Transcript
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Are Geodesic Dome Homes More Energy Efficient and Wind Resistant Because They

Resemble a Hemisphere?

by

Taralyn Fender

Presented to

THE FACULTY OF THE DEPARTMENT OF MATHEMATICS

In partial fulfillment of the requirements for the degree Master of Arts in Mathematics

JACKSONVILLE UNIVERSITY COLLEGE OF ARTS AND SCIENCES

April 2010

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Master of Arts in Mathematics Department of Mathematics

Jacksonville University

The members of the Committee approve the thesis of Taralyn Fender, titled “Are Geodesic

Dome Homes More Energy Efficient and Wind Resistant Because They Resemble a

Hemisphere?” defended on March 24, 2010 .

___________________________________________ Dr. Paul Crittenden Thesis Advisor ___________________________________________ Dr. Michael Gagliardo Committee Member Approved on: ____________________________ _____________________________________________ Dr. Pam Crawford Chair, Department of Mathematics

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ABSTRACT

Geodesic domes resemble hemispheres, which are considered to be one of the most

efficient geometric shapes. For this reason, it is said that geodesic domes are more energy

efficient and wind resistant than typical rectilinear homes. That hypothesis is tested in this

thesis using simple mathematical models, one for heat transfer and one for wind pressure.

Various geodesic domes are included in this study and were constructed from the platonic

solid, octahedron. The surface area and volume for various geodesic domes and rectilinear

homes were used to compute their sphericity, a measure of their roundness. The heat flux

ratio, a value that determines the relative energy efficiency of the models, was computed.

Finally, the wind resistance ratio, a value that determines the relative wind resistance of

each model was found. Once the computations of sphericity, heat flux, and wind resistance

ratios are found, an attempt will be made to show that as the frequency of the dome

increases, the sphericity of the geodesic dome approaches the sphericity of the hemisphere.

As the sphericity, ratio of the investigated home models approach the sphericity ratio of the

hemisphere, the data will show that the dome home is the most spherical, most energy

efficient, and on average most wind resistant structure of the models investigated.

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ACKNOWLEDGEMENTS

Investigating the geodesic dome has been an eye opening experience to encounter

all of the mathematics that surrounds us on a daily basis. Thanking those of you who have

helped me with this endeavor seems so inadequate.

I must first thank Jesus for giving me wisdom on a moment by moment basis. There

were days when I came to a dead end in my research, but I would ask God to give me some

of the wisdom that he gave King Solomon, He always heard my plea, and gave me the

thought that I needed to complete the task at hand. Thank you, Jesus, for being my

personal Savior.

My husband, Paul, is my biggest supporter and the love of my life. He relinquished

his hold on me and allowed me to spend numerous hours in front of the computer day after

long day without complaining. I can always count on him for his support, which included

but was not limited to cooking our meals, washing dishes, vacuuming, and washing clothes

during the time spent on the research and then the writing of this paper. He truly takes

care of me. He is my prayer warrior, a true gift from God. Honey, God has richly blessed me

by allowing you to be a very big part of my life. I am so thankful to call you husband and

best friend.

My daughters, Christee and Joye have been wonderful supporters and cheerleaders

during this time. You are true blessings from God and I thank you for all that you do for me.

You allow me to tell you all about this paper at any time. I am so very proud of you and the

wonderful women that you have become. I love you so much

My grandchildren are the best in this world. Tyler and Sara helped me build a

geodesic dome model that I purchased from American Ingenuity, Inc. while Timothy,

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Nathan, Noah, Logan, Dylan, and Megan thought they were playing with the geo-sticks but

they were really helping me with various dome constructions. They helped me visualize

the different frequencies of various domes by constructing different models. Your G.G.

loves you for all of your help in making this study a visual success.

Dr. Paul Crittenden, my thesis advisor, has been a fountain of knowledge and this

study would not have come to fruition without his vast knowledge and expertise. Your

unending patience, tireless hours of reading submission after submission, thinking, editing,

guiding, and directing are to be commended. You are truly a brilliant mathematician and I

am very thankful to have been assigned to you through this learning process. I know that

this paper would not be what it is today without you. You are truly a gift from God and I

will forever be grateful for the time spent with you. I know I will never be able to repay

you for all that you have done for me. Thank you for taking me under your wing and never

giving up on me.

My dear friend, Michael Vasileff, has spent many tireless hours reading and checking

for any grammatical errors that I may have missed prior to each submission. Although the

statistical information was not readily available for geodesic dome homes and their ability

to withstand hurricane force winds, you continued to spend many hours searching. Thank

you for always being my true and steadfast friend. God has again blessed me with your

valuable friendship. I am so thankful for you and your valuable input.

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TABLE OF CONTENTS

Page SIGNATURE PAGE ......................................................................................................................................ii ABSTRACT ....................................................................................................................................................iii ACKNOWLEDGEMENTS ..........................................................................................................................iv INTRODUCTION .......................................................................................................................................... 1 LITERATURE REVIEW

Definition Comparison of the Geodesic Dome to the Rectilinear Home ................6 Domes to Geodesic Domes .......................................................................................................6 Geodesic Dome .............................................................................................................................9

Structure ..........................................................................................................................9 Aerodynamic Strength ................................................................................................10 Energy Efficiency ..........................................................................................................12 Sphericity ..........................................................................................................................14

ILLUSTRATIVE EXAMPLES ....................................................................................................................15

Forming the Geodesic Dome ...................................................................................................17 Surface Area ...................................................................................................................................20 Volume .............................................................................................................................................28 Sphericity ........................................................................................................................................33 Energy efficiency ..........................................................................................................................37

Heat loss ............................................................................................................................37 Wind resistance ..............................................................................................................42

CONCLUSION ...............................................................................................................................................51 APPENDICES ................................................................................................................................................57 Appendix A: Calculations for the One-Frequency Dome ...........................................................58 Appendix B: Calculations for the Two-Frequency Dome ..........................................................59 Appendix C: Calculations for the Four-Frequency Dome ..........................................................60 Appendix D: Calculations for the Six-Frequency Dome .............................................................64 Appendix E: MATLab Computer Program .......................................................................................72 Appendix F: Email permission to use photographs .....................................................................80 American Ingenuity Domes, Inc. .............................................................................................80 Natural Spaces Domes ................................................................................................................81 FEMA .................................................................................................................................................82 References ....................................................................................................................................................83

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Are Geodesic Dome Homes More Energy Efficient and Wind Resistant Because They Resemble a Hemisphere?

INTRODUCTION

In this paper, various geodesic dome homes are investigated and compared to

rectilinear homes to determine which home more closely resembles a hemisphere.

Geodesic domes are created by connecting a mesh of triangular panels together in order to

closely resemble a hemisphere. It has been said that the hemisphere is considered the

most “efficient geometric shape” (Geodesic Dome, 2008, pg. 1), because it has the minimum

surface area for a given volume. Both Fuller, the inventor of the geodesic dome, and Busick,

CEO of American Ingenuity, said that the geodesic dome home is more energy efficient and

wind resistant than typical rectilinear homes because of this fact.

The hypothesis to be tested is that because the geodesic dome more closely

resembles the hemisphere, then it is more energy efficient and wind resistant than typical

rectilinear homes. Sphericity, the ratio of the volume to surface area, gives a measure of

the roundness of the object. Thus to test this hypothesis, simple mathematical models are

used on various geodesic domes, rectilinear homes, and the hemisphere. Heat flux and

wind pressure are computed and compared to that of a hemisphere. In order to make

these comparisons the surface area and volume for each of the various models must be

computed.

To determine energy efficiency of homes with the same volumes, the investigator

applies a simple mathematical model for heat transfer by comparing various geodesic

domes to model rectilinear homes and to hemispheres. Using these computations, the

investigator determined that when the sphericity ratio of various models was close to the

hemisphere, then the structure was also more energy efficient.

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Similarly, a simple mathematical model for straight line wind impact on various

geodesic domes and rectilinear homes is applied to determine if more spherical models

were more wind resistant. The projected area of the hemisphere, geodesic dome, and three

different views of one and two-story rectilinear home models with the same volumes are

computed and compared to determine the wind resistance. On average, the geodesic dome

homes are shown to be more energy efficient and wind resistant during a hurricane than

the rectilinear home because of their near hemispherical shape.

Building a geodesic dome home is financially and environmentally efficient because

less building materials are needed to construct a dome home (Busick, 2008). The National

Dome Council commissioned Knauer, author of the article, The Futurist, to do a study that

compared the energy efficiency of geodesic dome homes with rectilinear homes and the

results showed that geodesic domes were more energy efficient (October 2008).

According to investigators from the Lawrence Berkeley National Laboratory,

Diamond and Moezzi (July 2009), electrical energy consumption in the United States for the

years of 1949 to 2001 has steadily increased to almost double the amount it was in 1949.

The importance of conserving energy has been on the minds of many consumers and the

data shows that some make a concerted effort to limit their consumption. However, those

who desire instant comfort continue consuming energy in ever increasing amounts.

Energy consumption changes in the home when normal weather conditions change. When

the outside temperature changes, the inside temperature reflects that change unless an

intervention occurs to achieve a level of comfort for its residents. For the home to be

deemed energy efficient, the transfer of heat must be minimized while maintaining a

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desired level of comfort. Home design, construction methods, insulation, and the correct

heating and cooling unit are essential for any home to be labeled as energy efficient.

Since the early 1900s, data about hurricane history has been recorded by the

National Hurricane Center. This data includes the human death toll and property damage

due to hurricanes and other natural phenomena (Hurricane History, March 2009). From

that time through 2005, there have been 35 major hurricanes and tropical storms that

made landfall on the United States and surrounding countries. These natural disasters

have claimed the lives of approximately 30,000 people and injured numerous others.

Property devastation from these storms has been estimated to cost the homeowner and

government more than $200 billion.

To reduce the cost of devastation after a severe storm or hurricane, it is essential for

the affected residents to live in more wind resistant homes. According to Smith, Physicist

at the University of Munich (November 2008), research to determine the severity of a

storm and estimate its location is necessary to offer assistance to residents in a timely

manner in areas most prone to the ensuing hurricane. Offering timely information to

residents that a severe storm is going to occur at a particular location and showing its

projected path would result in fewer lives lost. Given this information, residents can

prepare their homes for the severe wind and tornadoes which accompany a severe storm

or hurricane. As residents prepare for their safety, it may require evacuation of their

homes. However, some residents are not willing to evacuate their homes. A geodesic dome

home could provide residents an alternative to evacuation, given its lower profile to the

wind.

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When severe weather occurs such as a hurricane or other natural phenomena,

changes in energy output occur as a result. Physical human energy is also expended and

battery or gasoline engines are used to power various tools necessary to clear debris from

devastated areas. After a disaster, electrical crews spend extra hours replacing downed

power lines to restore electricity to consumers as soon as possible. Wherever hurricanes

are more prone to occur, alternative methods to reduce energy consumption and natural

resources must be explored by reviewing the history of energy efficient homes.

Historically, humans have lived in domed caves, coned shaped tepees, rounded

igloos, and a myriad of traditional, rectilinear structures which are the stereotypical choice

for homes today. The need to build more homes increases when the population increases.

As of August 2009, the United States Census Bureau recorded in US and World Population

Clock that there are around 300 million people living in the United States and that number

continues to increase. An increase in population indicates that the need to build quality

homes is also increasing. By designing and building homes that are energy efficient and

wind resistant, the environment and its precious natural resources will be protected and

ultimately the loss of human life would be greatly reduced during natural disasters. Public

dome structures could also be provided to keep residents safe during a natural disaster.

After the hurricane Katrina disaster, several television stations reported that the Louisiana

Superdome was the shelter to which the devastated public was transported for safety.

For this study, the following terms are defined here and will be developed further

during the course of the paper. A geodesic dome is a mesh of triangular panels connected

to closely resemble a hemisphere. A more precise definition of a geodesic dome can be

defined as a geometric construction. Every geodesic dome can be created by the following

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procedure. Choose a platonic solid. Next, each edge of the solid will be sub-divided into

equal parts. The number of times the sides are sub-divided is called the frequency of the

geodesic dome. Each face of the solid will be sub-divided into equilateral triangles using

the new vertices. The new vertices, which are defined after the sub-division, are stretched

using vector algebra to be equidistant (one unit) from the center of the base. This creates

the geodesic dome. The side of each triangular panel is called a strut. Sphericity is a ratio

of volume to surface area which measures the roundness of a geometric shape. These

ratios will be computed on various geodesic dome models, rectilinear models, and a

hemisphere to provide a measure of roundness on each model. It will be shown that as the

frequency of the dome increases the sphericity of the geodesic dome approaches the

sphericity of the hemisphere. Energy efficiency is the reduction of the consumption of

energy and will be approximated by computing the transfer of heat of various models

contained in this study using a simple mathematical model, the heat transfer is then

compared to a hemisphere with the same volume. Similarly, the ratio of wind resistance is

defined by comparing the projected area of the models that are directly impacted by the

wind. Once the transfer of heat and wind resistance ratios are computed, then they are

compared to the ratios of sphericity to determine if the most energy efficient and wind

resistant models are also the models which most closely resembles the hemisphere.

For the purpose of this study, the octahedron is the platonic solid chosen to

construct the geodesic dome. The original vertices of the octahedron are taken to be one

unit from the origin along the coordinate axes. When constructing the dome from this

platonic solid, the triangular faces are subdivided by the frequency and the original vertices

are stretched to be equidistant (one unit) from the center of the dome.

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LITERATURE REVIEW

Definition Comparison of the Geodesic Dome to the Rectilinear Home

Kenner, author of Geodesic Math and How to Use It (1976), defines the geodesic

dome as a domicile, shell-like structure that holds itself up without supporting interior

columns. Both Fuller (Introduction to Geodesic Domes and Structure, November 2008),

inventor of geodesic domes, and Knauer (October 2008) agree that the geodesic dome is

defined as an approximate hemisphere formed by connecting a mesh of triangles, which

provide a self-supporting structure, which offers an open interior for maximum space and

light. Self supporting is defined as a structure that requires no load-bearing interior walls

to bear the weight of the roof or dome. The dome structure is both stable and strong when

compared to a rectilinear shaped structure. Fuller (2008) was convinced that by applying

modern technology to the design and construction of homes, that geodesic dome homes

could also be built to ensure comfort, as well as economic and energy efficiency.

