The golden ratio is 1.618 to 1 or any two Fibonacci numbers next to each other.

I would just like to underscore that Fibonacci numbers has very little to do with it (see the video on page 1). Start with any two positive integers that you like, the series will NOT be the Fibonacci series, but the ratio of any two consecutive numbers in the series will converge on the 1.618...

Sample:

Lets start the series with two numbers: 12345 and 57890. Each new number is the sum of the previous two, and we'll calculate the ratio of each pair on the series:

n1=12345, n2=67890, and gr=5.49939246658566
n1=67890, n2=80235, and gr=1.18183826778612
n1=80235, n2=148125, and gr=1.84613946532062
n1=148125, n2=228360, and gr=1.54167088607595
n1=228360, n2=376485, and gr=1.64864687335786
n1=376485, n2=604845, and gr=1.60655803020041
n1=604845, n2=981330, and gr=1.62244872653324
n1=981330, n2=1586175, and gr=1.61635229739231
n1=1586175, n2=2567505, and gr=1.61867700600501
n1=2567505, n2=4153680, and gr=1.61778847558233
n1=4153680, n2=6721185, and gr=1.61812778066678
n1=6721185, n2=10874865, and gr=1.61799816550207
n1=10874865, n2=17596050, and gr=1.61804767231593
n1=17596050, n2=28470915, and gr=1.61802876213696
n1=28470915, n2=46066965, and gr=1.61803598514484
n1=46066965, n2=74537880, and gr=1.61803322619582
n1=74537880, n2=120604845, and gr=1.61803428001977
n1=120604845, n2=195142725, and gr=1.61803387749472
n1=195142725, n2=315747570, and gr=1.61803403124559
n1=315747570, n2=510890295, and gr=1.61803397251798
n1=510890295, n2=826637865, and gr=1.61803399494993
n1=826637865, n2=1337528160, and gr=1.61803398638169
n1=1337528160, n2=2164166025, and gr=1.61803398965447
n1=2164166025, n2=3501694185, and gr=1.61803398840438
n1=3501694185, n2=5665860210, and gr=1.61803398888187
n1=5665860210, n2=9167554395, and gr=1.61803398869948
n1=9167554395, n2=14833414605, and gr=1.61803398876915
n1=14833414605, n2=24000969000, and gr=1.61803398874254
n1=24000969000, n2=38834383605, and gr=1.6180339887527
n1=38834383605, n2=62835352605, and gr=1.61803398874882
n1=62835352605, n2=101669736210, and gr=1.6180339887503
n1=101669736210, n2=164505088815, and gr=1.61803398874974
n1=164505088815, n2=266174825025, and gr=1.61803398874995
n1=266174825025, n2=430679913840, and gr=1.61803398874987
n1=430679913840, n2=696854738865, and gr=1.6180339887499
n1=696854738865, n2=1127534652705, and gr=1.61803398874989
n1=1127534652705, n2=1824389391570, and gr=1.6180339887499
n1=1824389391570, n2=2951924044275, and gr=1.61803398874989
n1=2951924044275, n2=4776313435845, and gr=1.6180339887499
n1=4776313435845, n2=7728237480120, and gr=1.61803398874989
n1=7728237480120, n2=12504550915965, and gr=1.61803398874989
n1=12504550915965, n2=20232788396085, and gr=1.61803398874989

The numbers above (n1, n2) are NOT Fibonacci numbers. It's just a property of the series (the next number is the sum of the previous two).

Another poster mentioned that if you have a piece of paper in the 1 : 1.618 proportion and then fold it in half, you get the same proportion. Yep. If you look at the video on p1, the "golden ratio" is exactly this:

[1 + sqrt(5)] / 2

Now, imagine a piece of paper in this exact ratio. If you half the largest side, and then take the ratio again, taking the reciprocal of the fraction (because the previous shorter end is now the longer end), you end up with:

