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1. *Corresponding Author: Paul T E Cusack, Email: St-michael@hotmail.com
RESEARCH ARTICLE
Available Online at www.ajms.in
Asian Journal of Mathematical Sciences 2017; 1(4):187-193
The Geodesic Dome& GDP Growth
*Paul T E Cusack
*BScE, DULE, 1641 Sandy Point Rd., Saint John, NB, E2K 5E8 Canada
Received on: 07/06/2017, Revised on: 30/06/2017, Accepted on: 10/08/2017
ABSTRACT
This paper builds on our knowledge of AT Mathematics used to solve for variables in a geodesic dome
structural analysis. Temperature loads can simulate expansion of the GDP, or GDP growth. This can be
used to see how the entire dome, or economy, adjusts to the new forces. There are computer software
programs readily available which could be used to help economists model the entire economy from the
individual to the world.
Keywords: Geodesic Dome; GDP Equation; Stiffness Method, Econphysics
INTRODUCTION
Economists now use structural engineering mechanics to solve outstanding economic theory problem.
The theory for matrix structural analysis has been worked out since at least 1962. It provides a solution
for nodes interconnected where the forces on one node affect all other nodes. This is a good
approximation for either individuals in an economy or for nation states. Really, one can draw the
boundary that is governed by the well known GDP Equation viz. (Y=C+I+G+S+(Ex-Im). The stiffness
method, which allows for thermal loads, which are akin to an influx of money, can be used to calculate
the affect on the entire economic structure. The temperature load can be thought of elongation of an
individual member caused by investment, government spending or entrepreneurship. We therefore can
see the affect of these three things in causing growth of the GDP. There are wider implications such as
how wealth is transferred by government spending in a particular local or by education leading to new
ideas; or by financing of those ideas by financiers. We begin by examining the geodesic dome
Figure 1: The Geodesic Dome model of the Economy. Income Y or GDP is at the node; and its components are the slender members
connects to nearby nodes.
The Geodesic Dome has many nodes that interjected to each other. This structural model can be used to
slove economic problems where the node could be an individual of a nation. The GDP Equation, Y =
Consumption+ Investment+ Government Spending+ Savings+(Exports-Imports). Y can be thought of as
the energy is a system. We use our knowledge of structural engineering, which has already solved this
structural problem, to be applied to solve economic problems as well. There are many textbooks that

2. Paul T E Cusack et al. The Geodesic Dome& GDP Growth
© 2017, AJMS. All Rights Reserved. 188
provide the solution to the strains in a geodesic dime. There are papers as well by economists who have
used the geodesic dome to solve transportation problems for example. We begin with the stiffness
method. Please note, it is necessary to understand the author’s Astro-theology Mathematics to appreciate
the variables used in this paper for optimum conditions.
STIFFNESS METHOD:
The stiffness method relies upon the basic formula of the simple spring equation which has F=Force; k =
spring stiffness; and s=displacement.
F=ks (eq. 1)
k=F/s=Y/s
But we know k=0.4233 =cuz (Spring constant, or stiffness k is Pi-e)
&
Y=GDP
s=ΔGDP /annum (Growth)
0.4233=Y/ 2.1%
Y=88.5=ε0 =Permittivity of the universe
88.5=ΔL/L=0.22/L
L=0.402=Re (Reynolds’s number from fluid mechanics)
So how does growth occur? We can answer this question by taking growth as a thermal load increase in
the Geodesic Dome members.
The temperature Load is given by the following equation:
TL=(1+αT)dl eq.(2)
But TL=e-t
e-t
=δαT
0.402=δ(S)(G)
S=1/7 G=17%
δ=1655 ~1623 = Mass of a proton
1622(0.402) =6.52=Gravitational Constant (not to be confused with Government spending)
1623(1/7)(1/0.402)=1/√3=cot 60°
“The elastic strains, e are related to the stresses by means of the usual Hooke’s law for linear isothermal
elasticity. [2]
ε=2[1+ν]/E × σ eq.(3)
e= strain
nu=Poisson’s ratio
E=Young’s Modulus
0.402=2(1.27)(0.4233) × σ
σ=1/6.7=1.49~1.50=Mass Gap from Chemistry. The maximum mass that the universe can support.
σ=F/A=F/L²
F/0.402)=1.5
F=6
This is 6-sigma of the Bell normal curve
For a plane geodesic connection:
σ=Ee-EαT where T=G (Temperature Load = Government Spending in a local)
=(0.4233)(0.402)-(0.4233)(1/7)(G)
=1/6=1/F (Force is the money applied to the economy)
F=ks
6= (0.4233) (s)
s=√2

