Appendix D
Another Step into Fibonacci Space
The Fibonacci spiral suggests a view of spatial intersections.
The linear spiral intersects with regions of two dimensional space at geometrical points (that have zero dimension.)
A two-dimensional surface intersects with three-dimensional space as a line and three-dimensional space intersects with five-dimensional space as a surface.
Similarly, the only three-dimensionally-experienced intersection is a three-dimensional spatial intersection with 5 and 8 dimensional space in a three-dimensional geometry.
This 3-dimensional geometry is the possible definition of a “black hole.”
From calculations already shown, the barrier energy E-sub-B for 5-dimensions can be quickly arrived at ~ 10^-6 eV/kg.
In three dimensions, points (and small regions around them) exist with this low E-sub-B only where F-sub-B ~ 0 as at the cancellation points of F = ∑G(m1m2)/r^2 = 0 between large masses or at the centers of mass for very large masses.
The points then move on lines as the cancellation location moves. The lines move on surfaces, and so on.
Similarly, a region in deep enough space to have or have had ~ 0 F-sub-G due to large r^2 and even due to cancellation forces, would also have experienced near zero E-sub-B.
Saturday, March 3, 2012
Thursday, March 1, 2012
New Appendix C
Appendix C
A General Fibonacci Calculation:
The Fibonacci infinite sequence was referenced in Appendix A,
F(n) = F(n-1) + F(n-2) with seed values F(0) = 0 and F(1) = 1.
Ratios converge, and
lim(n infinity) F(n+1) / F(n) = φ = (1 + 5^1/2) / 2 = .618 … and
lim(n infinity) F(n-2) / F(n) = γ = .382 … and so on.
Writing an example expression for spatial dimension ≥ 3 per Appendix A
∫∫∫∫∫∫∫∫dV = ∫∫∫∫∫dV(0) x exp(rV x t)
where rV = r-sub-V = φ^(D(n+1) – D(n))
then
dx /dx(0) = exp(φ ^ (D(n+1) – D(n))^1/(n+1).
Except we are now doing math in another dimension, and while e = 2.718 in three dimensions, the base of natural logarithms should change in higher or lower dimensional space.
For example, in the case of 5 dimensions: e e^1 / γ^1.
We quickly find dx /dx(0) = 1.08.
A different example, for the case of spatial dimension < 3:
The base e must change as a function of the power of B, i.e. in three dimensional space B ~ sec^2 while in two-dimensional space B ~ sec^3/2.
The difference in power of physical events B
2 – 3/2 = 1/2 and the two-dimensional e = 2.718^1/2.
Then we quickly find dx /dx(0) = 1.08 similar to the previous mean calculations for the difference between b(min) and b-empirical.
The (Fibonacci) calculations hold true for any spatial dimension n moving through n+1 with a dimensional adjustment for e.
A General Fibonacci Calculation:
The Fibonacci infinite sequence was referenced in Appendix A,
F(n) = F(n-1) + F(n-2) with seed values F(0) = 0 and F(1) = 1.
Ratios converge, and
lim(n infinity) F(n+1) / F(n) = φ = (1 + 5^1/2) / 2 = .618 … and
lim(n infinity) F(n-2) / F(n) = γ = .382 … and so on.
Writing an example expression for spatial dimension ≥ 3 per Appendix A
∫∫∫∫∫∫∫∫dV = ∫∫∫∫∫dV(0) x exp(rV x t)
where rV = r-sub-V = φ^(D(n+1) – D(n))
then
dx /dx(0) = exp(φ ^ (D(n+1) – D(n))^1/(n+1).
Except we are now doing math in another dimension, and while e = 2.718 in three dimensions, the base of natural logarithms should change in higher or lower dimensional space.
For example, in the case of 5 dimensions: e e^1 / γ^1.
We quickly find dx /dx(0) = 1.08.
A different example, for the case of spatial dimension < 3:
The base e must change as a function of the power of B, i.e. in three dimensional space B ~ sec^2 while in two-dimensional space B ~ sec^3/2.
The difference in power of physical events B
2 – 3/2 = 1/2 and the two-dimensional e = 2.718^1/2.
Then we quickly find dx /dx(0) = 1.08 similar to the previous mean calculations for the difference between b(min) and b-empirical.
