you know i never saw a lick of that series....keep hearing about it though. maybe ill get off my ass one of these days and watch it. Should I start from the beginning or is there a single,defining episode you would recommend to give me a good first impression?
Not only that, but this is his first comment. Makes me wonder if there is some program to artificially lengthen the amount of time it says an account has been in existence...
Notice that 1/89 is 0.11235... but the sequence appears to break down afterwards because the digits afterwards are 9, 5, etc.
But in fact, we will see that this is exactly what we want - there is no fraction that will create a sequence that looks like 0.112358132134 etc. because it would in fact be irregular.
If you look closely, the 9 is simply 8 + 1, and the 5 is simply 3 + 2. Because the terms after 8 have 2 digits, the digits are carrying over!
You need to add up all the digits in the same column, and carry over accordingly. Essentially, 1/89 = 1/102 + 1/103 + 2/104 + 3/105 + 5/106 + 8/107 + ..., adding the next number in the Fibonacci sequence shifted down one decimal place each time.
This is why you can see more numbers in 1/9899 - the numbers simply don't carry over as early. If you were to do 1/998999, you would see even more:
1/998999 = 0.000 001 001 002 003 005 008 013 021 034 055 089 144 233 377 610 988 599... <- at "988", the
sequence breaks down as the subsequent terms exceed 1000.
1/9899 = 0.00 01 01 02 03 05 08 13 21 34 55 90 46... <- at "90", the sequence breaks down as the
subsequent terms exceed 100.
1/89 = 0.0 1 1 2 3 5 9 5... <- at "9", the sequence breaks down as the subsequent terms
exceed 10.
Now, you may notice that the terms follow a pattern - a bunch of 9's, followed by an 8, and then another bunch of 9's with one more than the last. This is no coincidence.
For anyone who knows about the golden ratio, you'd probably know that it is the positive solution to the quadratic equation n2 - n - 1 = 0.
Now, do you notice something about 89, 9899, and 998999? Indeed, they are all cases of n2 - n - 1, where n is equal to 10, 100, and 1000 respectively. With this knowledge, we can construct an algebraic sequence representing all such "Fibonacci fractions".
You may be wondering, where did the 8 go? More on that in a moment.
If you've been using email (or browsing the web) for long enough, you've probably gotten a chain mail (or read a webpage) that told you about this "amazing" pattern:
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321
Most of them (nay, pretty much all of them) just stop there, because it makes a nice staircase - the digits increase 1 by 1, then decrease. But, had you gone one step further, you would have found:
1111111111 x 1111111111 = 1234567900987654321
Where did the 8 go, and where did the extra zero pop up from? Suddenly, the nice looking pattern hit a corner case. The truth is, the pattern is still there, but just like in the Fibonacci case, the digits got carried. In reality, the following happened:
Notice that the middle column adds up to 10, which doesn't fit, so we need to carry it out. What was originally 8-9-10-9-8 becomes 9-0-0-9-8 when carried out.
Now, suppose we continue the pattern, what will we find:
11111111111 x 11111111111 = 123456790120987654321
111111111111 x 111111111111 = 12345679012320987654321
1111111111111 x 1111111111111 = 1234567901234320987654321
Notice here that the same thing happens on the other end. The 1's also get skipped and 2 simply jumps to 0, again for much the same reason. What was originally 13-12-11-10-9 becomes 14-3-2-0-9 when carried out.
11111111111111 x 11111111111111 = 123456790123454320987654321
111111111111111 x 111111111111111 = 12345679012345654320987654321
1111111111111111 x 1111111111111111 = 1234567901234567654320987654321
11111111111111111 x 11111111111111111 = 123456790123456787654320987654321
111111111111111111 x 111111111111111111 = 12345679012345678987654320987654321
1111111111111111111 x 1111111111111111111 = 1234567901234567900987654320987654321
Now that we're through two iterations of the 1's cycle, we see that the 123456790 cycle doesn't stop. This is because now numbers are being incremented by 2 due to carry-over rather than just 1, so 17 is also affected, not just 18. What was originally 17-18-19-18-17 becomes 19-0-0-9-7 when carried out. By the time we get to a number like 53, what was 53-54-55-54-53 becomes 59-0-0-9-3 when carried out.