Domes to Geodesic Domes

During the Roman Empire, arches were used to strengthen a structure, whereby a

“keystone” was placed in the center of the arch (Kenner, 1976, p.3). This is seen on the

Arch of Severus, a famous Roman structure (Great Buildings Online, August 2009). The

keystone in the center of the archway makes the entire structure stronger and allows for a

wider opening than buildings with a horizontal crossbeam which limits the distance of the

opening as gravity pulls downward.

The Pantheon, dedicated around 120 A.D., is the largest domed building ever

created out of concrete and is still considered a magnificent building (Great Buildings

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Online, March, 2009). Its age indicates its resilience. Fuller recognized that the

gravitational force on the arch’s keystone designed by the Romans caused the arches to

stay in place. He used that idea to create the design for the geodesic dome. However, his

dome was built by connecting triangular panels and he is thereby credited with the

invention of the geodesic dome in the late 1940’s. In Baldwin’s book, he calls Fuller a

“missionary” in the design revolution, and science fiction writer, Clark remarked, “Fuller

may be our first engineering saint” (1996, p.65).

Geodesic structures have been built in the modern age for a variety of purposes.

The Climatron at the Missouri Botanical Gardens was built in 1961 and was the first

geodesic dome with a transparent covering to admit light and heat (November 2008). It

contains a temperature and humidity controlled atmosphere for some 1200 species of

plants in a natural tropical setting. In addition to the numerous plants, the Climatron is

home to tropical birds and waterfalls.

In 1954, the USAF built fiberglass plastic domes for the Distant Early Warning

(DEW) stations because the domes were assembled quickly, invisible to microwave radar,

and capable of withstanding the brutal weather conditions in Canada and Alaska (Massey,

1997). During the Cold War, the United States relied on these stations to detect enemy

aircraft and dispatch fighter planes to intercept them.

The geodesic dome at Epcot in Disney World in Orlando, Florida, was designed by

Fuller and opened in October 1982 shortly before his death in July 1983 (Epcot, November

2008). This is the geodesic dome for which he is most famous. Fuller was convinced that a

geodesic dome home was the most energy efficient and structurally sound structure, so in

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1960, he designed and built a dome home for himself and his wife in Carbondale, Illinois

(Introduction of Geodesic Domes and Structure, November 2008).

His dome home was constructed on a cement pad on the ground with no exterior

vertical riser wall to support the dome. Since that time, other dome home companies have

used his dome idea, but have added a 4 ft exterior vertical riser wall to increase the

functionality of the home and thereby limiting the amount of wasted space in the home.

In 1976, Busick became founder and CEO of America Ingenuity, Inc. He agreed with

Fuller about the safety and efficiency of the dome home.

Currently, he and his team of engineering experts build

custom dome homes in many parts of the United States.

Figure 1 is a picture of a geodesic dome home under

construction which clearly shows the triangular panels of the

dome as they are joined together. Since 1976, Busick has

expanded his designs to include homes with adjoining dome garages and patios as well.

While Fuller was the original design engineer, dome home manufacturing

companies are constantly making changes to meet the needs of consumers. Their goal is to

create the best design for the most efficient structure and to customize it to suit the need of

the consumer. Busick’s dome home manufacturing company offers their homeowners a full

replacement guarantee if their home is destroyed by a tornado or a hurricane. Mandel

(2008, pg. 1) reported that the geodesic dome home plan is best unless you want to “see

your home gone with the wind” after a hurricane.

In a telephone interview in December, 2008 with Mara, a builder for Natural Spaces

Domes, Mara stated that geodesic dome homes are the “safest and most energy efficient

Figure 7

Figure 1. Construction of a concrete geodesic dome home. Used by permission.

American Ingenuity warrants only the structure and is in no way liable for the loss of personal property, life, or limb as a result of 225 mph winds or #4 tornadoes. In the event of natural disasters, the occupants should evacuate when advised to do so by local authorities. In no event shall American Ingenuity’s liability exceed the amount paid by the Buyer.

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homes” and extended an invitation to anyone who would like

to participate in the construction of a current dome home

construction, as seen in Figures 2 and 3.

Geodesic Dome

Structure

Knauer (October 2008) states that geodesic dome

structures are returning to the design table as more people

consider their efficiency and wind resistance during a

hurricane or natural disaster when considering building the

family home. Kenner (1976) discusses several aspects of the

structure of the geodesic dome that must be investigated to understand its design and

determine the efficiency of the structure. They include the strut length, frequency, and

faces, which are concrete triangular panels that form the surface of the dome. The joints or

seams determine the strength of the dome, which is necessary to ensure that the geodesic

dome will be able to more resistant to the fierce, horizontal winds associated with

hurricanes and other natural phenomena.

During a natural disaster, trees may be uprooted and then fall at a tremendous force,

landing on the nearest object or structure that is in their path. When trees hit the roof of a

rectilinear home during a hurricane with an 8-foot vertical wall, the house can be severely

damaged. However, damage to the dome home will be minimal because the near

hemispherical shape of the geodesic dome will gradually break the fall of the tree in varied

increments of degrees. The picture of the dome home in Figure 4 is from the gallery of

Natural Spaces Domes (March 2009) that shows a tree which has fallen on the dome home.

Figure 2. Construction of a geodesic dome made of wood. Used by permission.

Figure 3. A completed dome home. Used by permission.

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Figure 4. A tree has fallen on a dome home. Used by permission.

The tree looks as if it is merely leaning on the home.

However, this investigative study does not include

(treat) the amount of damage trees may cause to any

home during a hurricane.

Geodesic domes are constructed with struts,

which are the sides of the equilateral and isosceles triangular faces or panels. A strut is the

brace which connects two adjacent vertices of the triangular face or panels which

eventually form the geodesic dome (Kenner, 1976), as seen in Figure 5. As these struts are

connected, the resulting geodesic dome is very

strong and resembles the most efficient geometric

shape, the hemisphere (Geodesic Dome, November

2008). Typically, the triangular panels consist of

reinforced concrete enveloping a polystyrene

insulation. A galvanized steel mesh interlocks the

two adjacent panels. Hornas (2000) states that

concrete is his favorite building material because it is fireproof, waterproof, and termite

proof.

Aerodynamic Strength

For the geodesic dome to remain intact given an external force, triangles are

designed, connected, and strategically placed to create a more hemispherical and smooth

surface. As the number of triangles increase, the stability of the structure increases and the

shape becomes more hemispherical (Kenner, 1976). Hornas (2000) also states that a

Figure 5. A geodesic dome model.

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Figure 7. Another demolished house after Hurricane Ivan. Used by permission.

Figure 8. A dome home with little damage after Hurricane Ivan. Used by permission.

round dome is so aerodynamic that strong destructive winds have nothing to directly push

against and is therefore resistant to hurricane force winds.

Hurricane winds swirl in a slightly upward spiral fashion according to Encyclopedia

Britannica, during a tropical cyclone (February 2009). The impact is most severe when the

wind is at an angle where the projected area of the structure is greatest. Since a geodesic

dome home has a low profile, the areas exposed to the wind forces are minimal compared

to a rectilinear home. The damage sustained as a result of the impact from the wind is

minimized.

According to Federal Emergency Management

Agency (FEMA, 2008), a Category 4 hurricane would

have winds between 131 and 155 mph, thereby

destroying poorly constructed buildings. The pictures

of the homes that are shown in Figures 6 and 7 were

destroyed by Hurricane Ivan in Pensacola Beach,

Florida, in September, 2004 (FEMA, 2008). Figure 8

shows a dome home built in Pensacola Beach, Florida,

in 2003 that withstood the wrath of Hurricane Ivan as

reported by J. Reynolds (2004). The homeowner said

that while the waves washed around his home, their strength

was not sufficient to totally demolish his home as it did to

other homes in his neighborhood.

Parker, a reporter for the Post and Courier in

Charleston, SC, (October 2006) reported that a local builder

Figure 6. A demolished house after Hurricane Ivan. Used by permission.

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built a dome home and called it his “safe haven” from hurricane winds of 300 miles per

hour and earthquakes of up to 7.5 on the Richter scale. He stated that the triangular panels

for his dome home are made from layering and bonding wood chips, then surrounding the

panel with a thick slab of polystyrene foam before sealing it with an exterior layer of

concrete. This creates a dome home that is considered to be very strong and resistant to

hurricane-force winds.

One feature that gives geodesic dome homes an advantage in high winds over the

rectilinear home is their lower vertical profile. The wind resistance ratio will be computed

for the geodesic dome home of various frequencies and compared with the ratio of various

rectilinear homes. Mathematically, it will be shown that the lower profile home with

similar volume will be more resilient to the forces of wind that accompany hurricanes.

Energy Efficiency

Rourke (October 2000) reported that Khalili, an engineer, built environmentally

sound dome homes in the desert because the construction of adobe domes was fairly

simple and the materials used were native to their land. During the construction process,

strategically placed ventilation openings were inserted to ensure the home was energy

efficient. Those openings kept the domes’ interior cooler than conventional houses. In

January 2005, Dulley, a reporter for the Post and Courier in Charleston, SC, stated that the

spherical dome shape is very energy efficient. However, he also stated that as changes are

made to the spherical shape, energy efficiency of the structure decreases.

In Palm Beach, Florida, Dolan (October 2005, pg. 1) reported that hurricane

resistant dome homes cost about 50% less to heat and cool than traditional rectilinear

homes of approximately the same size. Dolan also reported that Safe Harbor Dome Home

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Company was awarded the “Build Smart Certificate” for building an energy efficient dome

home.

Parker (October 2006) also reported that an energy efficient three story geodesic

dome home requires a two and one-half ton heating and cooling unit to adequately keep

the home at a comfortable temperature which is the typical size unit required for a much

smaller rectilinear home. He stated that according to the homeowner, the electric power

bill for this dome home was approximately $61 per month, which was considerably less

than his smaller rectilinear home.

The American Ingenuity Company describes their dome homes as being very energy

efficient (Busick, 2008). They achieve this efficiency by building their homes with

triangular panels that are created by enveloping polystyrene insulation with a concrete

outer layer that will not degrade over time. The near hemispherical shape of these homes

means reduced exposed surface area. Less energy escapes through the roof because the

dome is virtually airtight. The only insulation breaks are around the doors and windows,

unlike the insulation breaks between the load bearing walls and the wooden studs of the

traditional, rectilinear home.

Since the surface area of a geodesic dome home is less than that of a rectilinear

home of similar volume, the geodesic dome would require less exterior maintenance.

Maintenance costs of a concrete dome home will be minimal because the amount of

materials necessary will be less than that of a rectilinear home. The external concrete

construction on any structure ensures that rotting wood, mold, and mildew will not be a

problem. Since every geodesic dome home will not be built with a concrete roof, regular

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maintenance is required on the exterior and interior to ensure that the above contaminants

are kept to a minimum. This ensures that the home is a healthier, allergy-free place to live.

Thermal behavior or heat loss must be considered when designing and building a

home. Only the efficiency gained from the geometry of the dome is treated here. The heat

loss is proportional to the difference between the inside and outside temperature.

According to an article written for Comfortable Low Energy Architecture (CLEAR) in July

2009, the home is considered to be more energy efficient if the heat loss ratio is minimized.

A heat transfer model will be used to show heat loss is proportional to surface area and

since the surface area of a geodesic dome is less than the surface area of a rectilinear home

with the same volume, then the heat loss will be less for a geodesic dome compared to a

rectilinear home.

Sphericity

Sphericity is defined as the ratio of volume to surface area and determines the

roundness of a geometric shape (June 2009). As the frequency of the geodesic dome

increases, the sphericity ratio of the dome gets closer to that of a hemisphere (Kenner,

2003). As the sphericity ratio gets closer to that of a hemisphere, the heat loss is less for

the geodesic dome due to the lower surface area for similar volume of a rectilinear home.

According to Beals, Gross, and Harrell (2009), heat loss in animals is proportional to their

size and volume, their sphericity. They said that a small animal will lose heat faster due to

its volume to surface area ratio, so they need a higher metabolism to reduce the effects of

heat loss. In this study, the sphericity will be used as a measure of how closely the models

resemble a hemisphere.

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ILLUSTRATIVE EXAMPLES

The superior wind resistance and energy efficiency of geodesic domes when

compared to rectilinear homes is said to be due to their near hemispherical shape. In this

section, that hypothesis is tested using two simple mathematical models. One model is

used to determine the ratio of the force imparted by a straight line wind upon a geodesic

dome versus that imparted by an equal wind upon a rectilinear home. The other model is

used to find the heat transfer ratio between the two structures under some assumptions.

The hypothesis is tested by comparing the sphericity ratios of the structures to see if they

correspond to the wind resistant and heat flux ratios.

The terms geodesic dome and dome will be used synonymously throughout this

paper. The geodesic dome is defined by a geometric deformation of a platonic solid. To

better understand what a geodesic dome is, this section will demonstrate the procedure for

several domes. The length of the struts, the volume, surface area, and sphericity will be

computed for domes of various frequencies. Also included are computations which

compare the heat loss and wind resistance of geodesic domes to model rectilinear homes

with the same volume.

For this study, only geodesic domes formed from

octahedrons are investigated. Due to symmetry, only one-

eighth of the octahedron needs to be considered as seen in

Figure 9. The calculations are initially performed using a radius

of one and later scaled to typical house sizes. The

vertices are labeled as for the upper most vertex and and for the lower most

vertices of a one frequency dome, where the first digit in represents the row of the

Figure 9. A one frequency dome, one-eighth of the octahedron.

P11(0,0,1)

P21 (1,0,0) P22 (0,1,0)

(0,0,0)

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point and the second digit represents the position in that row from left to right. A one

frequency dome will be denoted by 1v. This notation will be used throughout this paper

with the number representing the frequency. Each time the frequency of the dome, ,

increases, a new row of points is added which will be labeled in the same fashion. The

labeling of the points on the bottom row of the dome will always begin with a digit one

greater than the dome’s frequency. Since frequency affects the geometric properties of the

dome, its effect on the sphericity is investigated.