The (Fibonacci) calculations hold true for any spatial dimension n moving through n+1 with a dimensional adjustment for e.
Appendix A, B and C - New Physics for Pharma - to Pass Along
Manuscript = Mathematical Transformation of ...
Appendix A
The difference in value between b(min) derived from the Schrödinger equation and b calculated empirically from Earth-surface F-sub-G is 8.4E-19 meters and proves to be the mean factor 1.079 or 7.9%.
The concept of continuous time t leads to exponential growth-decay:
a = a-sub-0 x e^ (rate x time) where e = lim (n infinity) (1 + 1/n)^n = 2.718.
Applying the expression for time itself, then
T(new) / T(old) = e^(r x t) = e^(0) = 1.
For time t itself, the rate r = 0 and there is no change in continuous time t so that one “second” of “time t” does not change. Time t is absolute.
Differential equations for centuries, e.g. the Schrödinger equation, assume time t is a real and continuous variable.
In the transformation t = cB, we need to treat continuity of time as a slight-contiguity of space.
In that case, we find 3 x 10^8 met/sec to be a large enough frequency (number n) to continue using the calculated value of e = 2.718 = lim as ninfinity of (1+1/n)^n.
But in a directional spatial sequential model, then space itself advances or grows per some rate different from r = 0.
As a one dimensional chalk line curves in a two dimensional blackboard, and as a two-dimensional earth-surface curves in three-dimensional space, then a change in 3-dimensional space needs to take place in a mathematical dimension higher than 3.
Postulating the higher number of dimension (vertices) to be 5 as in the Fibonacci infinite sequence, and considering physical events B = 1 / b = sec^2, then we calculate the following for one second of time t:
(V-sub-S / V-sub-0)^1/5 = e^(rV x t)^1/5 = e^(.618^(5-3))^1/5 = 1.079
or a 7.9% decrease in physical events B (increase in one-dimensional size b) from the continuous-time model used to calculate b(min).
Appendix B
Another Calculation for difference b-empirical - b(min) = 7.9%:
In two dimensional physics,
F = ma = m x met-sec^-2 becomes F-sub-2 = m x a-sub-2 = m x met –sec^-3/2
and one physical event B would no longer have units of sec^2; instead, sec^3/2.
The uncertainty principle then has a transformed h-bar, and now
b(min) = (h-bar)^1/2 x c^1/2 = 1.779E-13 meters.
Similarly, E-sub-B / m = 9.8 / c^3/2 J-kg^-1 per square boundary = 1.886E-12 J-kg^-1 per boundary and
b = E-sub-B / m / 9.8 = 1.924E-13 meters.
Then we again have the mean factor
= 1.079 or 7.9%
between b(min) and b (from F-sub-G) in two-dimensional space exactly the same as in three-dimensional space.
Appendix C
A General Fibonacci Calculation:
The Fibonacci infinite sequence was referenced in Appendix A,
F(n) = F(n-1) + F(n-2) with seed values F(0) = 0 and F(1) = 1.
Ratios converge, and
lim(n infinity) F(n+1) / F(n) = φ = (1 + 5^1/2) / 2 = .618 … and
lim(n infinity) F(n-2) / F(n) = γ = .382 … and so on.
Writing an example expression for spatial dimension ≥ 3 per Appendix A
∫∫∫∫∫∫∫∫dV = ∫∫∫∫∫dV(0) x exp(rV x t)
where rV = r-sub-V = φ
then
dx /dx(0) = exp(φ ^ (D(n+1) – D(n))^1/(n+1).
Except we are now doing math in another dimension, and while e = 2.718 in three dimensions, the base of natural logarithms should change in higher or lower dimensional space.
For example, in the case of 5 dimensions: e e^1 / γ^1.
We quickly find dx /dx(0) = 1.08.
A different example, for the case of spatial dimension < 3:
The base e must change as a function of the power of B, i.e. in three dimensional space B ~ sec^2 while in two-dimensional space B ~ sec^3/2.
The difference in power of physical events B
2 – 3/2 = 1/2 and the two-dimensional e = 2.718^1/2.
Then we quickly find dx /dx(0) = 1.08 similar to the previous mean calculations for the difference between b(min) and b-empirical.