3. Paul T E Cusack et al. The Geodesic Dome& GDP Growth
© 2017, AJMS. All Rights Reserved. 189
σ=1/F
A=1
So what causes the temperature load like elongation? The answer is in three things:
 Government Transfers or spending results in transfer of wealth from one local to another;
 Creative Ideas that create new wealth.
 Financing of Investments that produce wealth creation.
THE GEODESIC DOME
Now consider the volume of the sphere:
V=4/3 π R³
ΔV=4/3 π (R2-R1)³
ΔV=ΔGDP =2.2%
Figure 2 The Earth /Economy as a Geodesic Dome. Every node could be an individual or a city or a nation.
ΔR=Δs (R is the radius of the sphere which increase as does the GDP)
Δs=0.8068~81=c4
ε=ΔL/L=Δs/L
0.402=0.8068/L
L=0.4983~0.5
Figure 3 Elongation of Memberin the geodesic dome with associated strain increasing the length of the member. The cross-sectional
area of the member is A=1. The original length of the member is L.
ASTROTHEOLOGY MATHEMATICS
L=0.5+0.8068=1.3051α13.0
= (1-sin 1)=moment=Fd=Fs
(Refer to Astrotheology Cusack’s Universe for further discussing on this moment= F x d)
F=1/L
s=1/L
S=L
Y=Y’
The telescopic equation of the universe is where the derivative of the function is equal to the function.
i.e., Y=e^x)
Now we already know

4. Paul T E Cusack et al. The Geodesic Dome& GDP Growth
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Y=FL
Y’= (Ma) (L)
Y'=dM/dt (da/dt)(dL/dt)
=2(da/dt)(0.8415)
dK/dt=1683(da/dt)
Aside:
The universe exists where the distance, s =the velocity (momentum) = the acceleration (Force) So,
s=v=a=sin 1
sin 1=!/M
da/dt=Ln M
=Ln 2
So,
dY/dt=1683 Ln 2
=116.65=M (Period table of the elements)
d²Y/dt²=dM/dt=2
But Y=FL
2=Y'=F' L'
2=F'a
F=0.4208~k=cuz (This is Pi -base e)
COMPATIBILITY EQUATION
“For a two-dimensional plane stress problem, the six equations of compatibility reduce to only one
equation”. [2] pg. 24 It is:
δ²εxy/δ²y + δ²εxy/δ²x= δ²εxy/δxδy eq. (4)
Y=FL
L=Y/F
Yσ=F/A ×Y
ε=ΔL/L=Δs/L=ΔL/[Y × F/A]
ε=ΔL × A[YF]
ε=c4
×(1)/kF
0.402=1/34
×1/0.4233F
F=1176=Mass
HOOKE’S LAW
Hooke’s Law is stress = Yong’s Modulus x strain, or:
σ=Eε eq. (5)
Y=ΔL/L (k)
Y=k
1=Y=ΔL/L
L=ΔL
Y=Y’
L=0.402=Re
Re=ρv/ν
ΔL=Re=1/T
ε=ΔL/L
ΔL=L²
ΔL=0.25=T
STRAIN ENERGY
And, finally, consider strain energy:
W=Work; P=Force’ u=displacement in the direction of the force.
W=1/2 Pu eq.(6)
P=F
u=s