To determine energy efficiency, the exterior of the structures, the thickness of the

exterior wall, and the R-value of the insulation of the structures are said to be identical for

the models with the same volume. This comparison does not include building materials,

construction methods, or the internal physics of the structures. The computation of the

heat flux ratio is the ratio of the heat used by a geodesic dome to that which is used by a

rectilinear home.

It will also be shown that geodesic dome homes have a lower profile to wind and are

more spherical when comparing the projected area of the various models included in this

study. Three different views of each rectilinear home are investigated and compared with

two different geodesic domes with the same volume and then compared to a hemisphere.

The wind speed is identical for all of the models. The ratio of the force from a wind

imparted on a geodesic dome to the force imparted on a rectilinear home will be used to

determine which structure is more resistant to wind.

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Forming the Geodesic Dome

Starting with a platonic solid, the edges of each face are subdivided by the desired

frequency, . For example, a frequency of four,

4v, means that the sides of the original faces

are divided into four equal parts. Next, these

new points along the edges are connected into

a mesh of equilateral triangles. One face of

an octahedron, the 4v dome, is shown in

Figure 10.

Next, the new points are moved radially outward until they are one unit from the

center of the base. As the vertices are moved outward, the triangular mesh is deformed

into a more spherical shape. In Figure 11, a

dome shape begins to appear after the original

points have been moved to be equidistant from

the center of the base. If the frequency is

increased, the dome appears to more closely

resemble a hemisphere. This will be shown to

be true using the sphericity of each dome.

A strut length is the length of one edge of the triangular panel which connects two

vertices. These triangular panels create a mesh of triangles that forms the geodesic dome.

As the frequency increases, the number of struts increase and their lengths decrease.

Recall Figure 1, which shows the struts of the triangular panels as they are joined together

during the construction of the dome.

Figure 11. A 4v dome after movement.

0 0.2 0.4 0.6 0.8 1

0

0.5

1

0

0.2

0.4

0.6

0.8

1

0

0.5

10 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Figure 10. A 4v dome with equilateral triangles before movement.

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Figure 12 is a drawing of an equilateral

triangular face of a one frequency 1v dome. For a 1v

dome, the edges or sides of the equilateral triangular

face are not subdivided. Therefore, the strut length of

each side of the dome is equal to . Since this is a 1v

dome, then the number of triangular panels on one

face is .

In Figure 13, the equilateral triangular face has

been subdivided into two equal parts at the

midpoints of the edges. Since this a 2v dome, there

will be four equilateral triangles on each face of the

dome, .

Table 1 Vertex Points for the 2v Dome

Point Original Coordinate Magnitude Stretched Coordinate

1

1

1

Figure 12. A 1v dome.

Figure 13. A 2v dome. 0 0.2 0.4 0.6 0.8 1

00.20.40.60.810

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

00.5

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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The magnitude, the distance from the center of the base (the origin) is computed for

each of the original points by finding the square root of the sum of the squares of the

coordinates. The coordinates of each original point are then divided by this magnitude to

determine the coordinates of the stretched point. The original points, the magnitude, and

the stretched points are listed in Table 1 and in Appendix B for the 2v dome.

Once the original points have been stretched

to ensure the distance from the center of the base is

one unit, the distance formula can be used to

determine the strut lengths. Figure 14 shows the

2v dome after the original points have been moved

outward.

For example, the length of the strut from the vertex at to the vertex at :

units.

This process can be used to find all of the strut lengths on one face of the dome. Through

the use of symmetry, the strut lengths can be determined on the other faces.

While there may be many different strut lengths for the given frequency, in practice only a

few of them are used to physically construct the dome. This may cause the dome to be

somewhat distorted. For the purpose of this study, all of the lengths are used. There are

only two different strut lengths for the 2v dome which are listed in Table 2. There are four

triangles on one face and the middle triangle is the only one that is equilateral with the

other three being isosceles.

Figure 14. A 2v dome after movement.

00.2

0.40.6

0.81

0

0.5

1

0

0.2

0.4

0.6

0.8

1

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Table 2 Strut Lengths for the 2v Dome

Table 3, on the next page, contains the coordinates of the original points, the

magnitude, and the stretched coordinates for the 4v dome. Since this is a 4v dome, there

are triangular faces on one side of the dome. Using the same procedure as the 2v

dome, the strut lengths are found for the 4v dome, which are listed in Table 4. The central

triangle is equilateral and the others are isosceles.

An EXCEL spreadsheet was created for all of the vertex points for all of the

triangular faces of one side of the dome for a limited number of dome frequencies.

Magnitude was computed for each point and the stretched points were listed. Additionally,

a MATLab computer program was created to compute the original points, magnitude, and

stretched points for any frequency. The MATLab computer program is listed in Appendix C.

Surface Area

The total surface area of a dome is the sum of the surface areas of the triangular

faces determined by the frequency of the dome. To compute the surface area, the vectors

defining two sides of the triangular face are computed by finding the difference between

each of the , , and coordinates of the vertices.

Vectors with the same strut length Strut Length

,

, ,

, ,

0.7654

,

,

1.0000

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Table 3 Vertex Points of the 4v Dome

Point Original Coordinate Magnitude Stretched Coordinate

1

1

1

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Table 4 Strut Lengths for the 4v Dome

Vectors with the same strut length Strut Length

, , , , , 0.3204

, 0.4472

, , , , , 0.4595

, , , , 0.4389

, , , , , 0.5176

, , 0.5774

Stewart, author of Calculus Concepts & Contexts (2004), defines the cross product of

two vectors to be a new vector with a magnitude equal to the product of the magnitude of

the two vectors and the sine of the angle between them. For example:

Geometrically, this is the area of the parallelogram defined by the two vectors. Thus the

area of the triangle would be

that value. Algebraically, the cross product is also

given by the matrix determinant.

where the coefficients and are given by

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.

The cross product, as a vector, is also a normal vector to the plane containing the triangular

panel, the surface area of which is one-half the magnitude of the cross- product.

(1)

.

The total surface area computations for the 1v, 2v, and 4v domes are shown in this

section. The total surface areas for the 6v and 12v domes are listed in this section and the

detail for the computations of the 6v dome can be found in Appendix D. The computations

for the 12v dome are not listed in the Appendix, due to their length, but can be quickly

computed using the MATLab computer program, which is in Appendix E.

Figure 15 shows a 1v dome. Using the points at the

vertices, and the vectors

from to the other two points are:

and

.

Their cross product is:

Figure 23

00.2

0.40.6

0.81

0

0.5

1

0

0.2

0.4

0.6

0.8

1

Figure 15. A 1v dome.

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After finding the coefficients, the surface area is

the magnitude of the cross product.

.

.

This is the surface area of one quarter of a 1v, one-frequency dome. Thus, the total surface

area of the 1v dome is ≈ 3.4641 square units.

The same procedure is followed as

above with the 1v dome to determine the

surface area for a 2v dome. Figure 16 shows

one face of the 2v dome after the original points

have been stretched. Table 1 lists the

coordinates of one face of the dome. Since this

is a 2v dome, there are four triangles on each

face of the original octahedron.

The surface area is computed for using the points at the vertices,

,

0

, and 0

. The vectors

and are:

0

1

and

0 .

Figure 16. A 2v dome after movement.

0

0.5

1

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1

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Their cross product is:

The surface area of this triangular panel is then

square units.

By symmetry, and have the same surface area.

The surface area is computed for using the points at the vertices,

,

, and

. The vectors

and are:

0 and

0

.

Their cross product is:

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The surface area of this triangular panel is then

square units.

Adding the surface areas for all of these triangles gives one fourth of the total

surface area the dome, which is approximately equal to 1.302218 square units. After

multiplying that value by four, the total surface area of the 2v dome is 5.2088758 square

units. The EXCEL spreadsheet containing these computations and calculations can be

found in Appendix B.

The same procedure is used to compute the surface area for all of the triangular

faces of the 4v dome as was used with the 1v and 2v domes. The total surface area for this

4v dome was computed using an EXCEL spreadsheet. The points used to determine the

vectors, the computations, and calculations are included in Appendix C. Since this is a 4v

dome, there will be 16 triangles on each face of the original octahedron. The total surface

area of the 4v dome is approximately equal to 5.9733266 square units.

Continuing the same procedure as for the previous domes, the total surface area for

this 6v dome was computed using the EXCEL program. The points used to determine the

vectors, the computations, and calculations are included in Appendix D. Since this is a 6v

dome, there will be 36 triangles on each face of the original octahedron. The total surface

area for the 6v dome is approximately equal to 6.1405485 square units.

The total surface area for this 12v dome was computed using the EXCEL program.

Due to its length, the table of points used to determine the vectors, the computations, and

calculations are not included in the appendix section. However, using MATLab, a computer

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27

program was written to compute the total surface area for a dome of any frequency, which

is more efficient and effective to use. The MATLab program is included in Appendix E. This

is a 12v dome, so there will be 144 triangles on each face of the original octahedron. The

total surface area for the 12v dome is approximately equal to 6.2467332 square units.

To summarize, the area of each triangular panel was calculated by taking one half of

the magnitude of the cross product of the two vectors. The EXCEL spreadsheet program

was used to calculate the surface area and volumes for domes with 1, 2, 4, 6, and 12

frequencies. Using EXCEL made the tedious computations easier to calculate for these

frequencies. However, there are infinitely many different frequencies of geodesic domes,

so a MATLab computer program was written to determine the surface area for any

frequency. The MATLab program can be found in Appendix E. A generic algorithm was

written to compute the surface area of any dome of any frequency. As the frequency of the

dome increases, as shown in Table 5, the surface area gets closer to that of a hemisphere of

radius one, which is 2 .

Table 5 Summary of Surface Area of Various Frequency Domes

Frequency Surface Area

1v 3.4641

2v 5.2088

4v 5.9733

6v 6.1406

8v 6.2019

10v 6.2309

12v 6.2467

16v 6.2626

20v 6.2700

Hemisphere 6.2832

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Volume

The volume of the geodesic dome is the sum of the areas under each triangular

panel. Recall from calculus, that the integral of a surface is used to find the volume under

that surface and over its projection in the -plane. Each triangle is contained in a plane

with the specific equation defined by the normal to the plane. Each triangular face consists

of three vertices connected by struts. For one face, let the points be defined as ,

, and .

From Calculus Concepts and Contexts (2004), is the equation for

the plane, where , , and correspond to the , , and components of the vector normal

to the plane. The value for is determined by substituting any point on the plane into the

above equation. After substituting a point into the equation, and solving

for , the resulting equation for the surface is to be integrated. Since the distance from the

base of the solid to each of the vertices is not equidistant, a double integral from calculus is

used to determine the area under the triangular panel.

The computer program Maple was used to determine the volume under a generic

triangular panel. Generic values, , were used for the vertices to get the formula for

the plane. The three vertices of the triangle are labeled , , and

. The vector from to is , while

to is . The normal, , is the cross product of the

vectors.

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As stated earlier, the coefficients , , and , for the vectors , , and are:

.

Using the point , is given by

.

The equation of the plane is then

.

The notation of is used for the slope of the line from the point , to the

other point , in the -coordinate plane. Hence, the following slope formulas are

used to determine the slopes of the boundaries of the projected region of each face of the

triangular panel onto the -plane (see Figure 17).

(2)

(3)

. (4)

The double integral for the volume under the triangular surface is as follows:

Using Maple these double integrals were simplified to:

. (5)

Figure 17. A projected region.

y

x

(x2, y2)

(x1, y1)

(x3, y3)

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Figure 18. A boundary of a projected region.

Now, Eq. (5) can easily be used in any spreadsheet program. For this paper, the

EXCEL spreadsheet was used to record the specific vertex points, input the above formula,

and compute the volume under each triangular face of the geodesic dome. After computing

the volume under each face, the volumes are then added to determine the total volume

under the dome for the given frequency.

The original points of a 1v geodesic dome do not get stretched because no division

has occurred. The original points are , , and

. Using Eq. (5) the volume under this one panel is

.

Since this is only one fourth of the 1v dome, then after multiplying that volume by four, the

total volume of the 1v dome is

cubic units.

To verify Eq. (5), the slopes were found to determine the

boundaries of the projected region of the sides of the triangle

onto the -plane, as defined above. Using the above points and

Eqs. (2), (3), and (4), the calculated slopes of the edges in the

projected region, as shown in Figure 18, are:

The coefficients , , and , of the normal vector to the plane of the triangular face

are calculated for the given points of the 1v dome:

y

x (x1, y1) (x2, y2)

(x3, y3)

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.

To find the constant value, , the point was used in Eq. (6):

. (6)

, then .

After solving Eq. (6) for , the equation to be integrated with respect to and is:

.

The integral for the volume is

(7)

Since one side, and is aligned with the -axis, the second integral in Eq. (7) is equal to

zero.

Integrating Eq (7):

.

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This is the same value as given by Eq. (5).

The volume was found using the vertices in a spreadsheet for a limited number of

dome frequencies. Since other frequencies must be considered, a MATLab computer

program was written in which the formula for volume was coded to compute the volume

for any frequency dome. The total volumes found are listed in Table 6. As the frequency of

the dome increases, the volume of the dome gets closer to the volume of a hemisphere of

radius one unit which is 2.094395 cubic units. The graph in Figure 19 shows that as the

frequency of the geodesic dome increases, the volume of the dome gets very close to the

volume of a hemisphere, defined by the horizontal asymptote of 2.094395.

Table 6 Volume of the Geodesic Dome

Frequency Volume

1v .666667

2v 1.471404

4v 1.909744

6v 2.008834

8v 2.045532

10v 2.062900

12v 2.072440

16v 2.082000

20v 2.086446

32v 2.091300

Hemisphere 2.094395

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Figure 19. Graph of volume given the frequency of the dome.

Sphericity

Sphericity is the roundness of any shape determined by the ratio of its volume to its

surface area. It provides a measure the closeness a geometric shape or an object is to a

sphere. The volume and surface areas are used to determine the sphericity of the dome for

the various frequency domes. For a dome to have perfect sphericity, it would have the

same ratio as that of the hemisphere. For a hemisphere, the sphericity is

, where is the radius of the hemisphere. For this study, the

radius used is one; therefore the sphericity of the domes of various frequencies should get

closer to

as the frequency increases.