The (Fibonacci) calculations hold true for any spatial dimension n moving through n+1 with a dimensional adjustment for e.
Appendix A
The difference in value between b(min) derived from the Schrödinger equation and b calculated empirically from Earth-surface F-sub-G is 8.4E-19 meters and proves to be the mean factor 1.079 or 7.9%.
The concept of continuous time t leads to exponential growth-decay:
a = a-sub-0 x e^ (rate x time) where e = lim (n infinity) (1 + 1/n)^n = 2.718.
Applying the expression for time itself, then
T(new) / T(old) = e^(r x t) = e^(0) = 1.
For time t itself, the rate r = 0 and there is no change in continuous time t so that one “second” of “time t” does not change. Time t is absolute.
Differential equations for centuries, e.g. the Schrödinger equation, assume time t is a real and continuous variable.
In the transformation t = cB, we need to treat continuity of time as a slight-contiguity of space.
In that case, we find 3 x 10^8 met/sec to be a large enough frequency (number n) to continue using the calculated value of e = 2.718 = lim as ninfinity of (1+1/n)^n.
But in a directional spatial sequential model, then space itself advances or grows per some rate different from r = 0.
As a one dimensional chalk line curves in a two dimensional blackboard, and as a two-dimensional earth-surface curves in three-dimensional space, then a change in 3-dimensional space needs to take place in a mathematical dimension higher than 3.
Postulating the higher number of dimension (vertices) to be 5 as in the Fibonacci infinite sequence, and considering physical events B = 1 / b = sec^2, then we calculate the following for one second of time t:
(V-sub-S / V-sub-0)^1/5 = e^(rV x t)^1/5 = e^(.618^(5-3))^1/5 = 1.079
or a 7.9% decrease in physical events B (increase in one-dimensional size b) from the continuous-time model used to calculate b(min).
Appendix B
Another Calculation for difference b-empirical - b(min) = 7.9%:
In two dimensional physics,
F = ma = m x met-sec^-2 becomes F-sub-2 = m x a-sub-2 = m x met –sec^-3/2
and one physical event B would no longer have units of sec^2; instead, sec^3/2.
The uncertainty principle then has a transformed h-bar, and now
b(min) = (h-bar)^1/2 x c^1/2 = 1.779E-13 meters.
Similarly, E-sub-B / m = 9.8 / c^3/2 J-kg^-1 per square boundary = 1.886E-12 J-kg^-1 per boundary and
b = E-sub-B / m / 9.8 = 1.924E-13 meters.
Then we again have the mean factor
= 1.079 or 7.9%
between b(min) and b (from F-sub-G) in two-dimensional space exactly the same as in three-dimensional space.
Appendix C
A General Fibonacci Calculation:
The Fibonacci infinite sequence was referenced in Appendix A,
F(n) = F(n-1) + F(n-2) with seed values F(0) = 0 and F(1) = 1.
Ratios converge, and
lim(n infinity) F(n+1) / F(n) = φ = (1 + 5^1/2) / 2 = .618 … and
lim(n infinity) F(n-2) / F(n) = γ = .382 … and so on.
Writing an example expression for spatial dimension ≥ 3 per Appendix A
∫∫∫∫∫∫∫∫dV = ∫∫∫∫∫dV(0) x exp(rV x t)
where rV = r-sub-V = φ
then
dx /dx(0) = exp(φ ^ (D(n+1) – D(n))^1/(n+1).
Except we are now doing math in another dimension, and while e = 2.718 in three dimensions, the base of natural logarithms should change in higher or lower dimensional space.
For example, in the case of 5 dimensions: e e^1 / γ^1.
We quickly find dx /dx(0) = 1.08.
A different example, for the case of spatial dimension < 3:
The base e must change as a function of the power of B, i.e. in three dimensional space B ~ sec^2 while in two-dimensional space B ~ sec^3/2.
The difference in power of physical events B
2 – 3/2 = 1/2 and the two-dimensional e = 2.718^1/2.
Then we quickly find dx /dx(0) = 1.08 similar to the previous mean calculations for the difference between b(min) and b-empirical.
The (Fibonacci) calculations hold true for any spatial dimension n moving through n+1 with a dimensional adjustment for e.
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