Intrinsic mathematics

[againtry]

Forgive me for interrupting your interruption but you have already siad that. In fact you have already said that a number of times. so you need to get on with something original or at least moves forward.

[helios]

However, in many cases the aesthetically pleasing consequence is a consequence of, while still conforming with it's laws, taking parts of the system and returning them in a certain order so as to make the particularly aesthetically pleasing element discernible, which can be described as manipulation.

Sometimes, yes. For example, the usual definition of the Fibonacci sequence is
f(1) = 1
f(2) = 1
f(n) = f(n - 1) + f(n - 2)
However, I much prefer this one:
f(0) = 0
f(1) = 1
f(n) = f(n - 1) + f(n - 2)
because here the value f(2) is deducible, while those of f(0) and f(1) are not.

I don't suppose you know a particular resource you feel is consummate in explaining the intrinsic elements of the golden ratio?

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.

Furthermore, I don't suppose you know other intrinsic mathematics which is of, to our senses, pleasant?

I don't know what you mean by "intrinsic mathematics", but there's many mathematical structures that a lot of people find pleasing. Fractals such as the Mandelbrot set are a usual favorite because they're infinitely complex shapes arising from very simple rules. The Mandelbrot set in particular is really simple in its definition, but its perimeter contains a vast variety of landscapes.

I'm afraid I don't understand what you mean.

Can you prove your original statement, that the sum of consecutive hexagonal numbers are cubic numbers? I.e. if [1, 7, 19, 37, ...] are the hexagonal numbers, that [1, 1+7, 1+7+19, 1+7+19+37, ...] are all cubic numbers?

[againtry]

I don't subscribe to the idea that there's anything special about phi.

Rest of world take note.

The Mandelbrot set in particular is really simple in its definition

Excluding the fact that it resides in the complex plane and that it has an infinite 'perimeter', in fact no perimeter whatsoever.

[helios]

You certainly seem to care enough about my opinion to comment on it.

[Ganado]

Rest of world take note.

Please, do go on. Tell us how important phi is to the world.

[againtry]

We wouldn't be talking about phi if it wasn't important. That's a good enough start.

[againtry]

Perhaps you too, @Gonads, would like to add a constructive contribution to the wonders of phi.

[againtry]

You certainly seem to care enough about my opinion to comment on it.

Not really because they are more uninformed prejudices and dogmas rather than opinions. It is always helps me to remain grounded by seeing how someone else can get matters so wrong so often simply due to to such a narrow focus.

Opinions come from rounded education and experience.

Take @Gonads for instance. The motivation for the lame question she poses is nothing more than Pick-A-Side. An equally narrow endeavor that advances nothing, interests nobody.

@astra sees beauty and wonder, Carol and Gonads see nothing, a monitor at best. A life of no linking that doesn't even compile let alone compute. A life which is best described as a syntax error.

[helios]

Man, if my life was so empty that all I had to do with my free time was hurl childish insults at strangers on the Internet for no reason I'd probably try to contract cancer just to have something going on in my life. Do you enjoy this? Do you jump with glee and think to yourself "ooh, I wonder what Carol wrote this time" every time you see I've replied?
Well, enjoy it. Just know that there's nothing you can say that will bother me. At the most I just find you pitiable.

We wouldn't be talking about phi if it wasn't important. That's a good enough start.

That's so stupid. "Well, if homeopathy doesn't work then how come so many people talk about it?"
We're talking about phi because other people have talked about it. You're not arguing that phi is culturally significant, you're arguing that there's some intrinsic property to it that connects it to aesthetics.

[zapshe]

Also, please prove that the ability to recognize beauty is intrinsic to humans

Beauty is an opinion. An opinion, as far as we know, is something only humans hold - animals to a more abstract level. Beauty isn't so much limited to humans as it is limited by not only cognitive ability (I doubt a fly has an idea of beauty), but also by the functionality of brain/processing unit (does this consciousness even know the concept).

So even if there was intelligent life else where, it's not impossible that they won't have an opinion of beauty. Or if they do, beauty to them might be something completely strange to us.