The sphericity was computed for the various frequency domes using the EXCEL

spreadsheet program. However, by using the MATLab computer program, the computation

of the sphericity ratio is possible for any frequency dome. Table 7 gives the frequency and

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the sphericity of various domes of radius one, and shows that as the frequency of the dome

increases, the sphericity ratio gets very close to the sphericity ratio of a hemisphere.

Table 7 Sphericity of Geodesic Domes with Radius One Unit

Dome Frequency Sphericity ratio Times by 3

1v .19245 .57735

2v .28248 .84744

4v .31971 .95914

6v .32142 .96427

8v .32982 .98947

10v .33108 .99323

12v .33176 .99529

16v .33245 .99734

20v .33277 .99830

Hemisphere .33334 1.0

This study will include four rectilinear home

models with different volumes. There are one and

two-story models. The dimensions of the rectilinear

models are 30 ft by 30 ft and 30 ft by 15 ft. The wall

height of the one-story home is 10 ft and 20 ft for the

two-story home. The pitch of the roof is 12, so the

roof height is 10 ft for the house and 5 ft for the house. Figure 20

shows a one-story house with volume of 5625 cubic feet and surface area of

1515.83 square feet. The volume of the two-story house with the same length and width is

Figure 20. A 30x15x10 Rectilinear home.

05

1015

010

2030

0

5

10

15

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10,125 cubic feet with surface area of 2415.83 square feet.

In Figure 21, this one-story rectilinear house

has length and width measurements of 30 ft and

height of 10 ft. The volume is 13,500 cubic feet and

the surface area is 2581.67 square feet. The volume

of the two-story house with the same length and

width is 22,500 cubic feet with surface area of

3781.67 square feet.

Geodesic domes of various frequencies with and without 4 ft risers are included in

this investigative study. To make the comparison as fair as possible, the same volume was

used for the models compared. Table 8 shows the ratio of the sphericity of geodesic domes

without risers compared to the sphericity of the hemisphere with the same volume. Four

Table 8 Dome Sphericity without Four Foot Riser Compared to Hemisphere Sphericity

Dome Hemisphere

Volume Freq Radius Surface

Area Sphericity

Radius Sphericity

Dome to Hemisphere

13500 4v 19.19 2200.20 6.14 18.61 6.20 0.9891

5625

14.33 1227.40 4.58 13.90 4.63 0.9891

22500

22.75 3092.80 7.28 22.07 7.36 0.9891

10125

17.44 1816.20 5.57 16.91 5.64 0.9891

13500 8v 18.76 2182.10 6.19 18.61 6.20 0.9973

5625

14.01 1217.30 4.62 13.90 4.63 0.9973

22500

22.24 3067.40 7.34 22.07 7.36 0.9973

10125

17.04 1801.30 5.62 16.91 5.64 0.9973

0 5 10 15 20 25 30

010

2030

0

5

10

15

20

Figure 21. A 30x30x10 Rectilinear home.

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and eight frequency domes with different volumes are included. The closer the ratio is to

one, the more closely the 4v and 8v domes resemble the hemisphere. Similarly, Table 9

shows the ratio of the sphericity of the geodesic dome with risers compared to the

sphericity of the hemisphere. As expected, all of the ratios of the 4v and 8v domes to the

hemisphere are very close to one. The ratios for the domes without risers are independent

of the volume, as all of the 4v domes without risers investigated have a ratio of 0.9891,

while 8v domes without risers have a ratio of 0.9973. Since these values are closer to one

than the corresponding domes with risers, they more closely resemble a hemisphere.

Table 9 Dome Sphericity with four foot riser compared to Hemisphere Sphericity

Dome Hemisphere

Volume Freq Radius Surface

Area Sphericity

Radius Sphericity

Dome to Hemisphere

13500 4v 17.28 2214.70 6.10 18.61 6.20 0.9826

5625

12.49 1243.30 4.52 13.90 4.63 0.9764

22500

20.81 3106.10 7.24 22.07 7.36 0.9849

10125

15.55 1831.40 5.53 16.91 5.64 0.9809

13500 8v 16.93 2202.10 6.13 18.61 6.20 0.9882

5625

12.24 1236.90 4.55 13.90 4.63 0.9815

22500

20.38 3087.50 7.29 22.07 7.36 0.9908

10125

15.23 1821.30 5.56 16.91 5.64 0.9863

Table 10 shows the sphericity of the rectilinear home compared the hemisphere of the

same volume. In this comparison, the one-story square, rectilinear home is closest to the

hemisphere of the same volume.

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Table 10 Sphericity of the rectilinear home

House Hemisphere

Volume Size Surface

Area Sphericity

Radius Sphericity

House to Hemisphere

13500 30x30x10 2581.67 5.23 18.61 6.20 0.8429

5625 30x15x10 1515.83 3.71 13.90 4.63 0.8009

22500 30x30x20 3781.67 5.95 22.07 7.36 0.8089

10125 30x15x20 2415.83 4.19 16.91 5.67 0.7436

The domes with risers have a wider ratio range and it is dependent on volume. The

greater the volume of the dome with risers, the closer it resembles a hemisphere. While

the ratios are close to one for the domes with and without risers, the ratio differences for

the various domes investigated are minimal. The sphericity shows that whether the domes

investigated had a riser wall or not, the geodesic domes more closely resemble a

hemisphere than any of the rectilinear homes investigated.

Energy Efficiency

Heat Loss

In this section, one measure of energy efficiency, the ratio of the conductive heat

loss of geodesic domes to that of a rectilinear home is considered. Only conductive heat is

considered, which means the air is not moving and there is no radiative heat transfer.

Under these assumptions, the calculations show the percentage of heat savings. Since the

geodesic dome closely resembles a hemisphere, the conjecture is that the heat loss ratio of

the dome home to the rectilinear home will be less than one. If, for example, the ratio is

0.85, then the geodesic dome would use 85% of the heat that the rectilinear home uses or

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15% less heat.

For the comparison to be fair between the geodesic dome homes and the rectilinear

homes, the volumes of the homes, the difference between the inside and outside

temperatures of the homes, the R-values of insulation, and the exterior wall thicknesses are

all taken to be the identical. The one dimensional steady state heat equation taken from

Introduction to Heat Transfer (1990), assuming only conductive heat transfer is

,

where is the temperature and is the distance from the inside of the exterior wall. The

temperature equation after integrating twice is:

where and are constants. If the temperature outside is 95 and the temperature inside

is 80 and the wall is 8 units thick, then

and

Therefore, the temperature is Note that

degrees

per unit. This equation depends only on the thickness of the wall and temperature

difference, so it will be the same for equal wall thicknesses.

The heat flux through the wall is given by

,

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where

, where is the insulation value for the “R” rating, and is the surface area of

the structure. Let the heat flux for the rectilinear home be given as

and the heat flux for the geodesic dome be given as

.

Let equal the heat flux ratio or efficiency then

.

If the insulation value and thickness of the exterior walls are said to be identical,

then , and

. Therefore, the heat flux ratio simplifies to the ratios of the

surface areas of the dome and the rectilinear home is

. (8)

Since the volumes are taken to be equal, Eq. (8) guarantees the most hemispherical homes,

by this measure, will be the most energy efficient. This is because Eq. (8) is also the ratio of

sphericities, if the volumes are equal.

The following computations compare the heat flux values for a 4v and 8v geodesic

domes with the model rectilinear homes. Table 11 records the computations of the heat

flux ratios when comparing 4v and 8v geodesic domes with and without a 4-foot riser wall

to rectilinear homes and a hemisphere with the same volumes.

When looking at the dome to home results in Table 11, one case to be considered is the

comparison of the 4v dome to the 8v dome to the same rectilinear home. The lower the

ratio value, the more energy efficient the dome is said to be. The table shows that the most

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Table 11 Heat Flux Ratios of Geodesic Domes to Rectilinear Home

Dome Frequency

Rectilinear Home

Volume Riser Wall

Dome

Surface Area

Rectilinear Surface Area

Dome to Home

4v 30x30x10 13500 No 2200.20 2581.67 0.8522

4v

Yes 2214.70 2581.67 0.8579

4v 30x15x10 5625 No 1227.40 1515.83 0.8097

4v

Yes 1243.30 1515.83 0.8202

4v 30x30x20 22500 No 3092.80 3781.67 0.8178

4v

Yes 3106.10 3781.67 0.8214

4v 30x15x20 10125 No 1816.20 2415.83 0.7518

4v

Yes 1831.40 2415.83 0.7581

8v 30x30x10 13500 No 2182.10 2581.67 0.8452

8v

Yes 2202.10 2581.67 0.8530

8v 30x15x10 5625 No 1217.30 1515.83 0.8031

8v

Yes 1236.90 1515.83 0.8160

8v 30x30x20 22500 No 3067.40 3781.67 0.8111

8v

Yes 3087.50 3781.67 0.8164

8v 30x15x20 10125 No 1801.30 2415.83 0.7456

8v

Yes 1821.30 2415.83 0.7539

energy efficient dome, when compared to the rectilinear home with the same volume, is the

8v dome without a riser wall. For example, the ratio for the 8v dome to the rectilinear two-

story home is 74.56% which shows that the geodesic dome uses 74.56% of the heat that

the two-story, rectangular, rectilinear home uses. The ratios in Table 11 also show that the

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4v dome without a riser wall when compared to the same house uses 75.18% of the heat

that the rectilinear home uses and is 24.82% more energy efficient. Between the two

domes without a riser wall, the ratios show that the 8v dome is more energy efficient.

Therefore, the conclusion in this case is that the greater the frequency of the dome, the

more energy efficient it is.

When comparing the 8v dome without a riser wall to the 8v dome with a riser wall

with the same volume, the ratios show that all of the domes without a riser wall are more

energy efficient than the domes with a riser wall for all of the volumes included in this

study. The greater the ratio the more heat the model loses and the less energy efficient it is.

The data shows similar results for the 4v dome with and without a riser wall. Therefore, in

this comparison, domes without a riser wall are more energy efficient.

When looking at the different rectilinear models included in this study and recorded

in Table 11, the square rectilinear one-story home is more energy efficient than the square

two-story home. Comparing the surface area of the one-story home with the

surface area of the two-story home, the ratio shows that the one-story home

is more energy efficient because it uses 68.268% less heat than the two-story home uses.

Comparing the surface area of the one-story home with the surface area of

the two-story home, the ratio shows that the one-story home is more energy

efficient because it uses 62.746% less heat than the two-story home uses. Therefore, the

one-story, , rectilinear home is said to be more energy efficient.

In conclusion, the ratios in Table 11 show that since the heat flux ratio is a value less

than one, the 8v geodesic dome home without a riser wall is said to be more energy

efficient than the rectilinear homes. Mathematically it is shown that as the frequency of the

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dome increases, the more energy efficient the geodesic dome. It has also been shown that

as the frequency increases, the sphericity ratio of the dome approaches the sphericity ratio

of the hemisphere. As the frequency of the dome increases, the geodesic dome becomes

more spherical and more energy efficient.

Wind resistance

The force imparted on a structure by a straight line wind is approximated by using

the formula:

where is the force imparted on the home, is the force per unit area of the wind, and

is the projected surface area of the structure perpendicular to the wind. For this paper,

the wind speed is said to be equal for both structures. Let equal the force imparted upon

the rectilinear structure and equal the force on the geodesic dome then

and .

The ratio of the force on the dome to the force on the rectilinear home is then

. (9)

A ratio value of less than one indicates that the geodesic dome home is more wind resistant

while a ratio value of greater than one indicates that the rectilinear home is more wind

resistant.

This study will limit the projected area computations to that of three views of one

and two-story rectilinear homes with a height of ten units per story. Since the visual view

of the one and two-story models are similar, only the one-story model is shown in the

figures. The frontal and left side view is shown in Figure 22 of the rectilinear home.

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View 1, shown in Figure 23, is at an angle perpendicular to the diagonal of the rectangular

base of the home. View 2 is the front of the house with the triangular gable end visible (see

Figure 24). View 3 is the side of the house and

lengthwise view of the roof and can be seen in

Figure 25. The figures show the height of 10

feet for the one-story house. Using the models

in the different views, the angle for each size

home is the same for each view, but the height

of the two-story house is 20 ft. While it is necessary

to consider multiple views of the rectilinear homes, only one view is necessary when

viewing the geodesic dome home. While there may be some variations to the geodesic

dome structure, the geodesic dome in this study closely resembles a hemisphere and will

produce approximately the same projected area at any angle. The projected area of the

geodesic dome will be computed with and without the four-foot riser wall.

To measure the wind resistance

ratio, the projected area of the structure is

the visible area seen at a fixed angle, like

taking a picture. The projected area of the

geodesic dome is calculated using the

MATLab program which computes the

projected area for the dome with a specific

radius either with or without a riser wall.

The radius is chosen so that the volume is the same as the rectilinear home.

Figure 22. A 30x30x10 Rectilinear home.

0 10 20 300102030

0

2

4

6

8

10

12

14

16

18

20

Figure 23. View 1: A 30x30x10 Rectilinear home.

0 5 10 15 20 25 30

010

2030

0

5

10

15

20

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Figure 22 shows a drawing of a one-story model rectilinear home used in this paper

with outside dimensions in feet, and with a roof pitch ratio of 8:12. The

volume of this one-story home equals cubic feet.

View 1 in Figure 23 is the left and front sides of the home visible at an angle

perpendicular to the diagonal of the rectangular base. The projected area of the

home is computed by adding the surface area of the two faces with lengths 30 ft,

multiplying by and adding that value to the area of the roof, which appears

trapezoidal at this angle. The height of the roof is 10 ft and lengths of 60 feet and 30 feet

multiplied by cosine of the angle ( ).

One-story:

square feet.

Two-story:

square feet.

The projected area of this view for the one-story house equals square feet. The

projected area of this view for the two-story house equals square feet.

