Calendar Year 2001 County Highpointing Coincidences
For those readers unfamiliar with the Fibonnacci series, a brief
description is given here so as to more fully appreciate the
remarkable numerical coincidences described in this note.
The Fibonnacci series is a sequence of integers that are generated by recursion, i.e. successive
members are determined from preceeding members by a simple rule. We begin by assigning the integers
0 and 1 as the first two elements. The next element is obtained as the sum of the last two elements,
i.e. 0 + 1 = 1 is the third element. By recursion of this rule one obtains as successive elements
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
and so forth. The elements of this series are given here up to 1000 (beginning with 1 and 2) -
{1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987}
Note how the series elements rapidly increase in magnitude.
It may be proven algebraically that the ratio of the n-th element to the previous element (n-1) rapidly
converges to the famous golden ratio of approximately 1.618 and is given exactly by the irrational
expression [1 + sqrt(5)] / 2). The golden ratio occurs in nature in unexpected and remarkable ways.
One example is the ratio of the number of whorls of a pineapple going clockwise along a given radial
from the center, to the number of whorls going counterclockwise.
Greek mathematicians appreciated this ratio as special -
and believed that a rectangle constructed with a length to width ratio of 1.618 was particularly beautiful.
Indeed, the Parthenon was constructed based on the golden ratio.
The Fibonnacci series in Calendar 2001 County Highpointing
In 2000 I set out to eventually complete the states of California, Arizona and Nevada seeing as this
tri-state area seems to "naturally" form a pleasing set of contiguous states. At the time, mid-December
of 2000, my overall county count was 55 - including counties outside of this tri-state region.
Note how 55 is a Fibonnacci number.
It is now exactly one year later, that is, mid-December 2001. During that one year span I managed to
accumulate 34 additional counties - all of which are in the tri-state region AZ-CA-NV.
Note how 34 is a Fibonnacci number. Thereby my overall county count at this time stands at 89. Since
this value must necessarily equal the sum of my previous count (55) plus additional counties (34),
and since these two values (55, 34) are both Fibonnacci numbers, it must be that 89 as well is a
Fibonnacci number since the series is generated by summing the two previous numbers.
Of the 34 counties collected in this twelve month span of time since mid-December 2000,
exactly 21 counties are in California. Note how 21 is a Fibonnacci number.
Thereby the number of counties collected outside of California must necessarily also be a Fibonnacci number,
i.e. as the difference between 34 and 21. Parenthetically I note that of these 13 extra-California counties,
ten are in Arizona (so completing that state), while 3 are in Nevada.
Now consider how many counties remain in the tri-state area for next summer. I have 45 of 58 counties in
California - so making 13 counties to complete that state.
Note how 13 is a Fibonnacci number.
I have 9 of 17 counties in Nevada - so making 8 counties to complete that state.
Note how 8 is a Fibonnacci number.
The sum, as 13 + 8 = 21, is the number of remaining counties (recall that Arizona has been completed).
As the sum of two successive Fibonnacci numbers, 21 necessarily must be a Fibonnacci number.
It gets more bizarre.
My friend Edward Earl began 2001 with a total of 34 California counties.
Note how 34 is a Fibonnacci number.
In addition to collecting counties elsewhere, Edward increased his California count by 21 during calendar 2001.
Note how 21 is a Fibonnacci number.
The sum, as 34 + 21 = 55, is the number of counties Edward has completed within California at this time.
As the sum of two successive Fibonnacci numbers, 55 (just three shy of a state completion!) is
necessarily a Fibonnacci number
Last Sunday the 16th of December 2001, Edward Earl and myself trivially hiked up Carpenter Hill*
(the highpoint of Sacramento County, California).
As noted previously my county count thereby increased to 89 - a Fibonnacci number. On that day
Edward's overall county count increased to a nice even 144 - again, a Fibonnacci number.
Finally, I began 2001 with a total of 122 ascents of all types. Since then I have
added 89 new summits (everything from trivial walkups to
Nevado Illimani in Bolivia (21,201 feet)).
Note how the 89 new summits is a Fibonnacci number.
Addendum
On Friday October 20, 2006 (i.e. "10/20") the Dow Jones Industrial Average (DJIA)
closed for the first time ever above 12,000 - in fact, at 12,002 - a most definite
"Illimani number" with nothing but the digits "0", "1", or "2".
The next trading session, Monday, had an opening value of 12,001.
* Not completely trivial: I climbed Carpenter Hill backwards!