The document presents a theory of Fibonacci dimensional expansion of space to explain the measured rate of spatial expansion (H0). Key points: 1) The theory predicts the empirical value of H0 (74.3 km/sec/mpc) to within 1% using axioms derived from the Fibonacci sequence. 2) Calculations show the theoretical H0 (75.0 km/sec/mpc) matches the measured value, providing evidence the universal volume is expanding proportionally to Fibonacci ratios. 3) While transforming variables, the theory correlates with existing physics equations and can accurately describe observed "dark" expansion forces.

1. Theory of
Fibonacci Dimensional Expansion of Space
Abstract
The rate of measured spatial expansion, as defined by the measured constant H0, is
predicted by a theory of sequential events to within 1% of the empirical centric value
74.3 (+/- 2.1) km sec-1 mpc-1.
Introduction
The measured (empirical) rate of spatial expansion is best described as the expansion
constant H0 1. This measured constant is accurately correlated to the theory of
Fibonacci dimensional spatial expansion. The mathematics of the prediction are
presented as evidence of correlation, and the calculated theoretical explanation is
shown to match remarkably well with observation. The theory’s principle axioms are
defined and explained. Calculations for H0 are justified and compared with
measurement. Mathematics defining the theory are shown in Appendix A.
Suggesting a directional sequence of expanding events requires a transformation of
variables, and a transformation of physical units, which can be reversely transformed
back into standard nomenclature and standard physical units. The theory
mathematically treats time as an effective expansion through mathematics of the
Fibonacci infinite sequence. Definitions (Appx. A) and units (Appx. C.)
Axiomatic Suggested Facts from Theory
Three results (axioms,) directly from the Fibonacci derived theory, are required for
calculation:
Axioms
𝐸
𝐸 𝐵
= 𝑚𝑉𝐵
lim
𝐷→∞
𝑅 𝐷 = 𝑅 𝐿𝑂𝐺 = (𝛾 -1)(1/𝛾) = 12.42
𝑟𝐶𝐿 = 𝑟𝐵𝑜ℎ𝑟 = 5.29E-11 meter

2. where:
rBohr = Bohr radius
D is a Fibonacci number
𝛾 = infinite Fibonacci limit Fn-2 / Fn = 0.382
EB and VB represent boundaries between sequential events
Definitions (Appx. A)
Units (Appx. C)
Mathematics
Calculus of Spatial Growth Rate
Result of calculation is detailed numerically in Appx. B
H0 (theory) = 75.0 km sec-1 mpc-1
H0 (measured) = 74.3 +/- 2.1 km sec-1 mpc-1
Math
𝐸
𝐸 𝐵
= 𝑚𝑉𝐵
𝑑
𝑑𝑡
(
𝐸
𝐸 𝐵
) =
𝑑
𝑑𝑡
( 𝑚𝑉𝐵)
Substituting:
mu c2
𝑑
𝑑𝑡
(
1
𝐸 𝐵
) = mu (
𝑑𝑉𝑢𝐵
𝑑𝑡
)
𝑑
𝑑𝑡
(
1
𝐸 𝐵
) =
1
𝑐2 (
𝑑𝑉𝑢𝐵
𝑑𝑡
)
For 𝑟𝑢 = 𝑟𝑢(t):
𝑑𝑉𝑢𝐵
𝑑𝑡
=
𝑑
𝑑𝑡
𝑑
𝑑𝑟 𝑢
[ 4
3
π (ru + Δr)3
-
4
3
π 𝑟𝑢
3
] (see boundary units below)

3. For large 𝑟𝑢:
𝑑𝑉𝑢𝐵
𝑑𝑡
=
𝑑
𝑑𝑡
𝑑
𝑑𝑟 𝑢
(4π𝑟𝑢
2
Δr )
𝑑𝑉𝑢𝐵
𝑑𝑡
= Δr
𝑑
𝑑𝑡
𝑑
𝑑𝑟 𝑢
(4π𝑟𝑢
2
)
𝑑𝑉𝑢𝐵
𝑑𝑡
= Δr 8π
𝑑𝑟 𝑢
𝑑𝑡
𝑑𝑉𝑢𝐵
𝑑𝑡
= Δr 8πK where K =
𝐻0
2
(nomenclature)
For infinite Fibonacci dimensional expansion:
Δr = RLOG
𝑑𝑉𝑢𝐵
𝑑𝑡
= RLOG 4πHo
and RLOG requires units both (meter3-2=1
) and (meter5-3=2
) for Fibonacci
dimensions 3 and 5, respectively, in a Fibonacci dimensional model.
Then:
𝑑
𝑑𝑡
(
1
𝐸 𝐵
) = RLOG
1
𝑐2 (
𝑑𝑉𝑢𝐵
𝑑𝑡
)
𝑑
𝑑𝑡
(
1
𝐸 𝐵
) = RLOG
1
𝑐2 4πH0
And:
𝑑
𝑑𝑡
(
1
𝐸 𝐵
) = minimum allowed ΔV3 (boundary volume) for mU
=
4
3
π(2𝑟𝐶𝐿)3
-
4
3
π(𝑟𝐶𝐿)3
= 7
4
3
𝜋 𝑟𝐶𝐿
3
7
𝟒
𝟑
𝝅 𝒓 𝑪𝑳
𝟑
= RLOG
𝟏
𝒄 𝟐
4πH0

