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From: Ken S. Tucker on 11 Sep 2005 04:35

Jim Black wrote:
> Jimmy wrote:
> > What is so special with Phi? What does it mean?
> >
> >
> > In an overwhelming number of plants, a given branch or leaf will grow out of
> > the stem approximately 137.5 degrees around the stem relative to the prior
> > branch. In other words, after a branch grows out of the plant, the plant
> > grows up some amount and then sends out another branch rotated 137.5 degrees
> > relative to the direction that the first branch grew out of.
> >
> >
> > All plants use a constant amount of rotation in this way, although not all
> > plants use 137.5 degrees. However, it is believed that the majority of all
> > plants make use of either the 137.5 degree rotation or a rotation very close
> > to it as the core number in their leaf or branch dispersion, sending out
> > each and every leaf or branch after rotating 137.5 degrees around the stem
> > relative to the prior branch.
> >
> > If we were to multiply the value of 1 over Phi to the second power
> > (0.3819659.) times the total number of degrees in a circle (360), we obtain
> > for a product nothing other than 137.50. degrees. As an alternate way to
> > look at the same idea, if we were to take the value of 1 over Phi
> > (0.6180339) and multiply it by 360, we obtain approximately 222.5 degrees.
> > If we then subtract 222.5 from 360 we again find 137.5 degrees - in other
> > words, the complimentary angle to 1 over Phi is 137.5 degrees, which also
> > happens to be the value of 1 over Phi to the second power times 360.
> >
> > So, if we have followed the described mathematics, it is clear that any
> > plant that employs a 137.5 degree rotation in the dispersion of its leaves
> > or branches is using a Phi value intrinsically in its very form. Considering
> > our discussion in the prior section on Phi in geometry, it is interesting to
> > note the off-balance five-pointed star that can be seen in the leaf pattern
> > of plants that employ a 137.5 rotation scheme when they are viewed from
> > above.
>
> Interesting indeed!
>
> One of John Baez's "This Week's Finds in Mathematical Physics" posts
> included an interesting discussion of the golden ratio:
>
> http://groups.google.com/group/sci.physics.research/msg/34abc79351261b27
>
> One of the properties of Phi most relevant to your question is its
> position as the "most irrational number," i.e., the hardest to
> approximate with fractions. More precisely:

This is a bit complicated, but anyway...

Converting a Black Hole to a Golden Hole.

I came across the Golden Number studying
BH's. I hypothesized the conventional GR equation

g_00 = 1 - 2m /r

would *look* like

g_00 = 1 - 2M*sqrt(g_00) /r

where m=M*sqrt(g_00).

The reason for this conjecture is found in the
red shift. Suppose a photons energy has a fixed
ratio to mass m when emitted by the gravitating
mass as measured by an observer at the surface.
The idea being, the surface observer calculates
the gravitating mass and Energy E, and then measures
the emitted photon to be energy e, and figures the
ratio of Energy/energy to be

ratio = E/e (invariant)

The photon itself is red shifted as it tends to infinity,
yet the ratio is preserved due to it's invariance, ie
it's in units of Energy/energy.

The observer at infinity receives the red shifted
photon and using the invariant ratio determines
the gravitating mass at that location to be

mass = Mass*sqrt(g_00).

Hence the observer at infinity will determine

g_00 = 1-2M*sqrt(g_00) /r = 1-2m /r,

So using the invariant ratio "E/e" every observer
will be able to agree to the value of g_00 at
every "r" using the shifting frequency of photons
as they change in altitude.

Every observer can set 2M/r=1 to determine a
relative *event horizon*. To do this, algebraically
set,

g_00 = X^2 = 1 - X and rearrange to,

X^2 + X -1 =0

Using the "quadratic equation" , X is,

sqrt(5) -1
X = ------------- if I'm not mistaken.
2

This is also known as the Golden Number "G" in
J Baez's notation, where,

G = sqrt(g_00) at the common event horizon

defined by 2M/r =1, for all altitudes "r".

G is invariant for a stationary observer at any "r"
hence a Golden Hole. ((I conditioned "stationary"
for CYA, the effects of motion aren't germain)).

There is a complex *hidden* caveat in my conjecture,
because it assumed gravitational field intensity (from
a Newtonian perspective and system of reference)
reduced in proportion to the red shift, to preserve
the invariant ratio.
Indeed this is how I used the same value of "r"
throughout this article, by allowing the assumption
that r2 = x2 + y2 + z2 is unperturbed by gravitation,
and coordiante light speed variations that are only real
and accountable phenomena in curved spacetime in a
GR field.
In defense, the "r" in g_00= 1-2m/r is usually defined
that way, and is used that way in the conjecture.

Regards
Ken S. Tucker

From: Schoenfeld on 11 Sep 2005 07:28

Jim Black wrote:
[remove]

Baez makes various elementary errors in that page. For starters, G is a
quadratic and has two solutions:

"1/G = G - 1
or after a little algebra,
G^2 = G + 1
so that
G = (1 + sqrt(5))/2 = 1.618033988749894848204586834365... "

Does it burn, dork?

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