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The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.[7][12] In the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long (L) syllables that are 2 units of duration, and short (S) syllables that are 1 unit of duration; counting the different patterns of L and S of a given duration results in the Fibonacci numbers: the number of patterns that are m short syllables long is the Fibonacci number Fm + 1.[8]
Susantha Goonatilake writes that the development of the Fibonacci sequence "is attributed in part to Pingala (200 BC), later being associated with Virahanka (c. 700 AD), Gopāla (c. 1135), and Hemachandra (c. 1150)".[6] Parmanand Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and cites scholars who interpret it in context as saying that the cases for m beats (Fm+1) is obtained by adding a [S] to Fm cases and [L] to the Fm−1 cases. He dates Pingala before 450 BC.[13]
However, the clearest exposition of the series arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):Variations of two earlier meters [is the variation]... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]... In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations].[14]The series is also discussed by Gopala (before 1135 AD) and by the Jain scholar Hemachandra (c. 1150).
Outside of India, the Fibonacci sequence first appears in the book Liber Abaci (1202) by Leonardo of Pisa, known as Fibonacci.[5] Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?
At the end of the first month, they mate, but there is still only 1 pair.
At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month (n − 1). This is the nth Fibonacci number.[15]
The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.[16]
originally posted by: christophoros
Everything is half truth because it's in someone else's perspective
“Sharing is the essence of teaching. It is, I have come to believe, the essence of civilization . . . Without it, the imagination is but the echo of the self...
originally posted by: TheConstruKctionofLight
a reply to: InTheLight
“Sharing is the essence of teaching. It is, I have come to believe, the essence of civilization . . . Without it, the imagination is but the echo of the self...
Of course, and a timely reminder for this thread. We are all god...finding our way home.
I dont like the use of the word civil as defn: "relating to ordinary citizens and their concerns, as distinct from military or ecclesiastical matters."
A citizen is a legal construct, as the OP mentioned in the OP its a way to keep humans in prison/matrix.
Civilization again defn: "A civilization (US) or civilisation (UK) is any complex society characterized by urban development, social stratification, symbolic communication forms (typically, writing systems), and a perceived separation from and domination over the natural environment"
I prefer the descriptor humanity over civilization.
Sounds like your admission that you do not know what it IS - because clearly the pen has an actual existence beyond any of our points-of-view of and about it.
This result came over thousands of years of thinking, from Archimedes to Newton. Let's look at the analogies behind it.
We all live in a shiny continuum
Infinity is a constant source of paradoxes ("headaches"):
A line is made up of points? Sure.
So there's an infinite number of points on a line? Yep.
How do you cross a room when there's an infinite number of points to visit? (Gee, thanks Zeno).
And yet, we move. My intuition is to fight infinity with infinity. Sure, there's infinity points between 0 and 1. But I move two infinities of points per second (somehow!) and I cross the gap in half a second.
Distance has infinite points, motion is possible, therefore motion is in terms of "infinities of points per second".
Instead of thinking of differences ("How far to the next point?") we can compare rates ("How fast are you moving through this continuum?").
It's strange, but you can see 10/5 as "I need to travel 10 'infinities' in 5 segments of time. To do this, I travel 2 'infinities' for each unit of time".
Analogy: See division as a rate of motion through a continuum of points
What's after zero?
Another brain-buster: What number comes after zero? .01? .0001?
Hrm. Anything you can name, I can name smaller (I'll just halve your number... nyah!).
Even though we can't calculate the number after zero, it must be there, right? Like demons of yore, it's the "number that cannot be written, lest ye be smitten".
Call the gap to the next number "dx". I don't know exactly how big it is, but it's there!
Analogy: dx is a "jump" to the next number in the continuum.
a reply to: Itisnowagain
originally posted by: artistpoet
Nothing ... no Earth or Sun or other stars ... no love ... no movement ... without life ... not even dead ... just nothing
No soul ... no thoughts, ideas ... nothing to share ... nothing to give or receive ... just nothing
How does such a belief help or uplift others ... there is much more to Reality than just nothing
You responded
"Movement is happening is it not?"
I responded
Give me your honest response here...Imagine your child is dying of cancer and undergoing chemotherapy...the child asks you why they are suffering.
You would really say to them "MOVEMENT IS HAPPENING IS IT NOT"? But I guess your responses will be
"what is there to imagine?"
Everything is an expression of emptiness
originally posted by: earthling42
a reply to: bb23108
A long time ago i've written about this matter, we are all filled with words and descriptions which limit us and divide us from what is.