4. For Fibonacci dimensional expansion:
We perceive 𝑥3 =
𝑥5
(1+𝑅 𝐸
5/3
)
but we have calculated with 𝑥5
and
𝑥5
𝑥3
= (1 + 𝑅 𝐸
5/3
).
We have also calculated (using H0) implying an effective diameter
= 2 𝑥 𝑟𝑎𝑑𝑖𝑢𝑠.
For comparison to measured (perceived) H0:
H0 (as perceived) =
𝐻0 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑
(1+𝑅 𝐸
5/3
𝑥 2)
Then H0 from theory (perceived) = 75.0 km s-1
mpc-1
H0 (empirical) = 74.3 +/- 2.1 km s-1
mpc-1
H0 (theory) = 75.0 km s-1
mpc-1
Correlation of Fibonacci Dimensional Expansion Theory to Existing Empirical Physics
The theory agrees with all existing physical equations and empirical results because the
theory can be precisely described as a mathematical transformation of variables.
Appendix C expands this assertion.
Physical Suggestion
Fibonacci dimensional spatial expansion, while correlating with existing physics, both
relativistic and quantum mechanical, also accurately predicts and explains the effects
of dark matter forces that have been observed.

5. Conclusion
The observed spatial expansion, attributed to a “dark” force, is significant evidence of
the main Fibonacci dimensional theory. The universal volume appears to be expanding
at a rate directly proportional to ratios of the Fibonacci infinite sequence.
Reference:
1. Speed of Universe’s Expansion Measured Better than Ever; Moskowitz,
SPACE.com; 10/03/2012.
Appendix A
For Fibonacci dimensional spatial growth, we need to rewrite the velocity
c = 2.998E+8 met sec-1
as:
2.998E+8 physical events per meter (using MKS units) where a single (one) physical
event has the one-dimensional spatial dimension b3 = 1.111E-17 meter.
Such that:
c (event 1/2
) =
1
√ 𝑏3
and 𝑏3(meter) =
1
𝑐2 (event -1
) (meter)
For radius r:
𝑟𝐷 =
𝑟(𝐷+1𝐷)
(1+𝑅 𝐸
(𝐷+1𝐷)
𝐷 )
e.g. 𝑟3 =
𝑟5
(1+𝑅 𝐸
5/3
)
where
𝑅 𝐿𝑂𝐺 = lim
𝑛→∞
𝑅 𝐷(𝑛) (Appx. D)
{𝑅 𝐷} = { (1/1)1/1, (1/1)5/2, (5/2)8/3, (8/3)13/5, ... }
𝑅 𝐿𝑂𝐺 = (𝛾 -1
)(1/𝛾) = 12.42 =
1
𝑅 𝐸
(nomenclature)

6. We know:
∫
1
𝑡
𝑑𝑡
𝑒
1
= 1 where lim
𝑛→∞
(1 +
1
𝑛
)
𝑛
= 𝑒
and where ϕ = lim
𝑛→∞
𝐹 𝑛−1
𝐹 𝑛
= 0.618
𝛾 = lim
𝑛→∞
𝐹 𝑛−2
𝐹 𝑛
= 0.382
and 1 = ϕ + 𝛾
Then:
𝑟𝐶𝐿 =
𝑐𝑏3
2𝜋3(1+𝑅 𝐸
5/3
)
= 𝑟𝐵𝑜ℎ𝑟 for a closed geometry, and
𝐸
𝐸 𝐵
= 𝑚𝑉𝐵 where 1 (radial) Boundary event =
1
𝑏3
(meter -1
)
Appendix B
Excel Result.pdf
<Calc for H0>
Excel Show Formula.pdf
<Calc for H0 Show Formula>

