Hacker News Reader

Comment by lifeformed | Hacker News Reader

One I found as a kid was 0.73908513321. It appears when you just press the Cos button on a calculator repeatedly (for any starting value). I can't find it used anywhere else.

View on HN

You can see that your constant is the solution to x=cos(x) (if the equality is true, you can replace the x on the right side with cos(x), and repeat this process).

The solution apparently has a name, Dottie constant [1].

[1] https://mathworld.wolfram.com/DottieNumber.html

Beautifully, iterating any continuous function will yield a fixed point of the function if it converges.

https://en.wikipedia.org/wiki/Fixed-point_iteration

And in case it neither converges to a single value nor diverges to infinity, a whole new worldof wondrous math emerges [1][2][3]

[1] https://en.wikipedia.org/wiki/Logistic_map

[2] https://en.wikipedia.org/wiki/Mandelbrot_set

[3] https://en.wikipedia.org/wiki/Attractor#Strange_attractor

Interestingly, I've found this kinda sorta occasionally works on functionals of a differential equation too if you start with a good enough guess. Someone has probably made this concept and the necessary/sufficient conditions rigorous but its getting into territory that is a good bit too advanced for me to follow.

As I recall from my fiddling, you are probably most likely to end up with a series solution of sorts, so it's a good idea to guess with polynomials or exponentials so you end up with component functions that form a basis for analytic functions.

It's not too bad to formalize :) The [https://en.m.wikipedia.org/wiki/Fixed-point_iteration](wikip... article on fixed point iteration) gives lip service to everything working fine in arbitrary metric spaces. It's easy to define a useful metric between functions (e.g. via integrating their difference, ignoring edge cases like occasional pointwise differences since a more careful treatment can give a presentation where those don't matter for the problem at hand), so the same kinds of theorems that work with cos(x) on the reals also work with differential operators on function spaces.

Its name is the "Dottie Number":

[0] https://en.wikipedia.org/wiki/Dottie_number

[1] https://mathworld.wolfram.com/DottieNumber.html

I found that one too. I don't think it arises in any "real" mathematics but it's a nice example.

Also, if you alternate hitting the reciprocal key and adding 1 you get the golden ratio 1.618033... pretty quickly.

Yeah that’s the limit of the repeated map of the cosine function for any initial value; it can be shown to be equal to x such that cos(x) – x = 0

I suppose that’s the limit value of repeatedly taking the cosine of any value and then the cosine of that iterating up to infinity.

Sounds easy enough to check, and easy enough to prove, if the hunch proves right.

Reply on news.ycombinator.com