Intrinsic mathematics

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I didn't miss it, I just hadn't gotten around to it yet.

I'm not a professional, though perhaps this hexagonal number nth term is proof, as the sum of consecutive numbers would strictly be n^3 if I am not mistaken.
n^3 - (n-1)^3
I'm not sure what you're trying to say here.

I don't reckon you're aware of a hidden gem?
Oh, there's many. If you want to know more I recommend you study mathematics.

what is the level of mental ability demonstrated when someone demonstrates a high level of interest and awe when seeing an example of the given intrinsic mathematics. Is he demonstrating a low mental ability, as in truth it is merely a particular intentional arrangement of already established laws? (Assuming he is not enthralled by the aesthetically pleasing element, rather he is astounded by the fact maths just possessed such intrinsic ideas) (admittedly, that was partially copied)
It's interesting, isn't it? It's not difficult to wow an idiot, but there are things that are so complex that an idiot isn't smart enough to understand enough to be amazed by. So, I guess a person is amazed by things that are slightly more complex than they can understand, and bored by things much less complex.
If your test for something special is based on its opposition to something else then you'll be waiting a long time.
If you have no interest in saying why you think phi is special, or the arguments for why the world thinks phi is special, so be it, I was trying to find out why.

Also, while we're at it, you apparently don't know what "definition" means, because the perimeter of the mandelbrot set is not its definition; the definition is simply the set of points C that make the iteration Z <= Z^2 + C not diverge.
I have a perfect understanding of the term ‘definition’ and what it means. I didn’t define the set as its perimeter. I did however comment on carols reference to its perimeter which due to its fractal nature is infinite in length. That in turn is based on Benoit Mandelbrot 101. cf his introductory comments on rulers and measuring the distance around the coastline of England.

Phi is special for all the historical, aesthetic, architectural, mathematical reasons well known to most people and easily discovered by anybody who doesn’t know who isn’t lazy. Try googling it perhaps? I already know and appreciate its intrinsic beauty and application
I have a perfect understanding of the term ‘definition’ and what it means. I didn’t define the set as its perimeter. I did however comment on carols reference to its perimeter which due to its fractal nature is infinite in length. That in turn is based on Benoit Mandelbrot 101. cf his introductory comments on rulers and measuring the distance around the coastline of England.
Perhaps it's just reading comprehension then. helios wrote that "The Mandelbrot set in particular is really simple in its definition, but its perimeter contains a vast variety of landscapes."
You then brazenly replied
Excluding the fact that it resides in the complex plane and that it has an infinite 'perimeter', in fact no perimeter whatsoever.
You're correct that the definition relies on the complex plane (which isn't that complex, it's just two numbers), but the definition does not rely on the perimeter.

Right, so the context with phi was that the discussion was the aesthetic part of it. The full quote that you originally replied to was:
I don't subscribe to the idea that there's anything special about phi. I've yet to see any convincing arguments for it being applicable in art and having any special aesthetic properties. I've seen far more evidence in favor of 3 being aesthetically pleasing.
https://en.wikipedia.org/wiki/Rule_of_three_(writing)
https://tvtropes.org/pmwiki/pmwiki.php/Main/RuleOfThree
https://en.wikipedia.org/wiki/Rule_of_thirds
This could be cultural, though, not an intrinsic property of the human mind.

To re-iterate on this point, there's nothing that shows that a ratio of phi is more aesthetic to people than other ratios, e.g. 1/3, 3/4.
In addition, a lot of the arguments I hear about the golden ratio involve it supposedly showing up in nature more often than other ratios, e.g. in mollusk shells, but my understanding is that this is confirmation bias/overfitting the data to match pre-existing beliefs, and that the proportions are often significantly off of being the golden ratio.
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Try googling it perhaps?
Why are you posting if you have nothing to say? I would assume Ganado has done that already and hasn't found anything he thought was convincing. Since you apparently did find something that convinced you that there's something special about phi, he probably wanted to know what that was.
The complex plane is 'just two numbers'. How absurdly wrong is that!

but my understanding is that this is confirmation bias/overfitting the data to match pre-existing beliefs,
What pre-existing beliefs would they be? Perhaps at a stretch intrinsic appreciation of beauty. Word salad more likely.

Of course if you mix the spiral mollusc shell observations, among many others related natural phenomena, to the Fibonacci series and its relationship in the limit with phi you end up with a completely chopped and tossed salad like the above Gonadian tosser philosophy complete with that bias and prejudice.
"Phi is beautiful. It just is you guys, trust me. Oh, you don't agree? Well, that's because you're dumb! Neener neener!"

So weak.
Carol is angry again because the hexagonal proof is beautifully simple.
Why would that make me angry?
Because you were hoping Astra couldn’t prove it.
First, I wasn't hoping anything. I want to know if he can. Anyone can find curious sequences that appear to match up. Skill in mathematics is all about proving statements.

Second, just because there's a simple proof doesn't mean Rascake can do it.
Well he can prove it, especially since I posted the proof.

Anyone can find such things? Well you didn’t.

It depends on which branch of mathematics what skills are required. Proving statements is only part of it.
Well he can prove it, especially since I posted the proof.
He can certainly copy the proof. That's not what I asked and I trust that he won't do that.