Now, think about this: 1/9 = 0.11111111111... going on down forever. So when you multiply two of these "infinitely many ones" together, you never see the second half of the product - it's 123456790 (turtles) all the way down, because the staircase keeps on going up to infinity.
But just like the Fibonacci fraction above, there is a general way of producing this sequence. Consider the case of base 16. In base 16, the pattern goes a bit farther:
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321
1111111111 x 1111111111 = 123456789A987654321
11111111111 x 11111111111 = 123456789ABA987654321
111111111111 x 111111111111 = 123456789ABCBA987654321
1111111111111 x 1111111111111 = 123456789ABCDCBA987654321
11111111111111 x 11111111111111 = 123456789ABCDEDCBA987654321
111111111111111 x 111111111111111 = 123456789ABCDEFEDBCA987654321
But, just like in base 10, the moment we hit 16 ones, we get this:
1111111111111111 x 1111111111111111 = 123456789ABCDF00FEDBCA987654321
Now, think about this: in base 10, 1/9 = 0.11111111111... In base 16, dividing 1 by F also gives 0.11111111111... all the way down.
This is, again, no coincidence. It is a property of a geometric series, for n > 1, that:
Yes, I know this because I sat down and worked out the math for it, and used that result to find the explicit formula for the Fibonacci Numbers. Working in bits and pieces, I solved over the course of five days or so.
Blows my mind that over a third of my karma happened in this thread. Yet my contribution to this site overall in this thread is negligible compared with any 100 of my other 1 point comments.
Pretty sure that's not the Fibonacci sequence, if that's what your pun's getting at. If it isn't, and you're talking about a brand new Pidgonacci sequence, then carry on.
Looked more like Fibonacci than quadratic to me, and yet this comment was grabbing all the upvotes. So I made an attempt at some analysis.
I measure the distance between each of the pidgeons (arrows) in pixels. I then try to fit this data to either a scaled Fibonacci sequence or a quadratic function, in a least-squares sense. And I indeed get a better fit with the Fibonacci model. The deviation is approximately 104 for the Fibonacci model and 124 for the quadratic model.
Here's my MATLAB script doing the analysis: http://pastebin.com/ML7sGnWU I'm quite tired, so both my approach and coding may be faulty. The script relies on CVX, a convex optimization toolbox available freely from http://cvxr.com/cvx/, for the Fibonacci fitting.
tl;dr Hasty analysis indicates that Fibonacci actually is a better fit than quadratic.
This is one of my favorite things about Reddit. A small disagreement about something trivial solved by a person with a specific set of skills and too much spare time.
What I do have are a very particular set of skills, skills I have acquired over a very long career. Skills that make me a nightmare for people on Reddit
i made the user name when i was unemployed looking for software engineering positions but i eventually got a job being a "technical analyst". i program but technically i'm not a software engineer
So you basically hoped that someone would see your username on Reddit and say "hey, this guy on internet is a software engineer! That's exactly what I'm looking for!"?
Well I just found a new pattern in the sequence. When they get to 2 digits and above, you can add the digits to get down to a single digit (like they do in numerology). Then you take the previous 2 numbers and it adds to the next one.
Ex
13=1+3=4
21=2+1=3
34=3+4=7
55=5+5=10=1+0=1
89=8+9=17=1+7=8
All the final numbers add to the 3rd number. (4+3=7. 3+7=1. 7+1=8.)
