From Osher Doctorow
We've already seen that the quantities:
1) 1 + sqrt(2) or 1 + sqrt(3)
2) 1 + sqrt(4)
3) 1 + sqrt(5)
are respectively related to the Pell Numbers, the "deterministic"
Probable Causation/Influence (up to sets/events of probability 0), and
the Fibonacci numbers and Golden Mean/Ratio, which in turn are related
to physics, biology, etc.
Daniel Kral, Ondrej Paigrac or Pangrac, and Jean-Sebastient Sereni and
Riste Skrekovsky (first 3 of Charles U. Czech Republic, 4th of U.
Ljubljana Slovenia), in "Long cycles in Fullerene graphs," arXiv:
0801.3854 v1 [math.CO] 24 Jan 2008, 12 pages, prove that every
balanced Fullerene graph on n vertices has a cycle of length at least
(5/6)n - 2/3, which is (1/6)(5n - 4), extending a previous result of
Jendrol' and Owens in Math. Chem. 18 (1998) pages 83-90 that gave the
lower bound as (4/5)n.
In both cases, the quantities 4 and 5 of (2) and (3) occur.
For those who think that 4 and 5 are just coincidental because (2) and
(3) involve square roots, remember that for fixed n there is a mapping
from a + bi to a + b sqrt(n), and that the magnitude of bi is b, while
the magnitude squared of b sqrt(n), which is b^2 n, has a similarity
to the magnitude squared of the Schrodinger wave function w which is a
probability.
I should remind readers that the Probable Causation Influence (PI)
derivation of 1 +/- sqrt(4) comes from "Quantum Gravity 222.7," Jan.
7, 2008, 11PM, in my thread of that Section, where I set:
4) P ' (A-->B) = 1 + y - x = y iff x = 1
and setting:
5) x = (1 - y)^2 = 1 - 2y + y^2 = 1 from (4)
we get:
6) 2y = y^2, so y = 0 or 2, x = 1
When y = sqrt(n), (6) implies:
7) n = 0 or 4, x = 1, y = sqrt(n) = 0 or 2
The equation (4) says that the second version of PI is "on target" or
that x is deterministic (x = P(A) = 1) up to set/events of probability
0. This is certainly a fundamental special case of PI.
Fullerene, and Buckminster Fullerene, and Geodesic Domes, are
extremely important in chemistry and building engineering but also
have interesting stability and strength properties for physics related
to carbon atoms. I'll try to discuss them more later, but you can
look up various summaries or papers on the internet under those
keywords.
Osher Doctorow
.
|