Anyone can find such things? Well you didn’t.
Obviously I did. Anyone can find such things if they play around with numbers long enough. It doesn't take any skill, it just takes free time.
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helios wrote:
Oh, there's many hidden gems. If you want to know more I recommend you study mathematics.
I do plan to do so at the time of writing, however, I believe this response may be slightly self-contradicting, because a hidden gem is "something extremely outstanding [as we agreed, only to the senses we may have developed due to socialisation] and not many people may know about" and if it is disseminated to several students, it no longer seems as much of a hidden gem. It was auspicious to read "there's many". Fairly delightful indeed, and I understand that it can be difficult to select a single idea from many, hence, if you do wish to share one example, would it help if I mentioned a category, such as, for instance, "number 4"'

I'm not sure what you're trying to say here.
The formula is the nth term for hexagonal numbers. I believe it's format strictly means the sum of the consecutive hexagonal numbers which precede "n" will be the sum of a cube number

So, I guess a person is amazed by things that are slightly more complex than they can understand, and bored by things much less complex.
This theory may even be of utility for social scenarios

I am yet to confirm online what I have been informed, in person, regarding phi, however, paying regards to pi, it is said colliding blocks compute pi. The following explanation is from here: https://codereview.stackexchange.com/questions/211882/simulating-a-two-body-collision-problem-to-find-digits-of-pi

https://imgur.com/a/ISASS4a
The setup is as above. A "small" body of unit mass is at rest on the left and a "large" body on the right is moving towards the left (the initial positions and velocities are irrelevant). Assuming perfectly elastic collisions and that the large body's mass to be the n-th power of 100, where n is a natural number, the total number of collisions is always pi*10^n rounded down.

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Well, most people don't anything about actual mathematics. They think all mathematics is is running arithmetic algorithms.

Here's a couple interesting tidbits I found while looking at old emails with a discrete mathematics classmate:
* For all natural or zero n, ((2^n)+2)! is divisible by 2^(2^n)
* Let f(x) = x^n + x^(n-1) + ... + x + 1. f(x) = 0 if and only if x != 0 and x^(n+1) = 1

EDIT:
The formula is the nth term for hexagonal numbers.
That formula was derived after proving that the sum of consecutive hexagonal numbers is a cubic number. The actual direct formula for hexagonal numbers is 1+3*(n^2+n).

https://codereview.stackexchange.com/questions/211882/simulating-a-two-body-collision-problem-to-find-digits-of-pi

https://imgur.com/a/ISASS4a
The setup is as above. A "small" body of unit mass is at rest on the left and a "large" body on the right is moving towards the left (the initial positions and velocities are irrelevant). Assuming perfectly elastic collisions and that the large body's mass to be the n-th power of 100, where n is a natural number, the total number of collisions is always pi*10^n rounded down.
Weird. You see? The only times phi comes up in surprising ways like that is because someone put it there. Even sqrt(2) has more significance.
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Once again Carol has completely blitzed herself in her foot. The few times phi crops up in comparison with pi just goes to show how special and intrinsically beautiful phi is.

There is no Buffon Needle Problem for phi, But for pi, like the collider, there is. Carol can keep knitting. Ha, ha ha!
I think I've had just about enough of you. You're not interesting enough to warrant seeing your stupidity, so I'm going to bring back that old GreaseMonkey script that let me hide users by name. I hope you had your fun, because this will be my last response to you.
closed account (E8A4Nwbp)
* For all natural or zero n, ((2^n)+2)! is divisible by 2^(2^n)
* Let f(x) = x^n + x^(n-1) + ... + x + 1. f(x) = 0 if and only if x != 0 and x^(n+1) = 1
Magnificently enthralling, thank you.

Weird. You see? The only times phi comes up in surprising ways like that is because someone put it there.
I suppose some initiative may been essential for encountering the occurrences, however, it is these initiatives which make the encountering of reward, often.

Even sqrt(2) has more significance.
In regards to practicality, perhaps not. As there may be copious amounts of phenomena in nature which efficiently take place either due to sqrt(2) or phi. We are simply oblivious, and can't confirm which is of greater significance
I suppose some initiative may been essential for encountering the occurrences, however, it is these initiatives which make the encountering of reward, often.
Nah. That's just a glorified treasure hunt. It's more interesting to find that nature spontaneously constructs mathematical structures. For example that the structure of tetrafluoromethane is a tetrahedron because of the way fluoride atoms repel each other, or how ice forms hexagonal lattices because of the hydrogen bonds.
More relevantly for computer scientists, how DNA and the proteins that process it are analogous to a program tape and the reading head of a Turing machine, and how the cell as a whole is analogous to a von Neumann universal constructor. In a weird form of synchronicity, molecular biology and computer science were being almost at the same time in the early twentieth century.

We are simply oblivious, and can't confirm which is of greater significance
Phi is too weird a number to come up spontaneously. I guarantee sqrt(2) is a root of more polynomials with rational coefficients than phi is.
The turing machine and Carol’s first shawl knitting project are another example of very weird synchronicity.
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