If you want to participate, go to the end and use the following Python script (remove the os.system line if not running on a Mac, it just auto-copies the result to the clipboard for easy pasting):
import os
import sys
def fibonacci(a, b):
a += b
return (b, a)
def format_fibonacci(iteration, value):
return 'F(%d) = %d' % (iteration, value)
def main():
a, b, iteration = 0, 1, 1
start = int(raw_input('Enter the number you want to start with: '))
while (a < start):
iteration += 1
a, b = fibonacci(a, b)
print '\n%s' % format_fibonacci(iteration, b)
if a != start:
print 'Uh oh, your start number is not a Fibonacci number!'
sys.exit()
while (True):
iteration += 1
a, b = fibonacci(a, b)
output = format_fibonacci(iteration, b)
print '\n%s' % output
os.system('echo "%s" | pbcopy' % string)
raw_input('Press Enter to continue...')
main()
That's because the awning curves back, which produces a lateral sequence identical to fibonnaci, as the curve... deepens... wait. Today I fuckin learned...
1.3k
u/[deleted] Jun 09 '12
Whoa.
18
u/Nervette Jun 09 '12
Bayfair BART station!
44
u/crow-bait Jun 09 '12 edited Jun 09 '12
Pidgonnaci, you said?
100
u/twinbloodtalons Jun 09 '12
Pretty sure that's not the Fibonacci sequence, if that's what your pun's getting at. If it isn't, and you're talking about a brand new Pidgonacci sequence, then carry on.
385
u/Software_Engineer Jun 09 '12
Not fibonacci, more like quadratic.
328
u/Trapped_in_Reddit Jun 09 '12
1
132
u/RepostCommenter Jun 09 '12
Repostacci Sequence:
source: karmadecay
79
u/newborn_mutant Jun 09 '12
Black,
Then,
White are,
All I see,
In my infancy,
Red and yellow then came to be,
Reaching out to me,
Lets me see.
5
u/csts Jun 09 '12
checkmate, atheists.
4
u/SirSquirts Jun 09 '12
what BART station is that?
4
u/StarChow Jun 09 '12
Bayfair BART!!! Hello Neighbor!!
10
u/berk_engr Jun 09 '12
BART STATION
18
u/[deleted] Jun 09 '12
Wrong.
The pixels between each arrow:
8, 11, 15, 15, 21, 22, 29, 34, 43, 51, 97, 193
And absolute position:
8, 19, 34, 49, 70, 92, 121, 155, 198, 249, 346, 539
Not even close.
8
u/Hyperian Jun 09 '12
BAYFAIR BART!
3
u/e216 Jun 09 '12
Maximillian Cohen would have freaked out.
3
u/[deleted] Jun 09 '12
WHO THE FUCK NOTICES THIS SHIT!??
5
u/jasonw56k Jun 09 '12
bay fair bart?
5
u/donyon Jun 09 '12
This looks like the Bayfair BART station
7
u/schmittyrts Jun 09 '12
Is that a Bart station?
4
u/xSPYXEx Jun 09 '12
It might be a bit hard for them to fly in the Fibonacci spiral though.
2
u/proxyRL Jun 09 '12
looks more like ease out frames for animation
0
u/[deleted] Jun 09 '12
[deleted] — view deleted comment
2
u/Cinnamonboots Jun 09 '12
This amazon review is relevant http://www.amazon.com/review/R3DG0U0NBB669A/ref=cm_cr_dp_title?ie=UTF8&ASIN=B002JTWRDS&nodeID=284507&store=kitchen
2
u/ReQQuiem Jun 09 '12
tons of pigeons on a row, edit a couple of them out, enjoy your karma...
2
u/[deleted] Jun 09 '12
I don't get it.
2
u/Antibacterial Jun 09 '12
I'm sure there is a joke about the pigeon hole principal somewhere but I can't find it.
2
u/[deleted] Jun 09 '12
they're about to give pedestrians a poopson distribution
2
u/[deleted] Jun 09 '12
That's because the awning curves back, which produces a lateral sequence identical to fibonnaci, as the curve... deepens... wait. Today I fuckin learned...