Using View 1 in Figure 23, the left and front sides of the home are visible at a 45

degree angle, but for the rectilinear home the projected area of the roof is visible at

a degree angle. At this angle, the roof appears to be trapezoidal. The projected

area is computed by taking the square root of the sum of the squares of the length, 30 ft and

width, 15 ft and multiplying that value by the height of 10 added to the area of the

trapezoidal roof. At this angle, the trapezoidal roof has lengths of the square root of the

sum of the squares of the length, 30 ft and width, 15 ft plus length, 30 ft multiplied by

. After adding the base lengths together, divide by two and multiply by the

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roof height of 5.

One-story:

square feet.

Two-story:

square feet.

The projected area of this view for the one-story house equals 486.344756 square feet.

The projected area of this view for the two-story house equals 821.75495 square feet.

View 2, as seen in Figure 24, is the

frontal view of the home. The projected area

equals 450 square feet for the one-story

square house which includes the

rectangular wall and the triangular gable end.

The projected area of this view for the two

story house equals 750 square feet.

One-story:

square feet.

Two-story:

square feet.

The projected area of the one-story rectilinear home in Figure 24 is 187.5

square feet and the projected area of this view for the two-story house equals 337.5 square

feet. Since the gable end is on the side of the home where the length is 15 ft, the roof height

is only 5 feet.

One-story:

square feet.

Two-story:

square feet.

Figure 24. View 2: A 30X30X10 Rectilinear home.

0 5 10 15 20 25 30020

40

0

2

4

6

8

10

12

14

16

18

20

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View 3 in Figure 25 shows the projected area

of the left side of the home which includes the left

side of the home and the lengthwise side of the roof.

The projected area of this view for the one-story

house equals 600 square feet and 900

square feet for the two-story house.

One-story: square feet.

Two-story: square feet.

The projected area of the rectilinear homes from this view are

One-story: square feet.

Two-story: square feet.

Tables 12 and 13 show the wind resistance ratios when comparing the projected

area of 4v and 8v domes with and without risers to three different views of rectilinear

homes, and then to hemispheres with the same volumes. The discussion of this study is

limited to the following projected area ratio comparisons: the 4v dome compared to the

different rectilinear views, the 8v dome compared to the different rectilinear views, the 4v

dome compared to the hemisphere, the 8v dome compared to the hemisphere, the different

views of the rectilinear home compared to the hemisphere of same volume, and rectilinear

homes compared with one another.

When comparing the 4v dome to the different views of the rectilinear home,

Table 12 shows that the 4v dome without a riser wall is most wind resistant when

compared to View 1 of the homes with volume 10,125 cubic feet, which results in a ratio of

0.565. Therefore, the 4v dome is 56.5% more wind resistant than the two-story

020

40

051015202530

0

2

4

6

8

10

12

14

16

18

20

Figure 25. View 3: A 30x30x10 Rectilinear home.

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rectangular home. When comparing the 4v dome with a riser wall to the same rectilinear

home in View 1, the ratio is 0.6003, which is slightly different, but greater than the ratio

without the riser wall. The dome is 60% more wind resistant than the rectilinear home.

Therefore, the 4v dome without a riser wall is slightly more wind resistant than the dome

with a riser wall.

When comparing the 8v dome to the different views of the rectilinear home,

Table 13 shows that the 8v dome without a riser wall is most wind resistant when

compared to View 1 of the homes with volume 10,125 cubic feet, which results in a ratio of

0.5512. Therefore, the 8v dome is 55.12% more wind resistant than the two-story

rectangular home in View 1. When comparing the 8v dome with a riser wall to the same

rectilinear home in View 1, the ratio is 0.58868. In general, not having a riser wall makes

the dome more wind resistant. Also, it can be said that the 8v dome without a riser wall is

more wind resistant than the 4v dome.

To briefly mention View 2, Table 12 shows the ratios for the 4v dome without a

riser wall and Table 13 shows the ratios for the 8v dome without a riser wall are similar

and are close to but greater than one. The ratio for the 4v dome without a riser wall is

1.67328 and with a riser wall the ratio is 1.80295. The ratio for the 8v dome without a

riser wall is 1.63260 and with a riser wall the ratio is 1.76934. This means that in this case

for this view, the rectilinear home is more wind resistant because all of the ratios are

greater than one. According to this comparison for this view, the best rectilinear home

choice said to be more wind resistant is the rectangular, one-story rectilinear home with

volume 5625 cubic feet.

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Table 12 Projected Area Ratio Computations of 4v Domes to Rectilinear Homes.

Rectilinear Home

Volume Riser Wall

PA Dome Rectilinear

Home PA Home

PA Hemisphere

Dome to Home

30x30x10 13500 No 562.40 View 1 742.46 544.05 0.7575

30x30x20 22500 No 790.58 View 1 1166.73 764.78 0.6776

30x15x10 5625 No 313.74 View 1 486.34 303.50 0.6451

30x15x20 10125 No 464.26 View 1 821.75 449.10 0.5650

30x30x10 13500 No 562.40 View 2 450.00 544.05 1.2498

30x30x20 22500 No 790.58 View 2 750.00 764.78 1.0541

30x15x10 5625 No 313.74 View 2 187.50 303.50 1.6733

30x15x20 10125 No 464.26 View 2 337.50 449.10 1.3756

30x30x10 13500 No 562.40 View 3 600.00 544.05 0.9373

30x30x20 22500 No 790.58 View 3 900.00 764.78 0.8784

30x15x10 5625 No 313.74 View 3 450.00 303.50 0.6972

30x15x20 10125 No 464.26 View 3 750.00 449.10 0.6190

30x30x10 13500 Yes 594.13 View 1 742.46 544.05 0.8002

30x30x20 22500 Yes 827.69 View 1 1166.73 764.78 0.7094

30x15x10 5625 Yes 338.05 View 1 486.34 303.50 0.6951

30x15x20 10125 Yes 493.33 View 1 821.75 449.10 0.6003

30x30x10 13500 Yes 594.13 View 2 450.00 544.05 1.3203

30x30x20 22500 Yes 827.69 View 2 750.00 764.78 1.1036

30x15x10 5625 Yes 338.05 View 2 187.50 303.50 1.8030

30x15x20 10125 Yes 493.33 View 2 337.50 449.10 1.4617

30x30x10 13500 Yes 594.13 View 3 600.00 544.05 0.9902

30x30x20 22500 Yes 827.69 View 3 900.00 764.78 0.9197

30x15x10 5625 Yes 338.05 View 3 450.00 303.50 0.7512

30x15x20 10125 Yes 493.33 View 3 750.00 449.10 0.6578

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Table 13 Projected Area Ratio Computations of 8v Domes to Rectilinear Homes

Rectilinear Home

Volume Riser Wall

PA Dome Rectilinear

Home PA Home

PA Hemisphere

Dome to Home

30x30x10 13500 No 548.72 View 1 742.46 544.05 0.7391

30x30x20 22500 No 771.34 View 1 1166.73 764.78 0.6611

30x15x10 5625 No 306.11 View 1 486.34 303.50 0.6294

30x15x20 10125 No 452.97 View 1 821.75 449.10 0.5512

30x30x10 13500 No 548.72 View 2 450.00 544.05 1.2194

30x30x20 22500 No 771.34 View 2 750.00 764.78 1.0285

30x15x10 5625 No 306.11 View 2 187.50 303.50 1.6326

30x15x20 10125 No 452.97 View 2 337.50 449.10 1.3421

30x30x10 13500 No 548.72 View 3 600.00 544.05 0.9145

30x30x20 22500 No 771.34 View 3 900.00 764.78 0.8571

30x15x10 5625 No 306.11 View 3 450.00 303.50 0.6803

30x15x20 10125 No 452.97 View 3 750.00 449.10 0.6040

30x30x10 13500 Yes 582.38 View 1 742.46 544.05 0.7843

30x30x20 22500 Yes 810.86 View 1 1166.73 764.78 0.6950

30x15x10 5625 Yes 331.75 View 1 486.34 303.50 0.6821

30x15x20 10125 Yes 483.75 View 1 821.75 449.10 0.5887

30x30x10 13500 Yes 582.38 View 2 450.00 544.05 1.2942

30x30x20 22500 Yes 810.86 View 2 750.00 764.78 1.0812

30x15x10 5625 Yes 331.75 View 2 187.50 303.50 1.7693

30x15x20 10125 Yes 483.75 View 2 337.50 449.10 1.4333

30x30x10 13500 Yes 582.38 View 3 600.00 544.05 0.9706

30x30x20 22500 Yes 810.86 View 3 900.00 764.78 0.9010

30x15x10 5625 Yes 331.75 View 3 450.00 303.50 0.7372

30x15x20 10125 Yes 483.75 View 3 750.00 449.10 0.6450

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The same conclusion is made about View 3 of the rectilinear home as was made

about View 1 when comparing it to the domes investigated. When comparing the 4v dome

to View 3, Table 12 shows that the 4v dome without a riser wall is most wind resistant

when compared to View 3 of the homes with volume 10,125 cubic feet, which results in a

ratio of 0.61902. With this ratio, the 4v dome is 61.9% more wind resistant than the two-

story rectilinear home. When comparing the 4v dome with a riser wall to the same

rectilinear home in View 3, again, the 4v dome without a riser wall is more wind resistant.

Table 13 shows that the 8v dome without a riser wall is most wind resistant when

compared to View 3 of the homes with volume 10,125 cubic feet. The 8v dome is 60.395%

more wind resistant than the two-story rectangular home in View 3. With the riser wall,

the dome is 64.5% more wind resistant than the rectilinear home. Again, the 8v dome

without a riser wall is said to be more wind resistant than the 8v dome with a riser wall.

Also, it can be said that the 8v dome with or without a riser wall is more wind resistant

than the 4v dome for View 3.

According to the ratios in Tables 12 and 13, the conclusion made is that the 4v dome

without a riser wall is more wind resistant than the rectilinear homes as seen in views one

and three, the 8v dome without a riser wall is more wind resistant that the 4v dome.

Earlier, it was shown that the sphericity of the 8v dome is closer to the sphericity of the

hemisphere than the rectilinear home which implies the dome is more spherical. Likewise,

the same pattern is followed, so in general, it can be said that the more closely a structure

resembles a hemisphere, the more wind resistant it is. However, this is only true “on

average” since in one of the views (View 2) the rectilinear home was more wind resistant.

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CONCLUSION

Since verifiable statistical data do not exist, the investigator cannot make an

inference that geodesic dome homes are more resistant to hurricanes and other natural

phenomena. However, simulations and observations from hurricane disaster scenes do

suggest that the geodesic dome structures suffer far less destruction than rectilinear

structures. The hypothesis is that geodesic domes are more energy efficient and more

wind resistant because they more closely resemble a hemisphere.

The frequency of the geodesic dome was a very vital part of this study, it was

revealed that by letting the frequency of the dome equal , and then the number of

triangles on one face of the dome equals . As the frequency increased, the number of

triangles increased, and the dome becomes more hemispherical due to this increase.

Sphericity ratios of various geodesic domes with and without a riser walls and

rectilinear models were computed and compared with the sphericity ratios of various

hemispheres. These ratios are listed in Table 14. The sphericity of the 8v dome without a

riser wall is 99.7% and is closest to the sphericity of the hemisphere of 100%, which shows

the 8v dome to be the most spherical of all of the models investigated. When the dome

includes a riser wall, then volume is a factor that must be considered. The largest 8v dome

with a riser wall is next closest with a ratio of 99.08%. The 8v domes with and without a

riser wall are more spherical than the 4v dome, then this demonstrates that the greater the

frequency of the dome, the more spherical it is. When volume is a factor, then the greater

the volume of dome with the riser wall, and the more spherical the dome is. The data from

this part of the investigation as shown in Table 14, shows that domes with or without riser

walls more closely resemble a hemisphere than the rectilinear homes of the same volume.

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Various rectilinear models were included in this investigation. The sphericity ratios

of the rectilinear homes are recorded in Table 14 which shows the results that range from

74.36% to 84.29%. The one-story, square rectilinear home is shown to be most spherical

than any of the rectilinear models investigated with a ratio of 84.29%. The least spherical

is the , two-story rectangular home with a ratio of 74.36%.

Table 14 Most to Least Spherical Model

Most spherical Dome Frequency Volume/Riser Wall Ratio to Hemisphere

1 8v No 0.9973

2 8v 22500/yes 0.9908

3 4v No 0.9891

4 8v 13500/yes 0.9882

5 8v 10125yes 0.9863

6 4v 22500/yes 0.9849

7 4v 13500/yes 0.9826

8 8v 5625/yes 0.9815

9 4v 10125yes 0.9809

10 4v 5625/yes 0.9764

11 30x30x10 13500 0.8429

12 30x30x20 22500 0.8089

13 30x15x10 5625 0.8009

14 30x15x20 10125 0.7436

All of the calculations to determine energy efficiency can be seen in Table 11, and

Table 15 which shows the most to least energy efficient structure was created using those

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calculations. In Table 15, the ratio of the 4v dome without a riser wall to the hemisphere is

1.011, but the ratio of the 8v dome without a riser wall is 1.0027. While both of these

ratios are very close to one, the ratio of the 8v dome is closer to one and is said to more

energy efficient. A ratio of 1.0027 means the dome would use 0.27% more energy than a

hemisphere of the same volume. Table 15 also shows that all of the domes with or without

a riser wall are more energy efficient than any of the rectilinear homes. When comparing

the 8v domes with the 4v domes of the same volume, the 8v domes with and without riser

walls are more energy efficient. Therefore, the 8v dome is shown to be the most energy

efficient of all of the models investigated. Since Tables 14 and 15 are in the same order,

this means that the hypothesis that the more spherical implies more energy efficient is

true. In fact, this is a direct consequence of Eq. (8).

In this study, the wind resistance ratio was calculated for various geodesic dome

models compared to various rectilinear models. Only one view of the geodesic dome and

hemisphere were investigated, but there are three different views that are included in this

investigation of the various rectilinear models. All of the calculations to determine the

wind resistance ratios can be seen in Table 12 for the 4v dome and Table 13 for the 8v

dome. Table 16 shows the structures which are arranged from most wind resistant to least

wind resistant when comparing all of the investigated models with the wind resistance of a

hemisphere.