7. Appendix C
Agreement with Existing Physics
The definitions in Appendix A can be rewritten as a transformation of variables:
t(sec) = c(event) x B(sec event -1)
and the resulting expression
E / EB = mVB
where:
t represents time in seconds
c is the numerical value of the “speed of light” using the dimension (event)
B is the number of seconds per event, and
EB has the units; energy mass -1
meter -3
(we choose the MKS system of measure.)
This is a difficult and abstract concept, but it is nothing more than a mathematical
transformation of variables.
In the transformation, one second of time t is replaced by c events. In other words, time
is not fully continuous in this model, and one second of time t is treated as c
(2.998E+08) discrete events in MKS units.
To the extent that c (2.998E+08) is a large enough number to provide mathematical
continuity for the quantity e = lim(ninfinity) (1 + 1/n)n
, then the expression t=cB is
nothing more than a mathematical transformation of variables. Every physical equation
regarding time t could undergo a transformation of the variable t, and subsequently go
through the transformation back into t, with exactly the same physical result whether it
be a classical, quantum mechanical, or relativistic expression of physics.
There is no deviation from any existing empirical result while the Fibonacci dimensional
theory predicts the value of observed spatial expansion.

8. Appendix D
The reason for exploring this Fibonacci model of time is that the model should logically
lead to the physical expansion of our 3-dimensional space (universe) as a function of
time t.
It seems the concept of time could also be described as a directional motion of the
(Fibonacci number) 3 dimensions of space progressing through the (Fibonacci number)
5 dimensions of space. The physical units would need to be transformed, but physical
results would be exactly the same in this model and could be transformed back into the
variable t anytime as desired.
This Fibonacci model of time could be further explored by postulating that 5
spatial dimensions would then mathematically require an alternate value of:
e = lim
𝑛→∞
(1 +
1
𝑛
) 𝑛
because the corresponding statement:
∫
1
𝑥
𝑒
1
𝑑𝑥 = 1
suggests that the number 1 in 3-dimensions should be different from the number 1 in
5-dimensions and should no longer represent a 2-dimensional area but perhaps would
represent a 3-dimensional volume instead.
If we continue the mathematical exercise for time t, we would need to assume that, if
3-dimensions were mathematically traversing through 5-dimensions, then 5-
dimensions would need to be similarly traversing through 8-dimensions, and so on into
Fibonacci infinity.

9. In the infinite Fibonacci limit, then (as shown below) the logarithmic base should be
continually increasing at the rate:
(1/𝛾)^(1/𝛾) = 12.42
and to continue the mathematical model, then the number 1 would appear to be
increasing throughout higher Fibonacci dimensions by the factor:
(
1
12.42
)F(n+1)/F(n)
= (0.08)F(n+1)/F(n)
or about (8%) x [(F(n+1)/ F(n)] or 8% (Fn+1 / Fn) for each successive higher dimension
from the dimension Fn ( e.g. F3+1 / F3 =
5
3
)
where, regarding each adjacent dimension to Fn
𝛾 = lim
𝑛→∞
𝐹 𝑛−2
𝐹 𝑛
= 0.382
and, e.g.
F2 / F4 =
2
5
adjacent to F3 (where Dn = 3)
F3 / F5 =
3
8
adjacent to F4 (where Dn = 5)
and so on.
Then the figurative 5-dimensional number 1 should look more like the number 1.08 as
mathematically perceived in 3-dimensions, and we would determine:

10. e1 = 0
e2 = e0
e3 = e1
e5 = e3
(5/2)
e8 = e5
(8/3)
e13 = e8
(13/5)
and the sequence of dimensional logarithmic base:
{LD} = {0, 1, e, e5/2
, e5
8/3
, e8
13/5
, … }
where the rate of dimensional increase for the logarithmic base:
RLOG = lim
𝑛→∞
𝑅 𝐷(𝑛)
{RD} = { (1/1)1/1, (1/1)5/2, (5/2)8/3, (8/3)13/5, ... }
RLOG = (1/𝛾)^(1/𝛾) = 12.42
The reason for exploring this Fibonacci model of time is that the model should logically
lead to the physical expansion of our 3-dimensional space (universe) as a function of
time t.
To be a worthwhile mathematical exercise, there would need to be a correlation
between the mathematical growth rate of the logarithmic base and the observed
physical expansion of 3-dimensional space as perceived through telescopes, and the
like, while we attempt to measure the Hubble value H0.