The argument I read before seemed almost backwards. Againtry was arguing that beauty is a human concept, yet also that Phi is universally accepted.

[againtry]

homeopathy

Really? Another change of direction in yet another failed attempt at saving face.

... there's some intrinsic property to (phi) that connects it to aesthetics.

So there you go, we agree. And that is another contradictory statement you have made. Your dogma is in total collapse, Carol.

No wonder your only defence is to do the bully stuff and wish cancer on me. Guess who's the pitiable one here.

[E8A4Nwbp]

but there's many mathematical structures that a lot of people find pleasing. Fractals such as the Mandelbrot set are a usual favorite

Magnificent. I don't reckon you're aware of a hidden gem?

Can you prove your original statement, that the sum of consecutive hexagonal numbers are cubic numbers? I.e. if [1, 7, 19, 37, ...] are the hexagonal numbers, that [1, 1+7, 1+7+19, 1+7+19+37, ...] are all cubic numbers?

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

If it this was not the case, it is still an occurrence many may find pleasant, and causes me to wonder 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)

[againtry]

@Astra

You are touching on the wonderment and amazement which result from the even more fundamental studies and roots in Number Theory. Some of the greatest contributors to human civilization, philosophy and ... yes, mathematics, have made their eternal mark over thousands of years. Often creating whole new worlds of knowledge and application. All intrinsically beautiful, positive and edifying.

And then we have Carol Helios and Gonads ... :(

Take your pick - but never forget it would be an insult to Gauss and Euler, to name a few to, even contemplate that ugly choice.

[Ganado]

There is a lot of beauty in mathematics. Still waiting to know what is special about phi, as opposed to other ratios or logarithmic spirals.

[againtry]

There is a lot of beauty in mathematics.

Well bless my soul. Another contradictory statement to add to the crumbling pseudo-logic of the self appointed arbiters of intrinsic (or even extrinsic) beauty.

Still waiting to know what is special about phi, as opposed to other ratios or logarithmic spirals.

If your test for something special is based on its opposition to something else then you'll be waiting a long time. But it's just another contradictory straw man. Why stop at 'other ratios and logarithmic spirals' or other word salad. Why not pick e, pi, i, 1 or zero? They are all intrinsically special but neither oppose or support phi. They in fact bare little or no direct relation.

The 'opposed to' special-ness test is lazy nonsense - born of Carol's myopic cynicism.

[helios]

So there you go, we agree. And that is another contradictory statement you have made. Your dogma is in total collapse, Carol.

Your inability to defend your position is truly pathetic and it's patently obvious to anyone who's been following along that you have no idea what you're talking about. By all means, keep going. You're just proving my point with each of your posts.

No wonder your only defence is to do the bully stuff and wish cancer on me.

That's not my defense. It has nothing to do with the argument at hard. That's just to amuse myself.

Guess who's the pitiable one here.

It's you, buddy. You're the moron who thinks he's hot shit.

[againtry]

Oh Carol, you're beautiful when you're angry.
But just look at the words you use - they are always angry. You aren't accustomed to being questioned are you?

[againtry]

That's intrinsically angry and, extrinsically beautiful as a result.

[E8A4Nwbp]

You may have missed the answer to your question helios, while responding to againtry.
In the same post are two other questions, if it's no trouble.

I'm merely saying this to, if this is nothing undesirable, yield the maximum possible amount of answers. These discussions are truly enlightening.

helios wrote:

Can you prove your original statement, that the sum of consecutive hexagonal numbers are cubic numbers? I.e. if [1, 7, 19, 37, ...] are the hexagonal numbers, that [1, 1+7, 1+7+19, 1+7+19+37, ...] are all cubic numbers?

astra wrote:

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

AstraAsAnEnquiry wrote:

I don't reckon you're aware of a hidden gem?

AstraAsAnEnquiryAlso wrote:

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)

[againtry]

@Astra

The proof, which is quite simple, intrinsically beautiful, some might say as much as the graphical presentation, is at:

https://proofwiki.org/wiki/Cube_as_Sum_of_Sequence_of_Centered_Hexagonal_Numbers