According to Table 16, View 2 of the one-story rectangular home is shown to have

the smallest ratio when compared to the hemisphere which shows that this view is more

wind resistant than the other models investigated. The side of the rectangular home visible

in View 2 has the smallest amount of projected area of all of the models investigated.

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Table 15 Most to Least Energy Efficient Model

Most energy efficient

Dome/Home Volume/

Riser Wall Dome/Home Surface Area

Hemisphere Surface Area

Ratio to Hemisphere

1 8v No 3067.40 3059.12 1.0027

2 8v 22500/yes 3087.50 3059.12 1.0093

3 4v No 3092.80 3059.12 1.0110

4 8v 13500/yes 2202.10 2176.19 1.0119

5 8v 10125/yes 1821.30 1796.41 1.0139

6 4v 22500/yes 3106.10 3059.12 1.0154

7 4v 13500/yes 2214.70 2176.19 1.0177

8 8v 5625/yes 1236.90 1214.01 1.0189

9 4v 10125/yes 1831.40 1796.40 1.0195

10 4v 5625/yes 1243.30 1214.01 1.0241

11 30x30x10 13500 2581.67 2176.19 1.1863

12 30x30x20 22500 3781.67 3059.12 1.2362

13 30x15x10 5625 1515.83 1214.01 1.2486

14 30x15x20 10125 2415.83 1796.40 1.3448

Only View 2 of the rectilinear homes investigated will fare better when experiencing a

straight line wind than the geodesic dome.

When comparing the investigated domes and homes to the hemisphere, the ratio of

the 8v dome without a riser wall is 1.0086, which is closest to one. Therefore, the geodesic

dome home with the greater frequency is said to be more wind resistant on average, in two

of the three views used than a rectilinear home. The ratio of 1.0086 means the dome would

experience a 0.86% greater force from a straight line wind, neglecting aerodynamics, than a

hemisphere of the same volume.

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Table 16 Most to Least Wind Resistant Model

Most wind resistant

Dome/Home Volume/Riser Wall or View

Projected Area Dome/Home

Projected Area Hemisphere

Ratio to Hemisphere

1 30x15x10 5625/V2 187.50 303.50 0.6178

2 30x15x20 10125/V2 337.50 449.10 0.7515

3 30x30x10 13500/V2 450.00 544.05 0.8271

4 30x30x20 22500/V2 750.00 764.78 0.9807

5 8v No 548.72 544.05 1.0086

6 4v No 313.74 303.50 1.0337

7 8v 22500/Yes 810.86 764.78 1.0603

8 8v 13500/Yes 582.38 544.05 1.0705

9 8v 10125/Yes 483.75 449.10 1.0771

10 4v 22500/yes 827.69 764.78 1.0823

11 4v 13500/Yes 594.13 544.05 1.0921

12 8v 5625/Yes 331.75 303.50 1.0931

13 4v 10125/Yes 493.33 449.10 1.0985

14 30x30x10 13500/V3 600.00 544.05 1.1028

15 4v 5625/Yes 338.05 303.50 1.1138

16 30x30x20 22500V3 900.00 764.78 1.1768

17 30x30x10 13500/V1 742.46 544.05 1.3647

18 30x15x10 5625/V3 450.00 303.50 1.4827

19 30x30x20 22500/V1 1166.73 764.78 1.5256

20 30x15x10 5625/V1 486.34 303.50 1.6024

21 30x15x20 10125/V3 750.00 449.10 1.6700

22 30x15x20 10125/V1 821.75 449.10 1.8298

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In conclusion, from Table 14 and earlier calculations, the higher the frequency of the

dome the more spherical is the dome. By comparing Tables 14 and 15, the more spherical

the model the more energy efficient is the model. Similarly, by comparing Tables 14 and

16, the more spherical the model the more wind resistant (on average) is the model.

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APPENDICES

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Appendix A Calculations for the One-Frequency Dome

Original Points

Magnitude New Points

x y Z

x y z

P11 0.00 0.00 1.00

1.00 0.00 0.00 1.00 P21 1.00 0.00 0.00

1.00 1.00 0.00 0.00

P22 0.00 1.00 0.00

1.00 0.00 1.00 0.00

Volume

Surface Area

P11 0.00 0.00 1.00

i j k P21 1.00 0.00 0.00

P11P21 1.00 0.00 -1.00

P22 0.00 1.00 0.00

P21P22 -1.00 1.00 0.00

Volume of one face=0.167 Surface Area of one face =0.866 Total Volume of 1v Dome=0.667 Total Surface Area of 1v Dome=3.464

Sphericity of the 1v dome=0.193

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Appendix B Calculations for the Two-Frequency Dome

Original Points Magnitude New Points

x y z

x y z

P11 0.00 0.00 1.00

1.0000

0.0000 0.0000 1.0000

P21 0.50 0.00 0.50

0.7071

0.7071 0.0000 0.7071

P31 1.00 0.00 0.00

1.0000

1.0000 0.0000 0.0000

P22 0.00 0.50 0.50

0.7071

0.0000 0.7071 0.7071

P32 0.50 0.50 0.00

0.7071

0.7071 0.7071 0.0000

P33 0.00 1.00 0.00

1.0000

0.0000 1.0000 0.0000

Volume of symmetric faces Surface Area of symmetric faces

P11 0.0000 0.0000 1.0000

i j k

P21 0.7071 0.0000 0.7071

P11P21 0.7071 0.0000 -0.2929

P22 0.0000 0.7071 0.7071

P21P22 -0.7071 0.7071 0.0000

One face =0.2012 Four faces=0.81 One face =0.2897 Four faces=1.16

P21 0.7071 0.0000 0.7071

i j k

P31 1.0000 0.0000 0.0000

P31P21 0.2929 0.0000 -0.7071

P32 0.7071 0.7071 0.0000

P32P31 -0.2929 0.7071 0.0000

Two faces =0.0244 Eight faces=0.2 Two Faces=0.2897 Eight faces=2.32

P21 0.7071 0.0000 0.7071

i j k

P22 0.0000 0.7071 0.7071

P22P21 -0.7071 0.7071 0.0000

P32 0.7071 0.7071 0.0000

P32P22 0.7071 0.0000 -0.7071

One face =0.1179 Four faces=0.47 One face=0.4330 Four faces=1.73

Total Volume of the 2v Dome=1.48 Total Surface Area of the 2v Dome=5.21

Sphericity of the 2v Dome=0.28

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Appendix C Calculations for the Four-Frequency Dome

Original Points Magnitude New Points

x y z

x y z

P11 0.00 0.00 1.00 1.0000 0.0000 0.0000 1.0000

P21 0.25 0.00 0.75 0.7906 0.3162 0.0000 0.9487

P22 0.00 0.25 0.75 0.7906 0.0000 0.3162 0.9487

P31 0.50 0.00 0.50 0.7071 0.7071 0.0000 0.7071

P32 0.25 0.25 0.50 0.6124 0.4082 0.4082 0.8165

P33 0.00 0.50 0.50 0.7071 0.0000 0.7071 0.7071

P41 0.75 0.00 0.25 0.7906 0.9487 0.0000 0.3162

P42 0.50 0.25 0.25 0.6124 0.8165 0.4082 0.4082

P43 0.25 0.50 0.25 0.6124 0.4082 0.8165 0.4082

P44 0.00 0.75 0.25 0.7906 0.0000 0.9487 0.3162

P51 1.00 0.00 0.00 1.0000 1.0000 0.0000 0.0000

P52 0.75 0.25 0.00 0.7906 0.9487 0.3162 0.0000

P53 0.50 0.50 0.00 0.7071 0.7071 0.7071 0.0000

P54 0.25 0.75 0.00 0.7906 0.3162 0.9487 0.0000

P55 0.00 1.00 0.00 1.0000 0.0000 1.0000 0.0000

Volume of Symmetric Faces

Surface Area of Symmetric Faces

P11 0.0000 0.0000 1.0000 AAB i j k

P21 0.3162 0.0000 0.9487 P11P21 0.3162 0.0000 -0.0513

P22 0.0000 0.3162 0.9487 P21P22 -0.3162 0.3162 0.0000

One face=0.0483 Four faces =0.19 One face=0.0513 Four faces=0.21

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Appendix C, continued

Volume of Symmetric Faces Surface Area of Symmetric Faces

P21 0.3162 0.0000 0.9487 CDF i j k

P31 0.7071 0.0000 0.7071 P31P21 0.3909 0.0000 -0.2416

P32 0.4082 0.4082 0.8165 P32P31 -0.2989 0.4082 0.1094

One face=0.0658 Four faces =0.26 One face=0.0949 Four faces =0.38

P22 0.0000 0.3162 0.9487 CDF i j k

P32 0.4082 0.4082 0.8165 P32P22 0.4082 0.0920 -0.1322

P33 0.0000 0.7071 0.7071 P33P32 -0.4082 0.2989 -0.1094

One face=0.0658 Four faces =0.26 One face=0.0949 Four faces =0.38

P21 0.3162 0.0000 0.9487 CCB i j k

P22 0.0000 0.3162 0.9487 P22P21 -0.3162 0.3162 0.0000

P32 0.4082 0.4082 0.8165 P32P22 0.4082 0.0920 -0.1322

One face=0.0716 Four faces =0.29 One face=0.0844 Four faces =0.34

P31 0.7071 0.0000 0.7071 CDF i j k

P41 0.9487 0.0000 0.3162 P41P31 0.2416 0.0000 -0.3909

P42 0.8165 0.4082 0.4082 P42P41 -0.1322 0.4082 0.0920

One face=0.0235 Four faces =0.09 One face=0.0949 Four faces =0.38

P33 0.0000 0.7071 0.7071 CDF i j k

P43 0.4082 0.8165 0.4082 P43P33 0.4082 0.1094 -0.2989

P44 0.0000 0.9487 0.3162 P44P43 -0.4082 0.1322 -0.0920

One face=0.0235 Four faces =0.09 One face=0.0949 Four faces =0.38

P31 0.7071 0.0000 0.7071 DDE i J k

P32 0.4082 0.4082 0.8165 P32P31 -0.2989 0.4082 0.1094

P42 0.8165 0.4082 0.4082 P42P32 0.4082 0.0000 -0.4082

One face=0.0537 Four faces =0.21 One face=0.1240 Four faces =0.50

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Appendix C, continued

Volume of Symmetric Faces Surface Area of Symmetric Faces

P32 0.4082 0.4082 0.8165 DDE i j k

P33 0.0000 0.7071 0.7071 P32P31 -0.4082 0.2989 -0.1094

P43 0.4082 0.8165 0.4082 P42P32 0.4082 0.1094 -0.2989

One face=0.0537 Four faces =0.21 One face=0.1240 Four faces =0.50

P32 0.4082 0.4082 0.8165 EEE i j k

P42 0.8165 0.4082 0.4082 P42P32 0.4082 0.0000 -0.4082

P43 0.4082 0.8165 0.4082 P43P42 -0.4082 0.4082 0.0000

One face=0.0454 Four faces =0.18 One face=0.1443 Four faces =0.58

P41 0.9487 0.0000 0.3162 ABA i j k

P51 1.0000 0.0000 0.0000 P51P41 0.0513 0.0000 -0.3162

P52 0.9487 0.3162 0.0000 P52P51 -0.0513 0.3162 0.0000

One face=0.0009 Four faces =0.003 One face=0.0513 Four faces =0.21

P44 0.0000 0.9487 0.3162 ABA i j k

P54 0.3162 0.9487 0.0000 P54P44 0.3162 0.0000 -0.3162

P55 0.0000 1.0000 0.0000 P55P54 -0.3162 0.0513 0.0000

One face=0.0009 Four faces =0.003 One face=0.0513 Four faces =0.21

P41 0.9487 0.0000 0.3162 CCB I j k

P42 0.8165 0.4082 0.4082 P52P41 -0.1322 0.4082 0.0920

P52 0.9487 0.3162 0.0000 P42P52 0.1322 -0.0920 -0.4082

One face=0.0050 Four faces =0.02 One face=0.0844 Four faces =0.34

P43 0.4082 0.8165 0.4082 CCB I j k

P44 0.0000 0.9487 0.3162 P44P43 -0.4082 0.1322 -0.0920

P54 0.3162 0.9487 0.0000 P54P44 0.3162 0.0000 -0.3162

One face=0.0050 Four faces =0.02 One face=0.0844 Four faces =0.34

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Appendix C, continued

Volume of Symmetric Faces Surface Area of Symmetric Faces

P42 0.8165 0.4082 0.4082 CDF i j k

P52 0.9487 0.3162 0.0000 P52P42 0.1322 -0.0920 -0.4082

P53 0.7071 0.7071 0.0000 P53P52 -0.2416 0.3909 0.0000

One face=0.0020 Four faces =0.01 One face=0.0949 Four faces =0.38

P43 0.4082 0.8165 0.4082 CDF i j k

P53 0.7071 0.7071 0.0000 P53P43 0.2989 -0.1094 -0.4082

P54 0.3162 0.9487 0.0000 P54P53 -0.3909 0.2416 0.0000

One face=0.0020 Four faces =0.01 One face=0.0949 Four faces =0.38

P42 0.8165 0.4082 0.4082 DDE i j k

P43 0.4082 0.8165 0.4082 P43P42 -0.4082 0.4082 0.0000

P53 0.7071 0.7071 0.0000 P53P43 0.2989 -0.1094 -0.4082

One face=0.0105 Four faces =0.04 One face=0.1240 Four faces =0.50

Total Volume of the 4v Dome=1.91

Total Surface Area of the 4v Dome=5.97

Sphericity of the 4v Dome=0.3197

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Appendix D Calculations of the Six Frequency Dome

