fvirexi's comments

It's not based on stereotypes of Texas and Texans, but rather of the Wild West - as in the location of cowboy movies. There exists equivalent phrases refering to the Wild West directly, and at one point it seems "Texas" became (or perhaps already was) synonymous with "the wild west".

Speculation: in addition to cowbody movies, it probably has something to do with the norwegian book series Morgan Kane[1] from the 60s, as the phrase is mostly heard from people from around that time and before. [1]: https://en.wikipedia.org/wiki/Morgan_Kane

The keyboard part will help quite a bit, at least. Being from Norway, the occasional æ-ø-ås are quite revealing.

EDIT3: 233/114 should be 233/144, and it's off by ~10^{-5}

I apologize for the vagueness. "Hardest to approximate" does require certain definitions first.

The reasoning behind this is simple continued fractions, ie. fractions like a_0 + 1/(a_1 + 1/(a_2 + ...)) = [a_0; a_1, a_2, ...], with a_i\in N. Every irrational number corresponds uniquely to an infinite continued fraction, and the finite "steps" of the SCF are the fractions that best* approximates the irrational numbers. * a/b is "best" at approximating x, if b|x-a/b|<=d|x-c/d| for all c/d\in Q such that d <= b. The name "best" is to distinguish these from the "good" (without the b&d- weights), and I think its also known as "best approximation of of second kind".

The convergents, the finite "steps" of the SCF, are exactly these "best" approximations[1]. Such convergents are include 355/113 for pi, and are used for many things, like pianos and most of the different systems of leap years. Fascinating stuff, really. [1] IIRC, http://www.math.hawaii.edu/~pavel/contfrac.pdf contains a full proof.

The size of the a_is determines when there's going to be a jump in denominator size. The 355/113 approx of pi is right before an 292, which is fairly large, and the next convergent is 103993/33102. Phi, being [1;1,1,...], never reaches any such jump in denominator size, and its sequence of convergents (its best approximations, and for phi it's the ratio of fibs) converge slower than any irrational not having a trail of ones at the end. From this, one may consider it the number "least like a rational", or even "the most irrational number".

Its properties are not directly related to the square root of five, as far as I can tell, but it is in this way the uniquely (at least as an infinite tail of a SCF) hardest irrational to approximate.

That being said, my initial comment was intended to point out that phi and the fibonacci numbers is quite special, and its special enough that it "should" occur frequently in nature. I never actually meant to comment too deep on the spiral-parts, because I know fairly little of them. My "reduce periodicity"-argument is only based on the thought that pi with its fourth convergent 355/113 would almost have a period of 113 (off by ~10^{-7}), while phi with its 11th-ish convergent 233/114 would have a not-very-almost-period 114 (off by 0.5). Phi's ~10^{-7}-almost-period would be 1597 from its 16th-ish convergent. While one could just take any number, say 123012/153281=[0; 1, 4, 15, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 6] and claim that has a longer period, if I calculated correctly, it has a ~10^{-7}-almost-period of 871 from its seventh convergent 699/871, which is fairly less than phis. Note: one should probably even multiply the error 10^{-7} with the period for a more correct result, but as I hinted to, this is not my strongest subject.

Now, to your example: 1.62 has SCF [1; 1, 1, 1, 1, 1, 2, 2], all quite low, so it should have a fair amount of this "irrationality" that the fibs-ratios have. It doesn't have to be the fibonacci-ratios exactly (though these would be the best choice), but most* numbers trying to optimize on this property will be close to them. * I won't say all, because [x,y,1,1,1,1...] could possibly inherit some properties, but the number itself, 1/(x+1/(y+1/phi)), could be far from phi.

EDIT: I only realized now that your plant was hypothetical. Anyway, a quick search yielded this https://www.mathsisfun.com/numbers/nature-golden-ratio-fibon... which (if you ignore the "for-kids" language) has fairly good display for one of the properties, and actually mentions the continued fraction. EDIT2: For a more serious article, see https://plus.maths.org/content/chaos-numberland-secret-life-... which, all the way down at the bottom, explains that the numbers of SCF ending in [1,1,...] are "noble", and occur frequently as they "are least susceptible to being perturbed into chaotic instability."

Good point. The reason why few to no plants are not black is a very interesting problem, which I haven't heard an answer to.

Still, the number phi has very unique properties, considering its SCF. And the sequence of fibonacci ratios is not an arbitrary sequence converging to it. Whether anything has evolved to utilize this or not, I can not say.

I only meant it as a possible explanation for why it wouldn't be unreasonable to think phi appears often in nature. I don't really have enough knowledge of nature to say if it's the case.

Distribute the seeds of a sunflower radially, placing one seed every 360*x degrees, gradually increasing radius. If x=a/b, after placing b seeds, you will be back to the initial position and the next b seeds will be (radially) shaded from the first b seeds (which I admit, might not be how shading works in practice). If x is irrational, but closely approximated by a/b, the seeds won't line up perfectly, but still enough to shade quite a bit. If x is phi, then they will shade as little as theoretically possible (I think.. This is by no means a proof, just some thoughts). IIRC, the "sunflower seed pattern" is actually achieved only if you simulate such a seed placing with x close to phi.

I don't know about limbs, but it could explain sunflower seeds and branches of trees. If they are distributed with a period, every n'th time around they will shade for other seeds (or branches). Approximating \phi is then a good way to reduce such shading.

The fibonacci spirals _are_ something special. The simple continued fraction of the golden mean is [1;1,1,...], and its convergents (the sequence of fractions that best approximates it) is ratios of fibonacci numbers. In short, due to the 1,1,1,... nature of the golden mean's SCF, it is _the_ number that is hardest to approximate by any rational number. This is why anything that evolves to reduce "periodicity", they will try to approximate this number, and it is best approximated by ratios of fibonacci.

[white] as a catchall is a bit misleading. Perhaps a different border color or an asterisk inside the square to clearly mark that it's not an actual white block?

Amazing game!

On the "Fort" level, the map [blue][white]->[blue] removes the [blue] block, rather than the [white]. This is a different behavior than the map [red][red]->[blue], which converts two [red] into one [blue].

Can't this be partially solved by just throwing multiple dices? E.g two 6-sided dices would then have a higher chance of 7 than 12.

It doesn't change the sleep patterns, but would you say it does not make a difference in judging at which hours other people sleep?

I find this a really good argument for it: how are you supposed to know if you call somebody in the middle of the night, if it's noon where you are, and the hour is the same? By having different time-zones, you need to look it up, but then you have an intuitive way of judging which stage of the day they are perceiving.

I read logicalee's comment as a reference to the way most (all?) branches of mathematics may be built using sets and their axioms. Set theory boils down to "in the set" and "not in the set", ie. black and white.

For your first suggestion, I dusted off some old code and modified it slightly:

http://pastebin.com/qV8fU1a4

It should keep the layout as is, while only adding color, but I have only tested it on wikipedia. The color period is wrong, but it will show you some approximation of how sites will look with a non-interrupting beeline bookmarklet.

In Norway, we have BankID as well (works well on mac/chrome, though), but we also have a choice to use non-java MinID ("MyID") for logging in to do taxes/healthcare/education/etc. Signing electronically is still only BankID and java, as far as I know.

From a convenience-viewpoint, I think it works quite well, but I don't know how secure it is if an adversary is determined enough. It's basically (birth number) + (personal password) + (either a key sent to your phone or a randomly chosen PIN from a paper you've been sent through regular mail)