5. Paul T E Cusack et al. The Geodesic Dome& GDP Growth
© 2017, AJMS. All Rights Reserved. 191
L=1/2
W=FL=Y=Pu=LFs
So theGDP is equal to the Strain energy.
The Incremental increase in Strain energy is given by Przemieniecki. “In a linear system, if displacement
is increased to u+du, the corresponding increase in W becomes”
ΔW=Pu+1/2 δPdu
Or, using our notation:
ΔGDP=Fs+ [L × dF× Δs] eq.(7)
This them is the Strain Energy for the GDP Growth of the Economy.
ε=ΔL/L
But ΔL=L
Therefore ε=1
εx=εy=1 & εxy=√2
This means that the strain is a unit force, since a=1. it measures the sensitivity of the Individual to Force
(Money).
Figure 4 Torque or moment applied to node.
STRAIN ENERGY AND THE MOMENT (Leverage in spending):
Strain energy =dU/dt=θ [2]
Strain ε=1=U=θ=1 rad. Eq. (7)
1 rad=2π cycles=0.1592~1-sin 1=moment
Multiplier =7X’s
G (Government Spending) =Moment / 7= (1-sin 1)/7=0.227
17%=G
G/22.4%=0.1335~4/3
Volume =4/3 Pi R^3
Note Space=s from Astrotheology Mathematics.
=sPi(R^3)
=c^4 Pi
=254~Period T

6. Paul T E Cusack et al. The Geodesic Dome& GDP Growth
© 2017, AJMS. All Rights Reserved. 192
Figure 5: The economic multiplier produces leverage of 7X’s
SUPPLY AND DEMAND CURVES
For Supply and Demand Intersection, we have mathematically:
S=D
K=e^-t eq. (9)
0.4266= e-t
-t=0.8519=1/117.38=1/Mass =1/M
And,
K= [F/A]/ [ΔL/ L]
Superforce = F=2.667
A=1
Δ L=GDP Growth
2.667/1/2.1%
=0.127
=ρ
ρ/c=1.27/3=0.4233=k
ε=1/l[-1 1] [u1 u2] eq.(10)
=1/(1/2)[ -1 1 ] [ΔL ΔL]
ΔL=1
ε=2
ε=ΔL/L=2
ε=1/(1/2)=2 = dM/dt [From Astrotheology Mathematics]
E=Mc² (Einstein’s Equation)
Y’=dM/dt × 2a
Y’=(2)(0.8415)(2)
=3.36
1/Y=2.971~c
c=1/Y
k=ρ/c eq.(11)
0.4233=1.27/3
Y’=ΔGDP
k × ΔGDP=0.4233(2.1%)=8.88~c²
Supply Curve Equation
k =c²/ΔGDP eq.(12)

7. Paul T E Cusack et al. The Geodesic Dome& GDP Growth
© 2017, AJMS. All Rights Reserved. 193
Figure 6.: The supply Curve is produced for each consumer depending upon his income Y. The Supply-Demand intersect at market
clearing prices. “k” is the stiffness of the supply curve.
The stiffness factor must be determined for each person, (or city, or Nation) to have the Supply Curve.
The Demand curve is always the exponential function to a negative power of t.
CONCLUSION
The Geodesic Dome can be used to model the any economy that the economist desires to draw a
boundary around, the free body diagram of physics. Y can be calculated for an individual or for a Nation
State. Temperature load increase when there is an injection of wealth from an increase of savings; from
government spending; or from entrepreneurial activity. GDP Growth can be modeled by the geodesic
Dome. The more data that is known about the individual transaction, the civic economy and the national
economy, the better geodesic model can be had.
REFERENCES
1. Cusack, P., Astrotheology Mathematics, Cusacks Universe., J of Physical Mathematics. OMICS,
Jan 2016.
2. Przemieniecki’s, J.S. Theory of Matrix Structural Analysis., Dover. New York. 1962