Original Points Magnitude New Points

x y z

x y z

P11 0.0000 0.0000 1.0000 1.0000

0.0000 0.0000 1.0000

P21 0.1667 0.0000 0.8333 0.8498

0.1961 0.0000 0.9806

P22 0.0000 0.1667 0.8333 0.8498

0.0000 0.1961 0.9806

P31 0.3333 0.0000 0.6667 0.7454

0.4472 0.0000 0.8944

P32 0.1667 0.1667 0.6667 0.7071

0.2357 0.2357 0.9428

P33 0.0000 0.3333 0.6667 0.7454

0.0000 0.4472 0.8944

P41 0.5000 0.0000 0.5000 0.7071

0.7071 0.0000 0.7071

P42 0.3333 0.1667 0.5000 0.6236

0.5345 0.2673 0.8018

P43 0.1667 0.3333 0.5000 0.6236

0.2673 0.5345 0.8018

P44 0.0000 0.5000 0.5000 0.7071

0.0000 0.7071 0.7071

P51 0.6667 0.0000 0.3333 0.7454

0.8944 0.0000 0.4472

P52 0.5000 0.1667 0.3333 0.6236

0.8018 0.2673 0.5345

P53 0.3333 0.3333 0.3333 0.5773

0.5774 0.5774 0.5774

P54 0.1667 0.5000 0.3333 0.6236

0.2673 0.8018 0.5345

P55 0.0000 0.6667 0.3333 0.7454

0.0000 0.8944 0.4472

P61 0.8333 0.0000 0.1667 0.8498

0.9806 0.0000 0.1961

P62 0.6667 0.1667 0.1667 0.7071

0.9428 0.2357 0.2357

P63 0.5000 0.3333 0.1667 0.6236

0.8018 0.5345 0.2673

P64 0.3333 0.5000 0.1667 0.6236

0.5345 0.8018 0.2673

P65 0.1667 0.6667 0.1667 0.7071

0.2357 0.9428 0.2357

P66 0.0000 0.8333 0.1667 0.8498

0.0000 0.9806 0.1961

P71 1.0000 0.0000 0.0000 1.0000

1.0000 0.0000 0.0000

P72 0.8333 0.1667 0.0000 0.8498

0.9806 0.1961 0.0000

P73 0.6667 0.3333 0.0000 0.7454

0.8944 0.4472 0.0000

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Appendix D, continued

Original Points Magnitude New Points

x y z x y z

P74 0.5000 0.5000 0.0000 0.7071 0.7071 0.7071 0.0000

P75 0.3333 0.6667 0.0000 0.7454 0.4472 0.8944 0.0000

P76 0.1667 0.8333 0.0000 0.8498 0.1961 0.9806 0.0000

P77 0.0000 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000

Volume of Symmetric Faces Surface Area of Symmetric Faces

P11 0.0000 0.0000 1.0000 ABA i j k

P21 0.1961 0.0000 0.9806 P21P11 0.1961 0.0000 -0.0194

P22 0.0000 0.1961 0.9806 P22P21 -0.1961 0.1961 0.0000

One face=0.0190 Four faces=0.08 One face=0.0194 Four faces=0.08

P61 0.9806 0.0000 0.1961 ABA i j k

P71 1.0000 0.0000 0.0000 P71P61 0.0194 0.0000 -0.1961

P72 0.9806 0.1961 0.0000 P72P71 -0.0194 0.1961 0.0000

One face=0.0001 Four faces=0.0005 One face=0.0194 Four faces=0.08

P66 0.0000 0.9806 0.1961 ABA i j k

P76 0.1961 0.9806 0.0000 P76P66 0.1961 0.0000 -0.1961

P77 0.0000 1.0000 0.0000 P77P76 -0.1961 0.0194 0.0000

One face=0.0001 Four faces=0.0005 One face=0.0194 Four faces=0.08

P21 0.1961 0.0000 0.9806 CDE i j k

P31 0.4472 0.0000 0.8944 P31P21 0.2511 0.0000 -0.0862

P32 0.2357 0.2357 0.9428 P32P31 -0.2115 0.2357 0.0484

One face=0.0278 Four faces=0.11 One face=0.0314 Four faces=0.13

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Appendix D, continued

Volume of Symmetric Faces Surface Area of Symmetric Faces

P22 0.0000 0.1961 0.9806 CDE i j k

P32 0.2357 0.2357 0.9428 P32P22 0.2357 0.0396 -0.0378

P33 0.0000 0.4472 0.8944 P33P32 -0.2357 0.2115 -0.0484

One face=0.0278 Four faces=0.11 One face=0.0314 Four faces=0.13

P51 0.8944 0.0000 0.4472 CDE i j k

P61 0.9806 0.0000 0.1961 P61P51 0.0861 0.0000 -0.2511

P62 0.9428 0.2357 0.2357 P62P61 -0.0378 0.2357 0.0396

One face=0.0030 Four faces=0.011 One face=0.0314 Four faces=0.13

P55 0.0000 0.8944 0.4472 CDE i j k

P65 0.2357 0.9428 0.2357 P65P55 0.2357 0.0484 -0.2115

P66 0.0000 0.9806 0.1961 P66P65 -0.2357 0.0378 -0.0396

One face=0.0030 Four faces=0.011 One face=0.0314 Four faces=0.13

P62 0.9428 0.2357 0.2357 CDE i j k

P72 0.9806 0.1961 0.0000 P72P62 0.0378 -0.0396 -0.2357

P73 0.8944 0.4472 0.0000 P73P72 -0.0861 0.2511 0.0000

One face=0.0002 Four faces=0.001 One face=0.0314 Four faces=0.13

P65 0.2357 0.9428 0.2357 CDE i j k

P75 0.4472 0.8944 0.0000 P75P65 0.2115 -0.0484 -0.2357

P76 0.1961 0.9806 0.0000 P76P75 -0.2511 0.0861 0.0000

One face=0.0002 Four faces=0.001 One face=0.0314 Four faces=0.13

P21 0.1961 0.0000 0.9806 CCB i j k

P22 0.0000 0.1961 0.9806 P22P21 -0.1961 0.1961 0.0000

P32 0.2357 0.2357 0.9428 P32P22 0.2357 0.0396 -0.0378

One face=0.0261 Four faces=0.11 One face=0.0275 Four faces=0.11

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Appendix D, continued

Volume of the Symmetric Faces Surface Area of the Symmetric Faces

P61 0.9806 0.0000 0.1961 CCB i j k

P62 0.9428 0.2357 0.2357 P62P61 -0.0378 0.2357 0.0396

P72 0.9806 0.1961 0.0000 P72P62 0.0378 -0.0396 -0.2357

One face=0.0005 Four faces=0.002 One face=0.0275 Four faces=0.11

P65 0.2357 0.9428 0.2357 CCB i j k

P66 0.0000 0.9806 0.1961 P66P65 -0.2357 0.0378 -0.0396

P76 0.1961 0.9806 0.0000 P76P66 0.1961 0.0000 -0.1961

One face=0.0005 Four faces=0.002 One face=0.0275 Four faces=0.11

P31 0.4472 0.0000 0.8944 DFG i j k

P41 0.7071 0.0000 0.7071 P41P31 0.2599 0.0000 -0.1873

P42 0.5345 0.2673 0.8018 P42P41 -0.1726 0.2673 0.0947

One face=0.0278 Four faces=0.11 One face=0.0430 Four faces=0.17

P33 0.0000 0.4472 0.8944 DFG i j k

P43 0.2673 0.5345 0.8018 P43P33 0.2673 0.0873 -0.0926

P44 0.0000 0.7071 0.7071 P44P43 -0.2673 0.1726 -0.0947

One face=0.0278 Four faces=0.11 One face=0.0430 Four faces=0.17

P31 0.4472 0.0000 0.8944 DFG i j k

P32 0.2357 0.2357 0.9428 P32P31 -0.2115 0.2357 0.0484

P42 0.5345 0.2673 0.8018 P42P32 0.2988 0.0316 -0.1410

One face=0.0339 Four faces=0.14 One face=0.0430 Four faces=0.17

P32 0.2357 0.2357 0.9428 DFG i j k

P33 0.0000 0.4472 0.8944 P33P32 -0.2357 0.2115 -0.0484

P43 0.2673 0.5345 0.8018 P43P33 0.2673 0.0873 -0.0926

One face=0.0339 Four faces=0.14 One face=0.0430 Four faces=0.17

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Appendix D, continued

Volume of the Symmetric Faces Surface Area of the Symmetric Faces

P41 0.7071 0.0000 0.7071 DFG i j k

P51 0.8944 0.0000 0.4472 P51P41 0.1873 0.0000 -0.2599

P52 0.8018 0.2673 0.5345 P52P51 -0.0926 0.2673 0.0873

One face=0.0141 Four faces=0.06 One face=0.0430 Four faces=0.17

P44 0.0000 0.7071 0.7071 DFG i j k

P54 0.2673 0.8018 0.5345 P54P44 0.2673 0.0947 -0.1726

P55 0.0000 0.8944 0.4472 P55P54 -0.2673 0.0926 -0.0873

One face=0.0141 Four faces=0.06 One face=0.0430 Four faces=0.17

P51 0.8944 0.0000 0.4472 DFG i J k

P52 0.8018 0.2673 0.5345 P52P51 -0.0926 0.2673 0.0873

P62 0.9428 0.2357 0.2357 P62P52 0.1410 -0.0316 -0.2988

One face=0.0071 Four faces=0.03 One face=0.0430 Four faces=0.17

P54 0.2673 0.8018 0.5345 DFG i J k

P55 0.0000 0.8944 0.4472 P55P54 -0.2673 0.0926 -0.0873

P65 0.2357 0.9428 0.2357 P65P55 0.2357 0.0484 -0.2115

One face=0.0071 Four faces=0.03 One face=0.0430 Four faces=0.17

P62 0.9428 0.2357 0.2357 DFG i j k

P63 0.8018 0.5345 0.2673 P63P62 -0.1410 0.2988 0.0316

P73 0.8944 0.4472 0.0000 P73P63 0.0926 -0.0873 -0.2673

One face=0.0013 Four faces=0.005 One face=0.0430 Four faces=0.17

P64 0.5345 0.8018 0.2673 DFG i j k

P65 0.2357 0.9428 0.2357 P65P64 -0.2988 0.1410 -0.0316

P75 0.4472 0.8944 0.0000 P75P65 0.2115 -0.0484 -0.2357

One face=0.0013 Four faces=0.005 One face=0.0430 Four faces=0.17

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Appendix D, continued

Volume of the Symmetric Faces Surface Area of the Symmetric Faces

P63 0.8018 0.5345 0.2673 DFG i j k

P73 0.8944 0.4472 0.0000 P73P63 0.0926 -0.0873 -0.2673

P74 0.7071 0.7071 0.0000 P74P73 -0.1873 0.2599 0.0000

One face=0.0003 Four faces=0.001 One face=0.0430 Four faces=0.17

P64 0.5345 0.8018 0.2673 DFG i j k

P74 0.7071 0.7071 0.0000 P74P64 0.1726 -0.0947 -0.2673

P75 0.4472 0.8944 0.0000 P75P74 -0.2599 0.1873 0.0000

One face=0.0003 Four faces=0.001 One face=0.0430 Four faces=0.17

P32 0.2357 0.2357 0.9428 GGH i j k

P42 0.5345 0.2673 0.8018 P42P32 0.2988 0.0316 -0.1410

P43 0.2673 0.5345 0.8018 P43P42 -0.2673 0.2673 0.0000

One face=0.0375 Four faces=0.15 One face=0.0516 Four faces=0.21

P41 0.7071 0.0000 0.7071 GGH i j k

P42 0.5345 0.2673 0.8018 P42P41 -0.1726 0.2673 0.0947

P52 0.8018 0.2673 0.5345 P52P42 0.2673 0.0000 -0.2673

One face=0.0243 Four faces=0.10 One face=0.0516 Four faces=0.21

P43 0.2673 0.5345 0.8018 GGH i j k

P44 0.0000 0.7071 0.7071 P44P43 -0.2673 0.1726 -0.0947

P54 0.2673 0.8018 0.5345 P54P44 0.2673 0.0947 -0.1726

One face=0.0243 Four faces=0.10 One face=0.0516 Four faces=0.21

P52 0.8018 0.2673 0.5345 GGH i j k

P62 0.9428 0.2357 0.2357 P62P52 0.1410 -0.0316 -0.2988

P63 0.8018 0.5345 0.2673 P63P62 -0.1410 0.2988 0.0316

One face=0.0065 Four faces=0.03 One face=0.0516 Four faces=0.21

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Appendix D, continued

Volume of the Symmetric Faces Surface Area of the Symmetric Faces

P54 0.2673 0.8018 0.5345 GGH i j k

P64 0.5345 0.8018 0.2673 P64P54 0.2673 0.0000 -0.2673

P65 0.2357 0.9428 0.2357 P65P64 -0.2988 0.1410 -0.0316

One face=0.0065 Four faces=0.03 One face=0.0516 Four faces=0.21

P63 0.8018 0.5345 0.2673 GGH i j k

P64 0.5345 0.8018 0.2673 P64P63 -0.2673 0.2673 0.0000

P74 0.7071 0.7071 0.0000 P74P64 0.1726 -0.0947 -0.2673

One face=0.0019 Four faces=0.01 One face=0.0516 Four faces=0.21

P42 0.5345 0.2673 0.8018 IIH i j k

P43 0.2673 0.5345 0.8018 P43P42 -0.2673 0.2673 0.0000

P53 0.5774 0.5774 0.5774 P53P43 0.3101 0.0428 -0.2244

One face=0.0343 Four faces=0.14 One face=0.0634 Four faces=0.25

P42 0.5345 0.2673 0.8018 IIH i j k

P52 0.8018 0.2673 0.5345 P52P42 0.2673 0.0000 -0.2673

P53 0.5774 0.5774 0.5774 P53P52 -0.2244 0.3101 0.0428

One face=0.0264 Four faces=0.11 One face=0.0634 Four faces=0.25

P43 0.2673 0.5345 0.8018 IIH i j k

P53 0.5774 0.5774 0.5774 P53P43 0.3101 0.0428 -0.2244

P54 0.2673 0.8018 0.5345 P54P53 -0.3101 0.2244 -0.0428

One face=0.0264 Four faces=0.11 One face=0.0634 Four faces=0.25

P52 0.8018 0.2673 0.5345 IIH i j k

P53 0.5774 0.5774 0.5774 P53P52 -0.2244 0.3101 0.0428

P63 0.8018 0.5345 0.2673 P63P53 0.2244 -0.0428 -0.3101

One face=0.0138 Four faces=0.06 One face=0.0634 Four faces=0.25

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Appendix D, continued

Volume of the Symmetric Faces Surface Area of the Symmetric Faces

P53 0.5774 0.5774 0.5774 IIH i j k

P54 0.2673 0.8018 0.5345 P54P53 -0.3101 0.2244 -0.0428

P64 0.5345 0.8018 0.2673 P64P54 0.2673 0.0000 -0.2673

One face=0.0138 Four faces=0.06 One face=0.0634 Four faces=0.25

P53 0.5774 0.5774 0.5774 IIH i j k

P63 0.8018 0.5345 0.2673 P63P53 0.2244 -0.0428 -0.3101

P64 0.5345 0.8018 0.2673 P64P63 -0.2673 0.2673 0.0000

One face=0.0090 Four faces=0.04 One face=0.0634 Four faces=0.25

Total Volume of the 6v Dome=2.01 Total Surface Area of the 6v Dome=6.14

Sphericity of the 6v Dome=.3274

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Appendix E MATLab computer program