**4/ [1 + sqrt(5)]**

or 1.2360679774997896964091736687313

To check this in practice, I modified the above test to take the "halfing ratio", and here is what you come out with:

n1=12345, n2=67890, and gr=5.49939246658566, half ratio: 0.363676535572249.
n1=67890, n2=80235, and gr=1.18183826778612, half ratio: 1.69227893064124.
n1=80235, n2=148125, and gr=1.84613946532062, half ratio: 1.0833417721519.
n1=148125, n2=228360, and gr=1.54167088607595, half ratio: 1.29729374671571.
n1=228360, n2=376485, and gr=1.64864687335786, half ratio: 1.21311606040081.
n1=376485, n2=604845, and gr=1.60655803020041, half ratio: 1.24489745306649.
n1=604845, n2=981330, and gr=1.62244872653324, half ratio: 1.23270459478463.
n1=981330, n2=1586175, and gr=1.61635229739231, half ratio: 1.23735401201002.
n1=1586175, n2=2567505, and gr=1.61867700600501, half ratio: 1.23557695116465.
n1=2567505, n2=4153680, and gr=1.61778847558233, half ratio: 1.23625556133356.
n1=4153680, n2=6721185, and gr=1.61812778066678, half ratio: 1.23599633100413.
n1=6721185, n2=10874865, and gr=1.61799816550207, half ratio: 1.23609534463186.
n1=10874865, n2=17596050, and gr=1.61804767231593, half ratio: 1.23605752427391.
n1=17596050, n2=28470915, and gr=1.61802876213696, half ratio: 1.23607197028968.
n1=28470915, n2=46066965, and gr=1.61803598514484, half ratio: 1.23606645239164.
n1=46066965, n2=74537880, and gr=1.61803322619582, half ratio: 1.23606856003954.
n1=74537880, n2=120604845, and gr=1.61803428001977, half ratio: 1.23606775498944.
n1=120604845, n2=195142725, and gr=1.61803387749472, half ratio: 1.23606806249118.
n1=195142725, n2=315747570, and gr=1.61803403124559, half ratio: 1.23606794503597.
n1=315747570, n2=510890295, and gr=1.61803397251798, half ratio: 1.23606798989987.
n1=510890295, n2=826637865, and gr=1.61803399494993, half ratio: 1.23606797276338.
n1=826637865, n2=1337528160, and gr=1.61803398638169, half ratio: 1.23606797930894.
n1=1337528160, n2=2164166025, and gr=1.61803398965447, half ratio: 1.23606797680876.
n1=2164166025, n2=3501694185, and gr=1.61803398840438, half ratio: 1.23606797776374.
n1=3501694185, n2=5665860210, and gr=1.61803398888187, half ratio: 1.23606797739897.
n1=5665860210, n2=9167554395, and gr=1.61803398869948, half ratio: 1.2360679775383.
n1=9167554395, n2=14833414605, and gr=1.61803398876915, half ratio: 1.23606797748508.
n1=14833414605, n2=24000969000, and gr=1.61803398874254, half ratio: 1.23606797750541.
n1=24000969000, n2=38834383605, and gr=1.6180339887527, half ratio: 1.23606797749764.
n1=38834383605, n2=62835352605, and gr=1.61803398874882, half ratio: 1.23606797750061.
n1=62835352605, n2=101669736210, and gr=1.6180339887503, half ratio: 1.23606797749948.
n1=101669736210, n2=164505088815, and gr=1.61803398874974, half ratio: 1.23606797749991.
n1=164505088815, n2=266174825025, and gr=1.61803398874995, half ratio: 1.23606797749974.
n1=266174825025, n2=430679913840, and gr=1.61803398874987, half ratio: 1.23606797749981.
n1=430679913840, n2=696854738865, and gr=1.6180339887499, half ratio: 1.23606797749978.
n1=696854738865, n2=1127534652705, and gr=1.61803398874989, half ratio: 1.23606797749979.
n1=1127534652705, n2=1824389391570, and gr=1.6180339887499, half ratio: 1.23606797749979.
n1=1824389391570, n2=2951924044275, and gr=1.61803398874989, half ratio: 1.23606797749979.
n1=2951924044275, n2=4776313435845, and gr=1.6180339887499, half ratio: 1.23606797749979.
n1=4776313435845, n2=7728237480120, and gr=1.61803398874989, half ratio: 1.23606797749979.
n1=7728237480120, n2=12504550915965, and gr=1.61803398874989, half ratio: 1.23606797749979.
n1=12504550915965, n2=20232788396085, and gr=1.61803398874989, half ratio: 1.23606797749979.

In summary, if a piece of paper is in the golden ratio, and you half it, then it is no longer in the golden ratio. Sorry.

I still don't see what the big deal is with this ratio. You are just noticing a pattern in mathematics. That is like saying: "I think it is amazing that a flower grew in a field of flowers."