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 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.
"it is _the_ number that is hardest to approximate by any rational number"
I'm not sure what you mean by this. Any irrational number can be approximated to an infinite number of different levels of precision by an infinite number of different rational numbers. And rational approximations to phi can be trivially generated by any Lucas sequence, starting from an infinite number of different possible seeds (not just the '1, 1' seeds of Fibonacci).
Approximating e, pi and the square root of 2 by a rational number is equally 'difficult'. Phi is exactly (1 + 5^1/2)/2 - or, to put it another way, a half, plus half the square root of five. Are you saying that the square root of five is 'uniquely' hard to approximate with a rational number?
If 'anything that evolves to reproduce periodicity' tries to approximate this ('hardest to approximate') number, then you surely have many specific examples of places in nature where close approximations to phi can be reliably found.
And that doesn't mean 'spirals that sort of look a bit fibonacci-ish even though the center in no way divides its diameters in the golden ratio'. That means, like, you can point to a plant and say 'the ratio of successive buds on the stem of Fooii Bariensis are always in a ratio of precisely 1.62'.
But then to further privilege Fibonacci, not just the golden ratio, you'd have to further show that that 1.62 ratio wasn't just a real approximation of the golden ratio but is actually 1.6181818..., a rational derived from the specific 89/55 approximation to phi produced by the Fibonacci sequence. And then show a mechanism whereby the plant actually uses the fibonacci recursion in its growth patterns somewhere to generate this precise ratio rather than some other ratio.
And then you'd need to find several such examples to back up your claim that this kind of pattern is a common attractor in evolutionary space.
It's just not there, sorry. There's just no reason for growth patterns to favor phi, or Fibonacci numbers.
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."
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.
That argument is too hand wavy to be convincing. Can you make it precise? How exactly are the seeds or branches being distributed in 2d/3d space, and how does that minimize shading? Why would sunflower seeds want to minimize shading in the first place?
There's also the trouble that sunflower seeds and branches of trees do not actually approximate phi in any meaningful way. Check out this picture: http://www.wingsdailynews.com/wp-content/uploads/2015/04/fib... No doubt it was cherry picked and the spiral placed in the best position, yet the match is abysmal (look at how far the center of the spiral is from the center of the flower). The seeds are just packed tightly together and this produces some patterns due to the seeds on the outside being more developed than those on the inside.
They actually do have a relationship to Phi, but not the one in the in picture you link, which is obviously bad.
If you draw a pattern of dots according by rotating points 137.5 degrees (which you see on a lot of seed heads and fruits, such as sunflowers, pineapples, pine cones, romanesco broccoli, various succulents, etc.), you create a pattern were certain spirals 'jump out' at you. If you count the number of 'arms' in each successive set of spirals, the numbers are the Fibonacci sequence. http://momath.org/home/fibonacci-numbers-of-sunflower-seed-s...
Somewhere along the line I read on some website that it might have to do with optimal packing theory -- distributing the maximum number of seeds over the seedhead, but I think it was just a guess. It does seem to show up in a variety of plants, more than just chance would lead you to expect.
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.
This is trying to find meaning where there is none. I don't think shading of seeds has any impact on the evolutionary fitness, even if we assume that light is coming in radially which is of course not true at all. Furthermore, even if we make the two (clearly incorrect) assumptions that sunflowers do care about shading of the seeds and that light does come in radially, that does not even constitute a convincing argument that the seeds grow in that pattern. Here is a far more convincing argument along the same lines. Plants care about getting energy. They get energy by absorbing sunlight. Black absorbs the most light. Ergo, plants are black.
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.
Thing is, Phi's not really all that interesting. It's just a simple root of a quadratic, no more 'mystical' than the square root of five. It doesn't 'emerge' from arithmetic the way e or pi do. It's a fixed point of the sequence of operations: 'take a number; invert it; add one; repeat'... that's, sort of interesting, but 'add one' isn't a very special operation - why not add 12? or add pi? or add phi?
Take a number, invert it, add two, repeat... eventually you get root 2 + 1. And the inverse of that is root 2 - 1! That's pretty magical! Kind of more magical than 'half plus root 5 over 2', anyway. Maybe root 2 + 1 is the platinum ratio!
[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?
Now that I've seen the update, I think just make it transparent, so it is the color of the button on which it is placed. The dashed border is nice though.
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].
IMO the problem can be best explained with a classroom experiment, almost like a magic trick. It is a built in human intuition which is at odds with the reality of randomness.
Split your class into 2. Give half A the players dice and a piece of paper. Give other half B just a piece of paper. Half B plays with imaginary dice, Half A with real dice. Everyone records 20 games of a piece of paper. You hand in the papers and the demonstrator (or maybe a new one that hasn't been in the class, for extra theatrics) looks at the papers. He will accurately place them into separate piles.
The trick is that fake randomness and real randomness are different enough that they are distinguishable with a high level of accuracy. Real dice will roll the same number in a row far more often than fake dice. The more games you play, the more accurately they can be distinguished.
You can look at the formulas to figure out the odds of rolling the same number 4 times in 100 rolls, but it doesn't drive home the message that your intuition is broken until you see intuition being beaten consistently and reliably by a better tool. It's fun too.
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.
No, time-zones alone can't give you an estimate of sleep-times (even ignoring individual sleep patterns for the moment).
Time-zones are related to the longitude of the location You also need to account for the latitude, and the time of the year, because the pattern of day and night depends on all three factors.
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.
Quite interestingly, the axioms mathematics is build upon do not let you decide every question that you might come up with. As such in mathematics not everything is black or white. For example, the continuum hypothesis is independent of the most commonly used axiom system ZFC. To explain what that means imagine ZFC would be describing an apple. Out of the definition of "apple" you can get theorems that tell you "if you start on any point of the apple and dig down in a straight line, at some future point in time you will reach the 'other side' of the apple" or some statements you can disprove "if you walk on the surface of the apple in a straight you will never come back to point where you started". What you will not get is the colour of the apple. Everything you know about the apple is consistent with it being green. But all that is also consistent with it being red. The colour of the apple is independent of everything that you are interested in in an apple. Therefore there is no reasonable answer to the question "which colour does the apple have", except maybe: It should not matter. The same is true of the continuum hypothesis. I remember reading that if you need the CH to prove something, then you should definitely reconsider the statement you are trying to prove :).
> I remember reading that if you need the CH to prove something, then you should definitely reconsider the statement you are trying to prove
This is not entirely true. There are many proofs that begin by assuming CH. There are also two ways of interpreting "need CH to prove". One way is that the thing you are trying to prove is equivalent to CH, which is an interesting/useful result. Another way to interpret it is that you can not come up with another way of proving it. In the latter case, the CH based proof justifies that your proposition is not inconsistent, and may even lead you (or others) to proof that it is true regardless of CH. If I recall correctly some statements have been proved by proving the statement when CH is true, and also when CH is false.
Ah but the real story is that like sophisticated math systems, in relation to any fixed axiomization, set theory involves provably, provably false and unprovable/independent statements. And in relation to any fixed model, set theory involves true and provable, true but unprovable, false but un-disprovable and false and disprovable.
And when you're math as a human endeavor, you can also add "provable but not yet proven" and "proven independent"
So, learn some stuff, see how far from black and white higher math can be.
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)
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