If we see a tree, we'll think "that is a tree" an oak or chestnut, a willow, and that's it, we'll move on because we know what it is. (we have the word and a description of it)
In this state of being, the magic is lost, it is cold and solid, ice is not seen for what it really is.
If we see that the word is not the object, the description is not the described, than what do we really know? the answer is that we do not really know anything, we are simply filled with secondhand knowledge and explanations.
If we can acknowledge this fact and empty our consciousness of its content, life becomes magical, a walk in the divine.
If i describe the sun, from a memory, or during an actual moment in the sun, it is simply a description, solidified music and not the actual.
One can read it and be thinking that he knows what the sun is, but this description which is knowledge describes something which is not actual anymore, therefore it is dead.
To actually see and feel the sun, the warmth, the brightness, the energy, that is being alive.
I take it that this is what you try to convey.
For me, i like to distinct between manmade tools such as a pen or a car, and so on, and that which is natural.
originally posted by: TheConstruKctionofLight
A pen is an approximation (think calculus and limits) of a pen. How can its existence be captured?
If we can acknowledge this fact and empty our consciousness of its content, life becomes magical, a walk in the divine.
Without civility would we not dehumanize others?
It cannot be. That was my point - we can never know it as it actually is in reality. Every point of view about it is incomplete
This feeling of separation is what reinforces the sense of "me", but the heart already understands that such separation is an illusion, that union or unity is already the case. But this endlessly thinking brain-mind cannot control this train of thoughts
originally posted by: TheConstruKctionofLight
I have many problems trying to see any understanding with Itisnowagain assistance in these discussions. Just when I think I can grasp the beauty of his words...it eludes me.
originally posted by: TheConstruKctionofLight
a reply to: Itisnowagain
Hello Itisnowagain, would you be so kind as to respond to my question to you afew pages back?
But this is the issue - IMAGINATION. Absolutely anything can happen in imagination but what is really happening is just happening.
The mind is constantly making horror movies or fairy stories. The mind imagines time and space where it can be the hero and hopefully not the villain. The stories centre around what I will say or do in a situation that is not happening.
The mind imagines that there are two things making up reality - but reality is what is appearing to happen - reality is not two.
originally posted by: TheConstruKctionofLight
Give me your honest response here...Imagine your child is dying of cancer and undergoing chemotherapy...the child asks you why they are suffering.
What would your honest response be to a child that asks why it is suffering?
originally posted by: TheConstruKctionofLight
a reply to: Itisnowagain
Everything is an expression of emptiness
I like this.
I like this better....Everything and/or nothing is an expression and/or concealment of emptiness that in the fullness of time becomes IS-ness.
Ouspensky has a cautionary perception in this regard. The communal life of ants and bees was of great interest to him.(3) "The wonderful completeness," he said, of their organization filled him with wonder and aversion because of the "invariably cold reasoning which dominates their life and by the absolute impossibility for an individual to escape from the wheel of life of the ant-hill and beehive." He theorized that the model for this organization was created in the remote past and "required reasoning and logical intelligence of great power." However, once created such reasoning and intelligence was no longer needed. This is, the anthill and beehive operate on a level of absolute mechanicality.
www.gurdjiefflegacy.org...
Ouspensky believed that this could have happened in only one way. "If ants and bees, or both, of course at different periods of time," he said, "had been intelligent and evolving beings and then lost their intelligence and their ability to evolve, this could have happened only because their 'intelligence' went against their 'evolution,' in other words, because in thinking that they were helping their evolution they managed to arrest it."
In effect, the ants and bees renounced the opportunity and privilege of evolving—"ceased," he said, "to send forth an evolving current." Nature, then, began a new experiment with them, first altering their size. "Nature possesses the power of changing the size of these living beings without altering anything in them, and can effect this change in one generation, that is, at once, simply by arresting their development at a certain stage. Over time, their capacity to think atrophied in ratio to the mechanicality of their lives and automatic habitual patterns of action reigned. Hence, with ants and bees we see in the anthills and beehives not only an incredible degree of intelligence and calculation in operation—but "the complete absence of intelligence in their activities."
Their mistake, Ouspensky intuits, was that "They must have become convinced that they knew what was good and what was evil, and must have believed that they themselves could act according to their understanding. They renounced the idea of higher knowledge and the inner circle of life and placed their faith in their own knowledge, their own powers and their own understanding of the aims and purposes of existence."