-(y/2)^2 for “nice” y to “good” accuracy is in fact on the quadratic; i.e., when a number is “approximately” 1-(y/2)^2 for a y that “could” come next, that y does in fact come next. BUT to make this so I am having to define “nice” and “good” fairly tightly; this in fact occurs for only 14 values early in the sequence, perhaps before roundoff error starts being a nuisance.
(I am conjecturing that the way this thing is constructed might be that we start with a few initial values and then iterate some process like: “if (condition) then next number is—previous number; else if (condition) then next number is 2sqrt(1-previous); else if …”. If so, then we might start with “nice” numbers—in fact the first two are 1.3 and 2.1 to machine precision—but that “niceness” might be eroded as we calculate. This is all very handwavy and there is at least one kinda-obvious objection to it.)
A few reasons for suspecting that this sort of iterative process is in play:
It would fit with the relationship to “That Alien Message” where the idea is that each frame (here, each number) is the “next” state of affairs as something evolves according to simple rules.
There are many cases in which it sure looks as if one number is derived from its predecessor. (Or maybe from its successor.)
It’s an obvious thing to do, if you want to set a challenge like this.
Note that I am not necessarily conjecturing that each number is a function of its predecessor. There might be other internal state that we don’t get to see.
Do you see how such an iteration can produce the long-distance correlations I mention in a message below, between floats at positions that differ by a factor of 2/e? It seems that this would require some explicit dependence on the index.
Its not exactly 2/e. Here is a plot of the “error” of those points. The x axis is the larger point. The y axis is the smaller point minus 2/e times the larger.
So its within about 1% of 2/e, suggesting it might be a real thing, or might just be a coincidence.
(I am conjecturing that the way this thing is constructed might be that we start with a few initial values and then iterate some process like: “if (condition) then next number is—previous number; else if (condition) then next number is 2sqrt(1-previous); else if …”. If so, then we might start with “nice” numbers—in fact the first two are 1.3 and 2.1 to machine precision—but that “niceness” might be eroded as we calculate. This is all very handwavy and there is at least one kinda-obvious objection to it.)
Have you tried writing programs that create sequences like the above to see how close they get? (Asking to avoid redoing work if the answer is yes.)
Making explicit something I said in passing elsewhere: it seems not-impossible that the sequence might be best looked at in reverse order, since mapping x to 1-(x/2)^2 or (1-x^2)/2 is somewhat more natural than going the other way. However, it does look rather as if there might be some accumulation of roundoff error in the forward direction, which if true would argue for reading it forwards rather than backwards.
Every number in the sequence which equals 1-(y/2)^2 for “nice” y to “good” accuracy is in fact on the quadratic; i.e., when a number is “approximately” 1-(y/2)^2 for a y that “could” come next, that y does in fact come next. BUT to make this so I am having to define “nice” and “good” fairly tightly; this in fact occurs for only 14 values early in the sequence, perhaps before roundoff error starts being a nuisance.
(I am conjecturing that the way this thing is constructed might be that we start with a few initial values and then iterate some process like: “if (condition) then next number is—previous number; else if (condition) then next number is 2sqrt(1-previous); else if …”. If so, then we might start with “nice” numbers—in fact the first two are 1.3 and 2.1 to machine precision—but that “niceness” might be eroded as we calculate. This is all very handwavy and there is at least one kinda-obvious objection to it.)
A few reasons for suspecting that this sort of iterative process is in play:
It would fit with the relationship to “That Alien Message” where the idea is that each frame (here, each number) is the “next” state of affairs as something evolves according to simple rules.
There are many cases in which it sure looks as if one number is derived from its predecessor. (Or maybe from its successor.)
It’s an obvious thing to do, if you want to set a challenge like this.
Note that I am not necessarily conjecturing that each number is a function of its predecessor. There might be other internal state that we don’t get to see.
Do you see how such an iteration can produce the long-distance correlations I mention in a message below, between floats at positions that differ by a factor of 2/e? It seems that this would require some explicit dependence on the index.
Its not exactly 2/e. Here is a plot of the “error” of those points. The x axis is the larger point. The y axis is the smaller point minus 2/e times the larger.
So its within about 1% of 2/e, suggesting it might be a real thing, or might just be a coincidence.
Have you tried writing programs that create sequences like the above to see how close they get? (Asking to avoid redoing work if the answer is yes.)
Making explicit something I said in passing elsewhere: it seems not-impossible that the sequence might be best looked at in reverse order, since mapping x to 1-(x/2)^2 or (1-x^2)/2 is somewhat more natural than going the other way. However, it does look rather as if there might be some accumulation of roundoff error in the forward direction, which if true would argue for reading it forwards rather than backwards.