function SurfaceArea % The purpose of this program is to find the SurfaceArea and Volume % for any geodesic dome given some defined frequency. % The frequency of the geodesic dome is defined by n. % Since the dome is created with equilateral triangles, % each triangle has three vertices. % These vertices will change for each iteration. % First, label the vertices of one eighth of the octahedron, the base % platonic solid. % x(1,1) represents the vertex on row one, point one. % x(2,1) represents the vertex on row two, point one, and so forth. % Frequency is defined by n. To change the frequency, change the n value. n=8; close all N=n+1; x(1,1)=0; y(1,1)=0; z(1,1)=1; x(N,1)=1; y(N,1)=0; z(N,1)=0; x(N,N)=0; y(N,N)=1; z(N,N)=0; delta=1/n; for k=2:n x(k,1)=x(1,1)+delta*(k-1); y(k,1)=0; z(k,1)=z(1,1)-delta*(k-1); x(k,k)=0; y(k,k)=y(1,1)+delta*(k-1); z(k,k)=z(1,1)-delta*(k-1); end for k=3:N for m=2:k x(k,m)=x(k,1)+(m-1)*(x(k,k)-x(k,1))/(k-1); y(k,m)=y(k,1)+(m-1)*(y(k,k)-y(k,1))/(k-1); z(k,m)=z(k,1)+(m-1)*(z(k,k)-z(k,1))/(k-1); end

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end figure hold on % Plot the lines connecting two of the vertices. for j=1:n for i=j:n plot3([x(i,j) x(i+1,j+1)],[y(i,j),y(i+1,j+1)],[z(i,j),z(i+1,j+1)],'-','Linewidth',3,'Color','Black') plot3([x(i,j) x(i+1,j)],[y(i,j),y(i+1,j)],[z(i,j),z(i+1,j)],'-','Linewidth',3,'Color','Black') end end % This plots one face of the dome before the stretch. for i=2:N for j=1:i-1 plot3([x(i,j) x(i,j+1)],[y(i,j),y(i,j+1)],[z(i,j),z(i,j+1)],'-','Linewidth',3,'Color','Black') end end % This defines the magnitude, L, by which the original points are stretched to % ensure they are equidistant to the center-base point. % L is divided by the radius of the dome to ensure the volume is close to % the volume of a rectilinear home. for i=1:N for j=1:i L=sqrt(x(i,j)^2+y(i,j)^2+z(i,j)^2)/15.835; x(i,j)=x(i,j)/L; y(i,j)=y(i,j)/L; z(i,j)=z(i,j)/L; end end x y z figure hold on % This plots the geodesic dome in 3D. for j=1:n for i=j:n plot3([x(i,j) x(i+1,j+1)],[y(i,j),y(i+1,j+1)],[z(i,j),z(i+1,j+1)],'-','Linewidth',3,'Color','Black') plot3([x(i,j) x(i+1,j)],[y(i,j),y(i+1,j)],[z(i,j),z(i+1,j)],'-','Linewidth',3,'Color','Black') end

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end for i=2:N for j=1:i-1 plot3([x(i,j) x(i,j+1)],[y(i,j),y(i,j+1)],[z(i,j),z(i,j+1)],'-','Linewidth',3,'Color','Black') end end % Use the cross product to find surface area of the geodesic dome. k=1; for i=1:N-1 for j=1:i D=(((y(i+1,j)-y(i,j))*(z(i+1,j+1)-z(i,j))-(y(i+1,j+1)-y(i,j))*(z(i+1,j)-z(i,j)))^2+((x(i+1,j)-x(i,j))*(z(i+1,j+1)-z(i,j))-(x(i+1,j+1)-x(i,j))*(z(i+1,j)-z(i,j)))^2+((x(i+1,j)-x(i,j))*(y(i+1,j+1)-y(i,j))-(x(i+1,j+1)-x(i,j))*(y(i+1,j)-y(i,j)))^2); Area(k)=.5*sqrt(D); k=k+1; end end for i=3:N for j=2:i-1 D=(((y(i-1,j-1)-y(i,j))*(z(i-1,j)-z(i,j))-(y(i-1,j)-y(i,j))*(z(i-1,j-1)-z(i,j)))^2+((x(i-1,j-1)-x(i,j))*(z(i-1,j)-z(i,j))-(x(i-1,j)-x(i,j))*(z(i-1,j-1)-z(i,j)))^2+((x(i-1,j-1)-x(i,j))*(y(i-1,j)-y(i,j))-(x(i-1,j)-x(i,j))*(y(i-1,j-1)-y(i,j)))^2); Area(k)=.5*sqrt(D); k=k+1; end end % This command shows the total surface area of the four faces of the % geodesic dome. NT=k-1; sum=0; for k=1:NT sum=sum+Area(k); end % This section computes the surface area of the dome with and without the % riser wall. Multiply LR is, the length of the wall by the height of % 4 when there is a riser wall and by 0 when there is no riser wall. RSA=0 for k=1:n LRis=sqrt((x(N,k+1)-x(N,k))^2+(y(N,k+1)-y(N,k))^2); RSA=RSA+4*LRis end sum=sum+RSA

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SA=4*sum % This command will find the volume of each of the triangular faces of the % geodesic dome pointed upward. Add 12 between the parentheses before % z(i+1,j)to compute the volume with 4 foot riser wall. % Delete the 12 when finding the volume of the dome without the % 4 foot riser wall. k=1; for i=1:N-1 for j=1:i V=(-1/6)*(12+(z(i+1,j)+z(i,j)+z(i+1,j+1)))*(-x(i+1,j+1)*y(i+1,j)+x(i+1,j+1)*y(i,j)+x(i,j)*y(i+1,j)+y(i+1,j+1)*x(i+1,j)-y(i+1,j+1)*x(i,j)-y(i,j)*x(i+1,j)); Volume(k)=abs(V); k=k+1; end end % This will compute the volume of each of the triangular faces of the % geodesic dome pointed downward. Add 12 between the parentheses before % z(i+1,j)to compute the volume with 4 foot riser wall. % Delete the 12 when finding the volume of the dome without the % 4 foot riser wall. for i=2:N-1 for j=1:i-1 V=(-1/6)*(12+(z(i,j+1)+z(i,j)+z(i+1,j+1)))*(-x(i+1,j+1)*y(i,j+1)+x(i+1,j+1)*y(i,j)+x(i,j)*y(i,j+1)+y(i+1,j+1)*x(i,j+1)-y(i+1,j+1)*x(i,j)-y(i,j)*x(i,j+1)); Volume(k)=abs(V); k=k+1; end end NT=k-1; sumV=0; for k=1:NT sumV=sumV+Volume(k); end TV=4*sumV % This computes the sphericity of the dome as a ratio of volume to % surface area. SP=TV/SA for i=1:n for j=i

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Trapezoid(k)=((y(i,j)+y(i+1,j+1))/2)*(z(i,j)-z(i+1,j+1)); k=k+1; end end Trap=k-1; sumTrap=0; for k=1:Trap sumTrap=sumTrap+Trapezoid(k); end SumTrapezoid=sumTrap Riser = 4*(y(n+1,n+1)) %PA = Projected area of dome PAR=2*(sumTrap+Riser); PA=2*sumTrap; ProjectedAreaRiser=PAR ProjectedAreaNoRiser=PA % The coordinates of the 30x30x10 rectilinear home are: % 1.(15,0,20) 2.(15,30,20) 3.(0,30,10) 4. (0,30,0) % 5.(30,30,0) 6. (30,30,10) 7. (0,0,0) 8. (30,0,0) % 9, (30,0,10) 10. (0,0,10). % Change to coordinates for the 30x15x10 to: % 1.(7.5,0,20) 2.(7.5,30,20) 3.(0,30,10) 4. (0,30,0) % 5.(15,30,0) 6. (15,30,10) 7. (0,0,0) 8. (15,0,0) % 9, (15,0,10) 10. (0,0,10). x1=7.5; y1=0; z1=20; x2=7.5; y2=30; z2=20; x3=0; y3=30; z3=10; x4=0; y4=30; z4=0; x5=15; y5=30; z5=0; x6=15; y6=30; z6=10;

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x7=0; y7=0; z7=0; x8=15; y8=0; z8=0; x9=15; y9=0; z9=10; x10=0; y10=0; z10=10; figure hold on % Shows the one-story 30x30x10 house in 3D. plot3([x1 x2],[y1 y2],[z1 z2],'-','Linewidth',3,'Color','Black') plot3([x2 x3],[y2 y3],[z2 z3],'-','Linewidth',3,'Color','Black') plot3([x3 x4],[y3 y4],[z3 z4],'-','Linewidth',3,'Color','Black') plot3([x4 x5],[y4 y5],[z4 z5],'-','Linewidth',3,'Color','Black') plot3([x5 x6],[y5 y6],[z5 z6],'-','Linewidth',3,'Color','Black') plot3([x3 x6],[y3 y6],[z3 z6],'-','Linewidth',3,'Color','Black') plot3([x2 x6],[y2 y6],[z2 z6],'-','Linewidth',3,'Color','Black') plot3([x6 x9],[y6 y9],[z6 z9],'-','Linewidth',3,'Color','Black') plot3([x5 x8],[y5 y8],[z5 z8],'-','Linewidth',3,'Color','Black') plot3([x8 x9],[y8 y9],[z8 z9],'-','Linewidth',3,'Color','Black') plot3([x7 x8],[y7 y8],[z7 z8],'-','Linewidth',3,'Color','Black') plot3([x7 x10],[y7 y10],[z7 z10],'-','Linewidth',3,'Color','Black') plot3([x9 x10],[y9 y10],[z9 z10],'-','Linewidth',3,'Color','Black') plot3([x1 x10],[y1 y10],[z1 z10],'-','Linewidth',3,'Color','Black') plot3([x1 x9],[y1 y9],[z1 z9],'-','Linewidth',3,'Color','Black') plot3([x7 x4],[y7 y4],[z7 z4],'-','Linewidth',3,'Color','Black') plot3([x10 x3],[y10 y3],[z10 z3],'-','Linewidth',3,'Color','Black') L96=sqrt((x9-x6)^2+(y9-y6)^2+(z9-z6)^2); L62=sqrt((x6-x2)^2+(y6-y2)^2+(z6-z2)^2); L63=sqrt((x6-x3)^2+(y6-y3)^2+(z6-z3)^2); L65=sqrt((x6-x5)^2+(y6-y5)^2+(z6-z5)^2); PAreaRoof=L96*10; Onehalfroof=.5*PAreaRoof; AreaFrontSideLeft=L96*L65; AreaFrontSideRight=L65*L63; RoofHeight=10; RtTriangleRoof=.5*L63*RoofHeight; SurfaceAreaView1=((L63+L96)*cos(pi/4))*L65+(1/2*(2*L96*cos(pi/4)+L63*cos(pi/4)))*RoofHeight

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SurfaceAreaView2=AreaFrontSideRight+RtTriangleRoof SurfaceAreaView3=AreaFrontSideLeft+PAreaRoof % The coordinates of the 30x30x20 rectilinear home are: % 1.(15,0,30) 2.(15,30,30) 3.(0,30,20) 4. (0,30,0) % 5.(30,30,0) 6. (30,30,20) 7. (0,0,0) 8. (30,0,0) % 9, (30,0,20) 10. (0,0,20). x1=15; y1=0; z1=30; x2=15; y2=30; z2=30; x3=0; y3=30; z3=20; x4=0; y4=30; z4=0; x5=30; y5=30; z5=0; x6=30; y6=30; z6=20; x7=0; y7=0; z7=0; x8=30; y8=0; z8=0; x9=30; y9=0; z9=20; x10=0; y10=0; z10=20; figure hold on plot3([x1 x2],[y1 y2],[z1 z2],'-','Linewidth',3,'Color','Black') plot3([x2 x3],[y2 y3],[z2 z3],'-','Linewidth',3,'Color','Black') plot3([x3 x4],[y3 y4],[z3 z4],'-','Linewidth',3,'Color','Black') plot3([x4 x5],[y4 y5],[z4 z5],'-','Linewidth',3,'Color','Black') plot3([x5 x6],[y5 y6],[z5 z6],'-','Linewidth',3,'Color','Black')

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plot3([x3 x6],[y3 y6],[z3 z6],'-','Linewidth',3,'Color','Black') plot3([x2 x6],[y2 y6],[z2 z6],'-','Linewidth',3,'Color','Black') plot3([x6 x9],[y6 y9],[z6 z9],'-','Linewidth',3,'Color','Black') plot3([x5 x8],[y5 y8],[z5 z8],'-','Linewidth',3,'Color','Black') plot3([x8 x9],[y8 y9],[z8 z9],'-','Linewidth',3,'Color','Black') plot3([x7 x8],[y7 y8],[z7 z8],'-','Linewidth',3,'Color','Black') plot3([x7 x10],[y7 y10],[z7 z10],'-','Linewidth',3,'Color','Black') plot3([x9 x10],[y9 y10],[z9 z10],'-','Linewidth',3,'Color','Black') plot3([x1 x10],[y1 y10],[z1 z10],'-','Linewidth',3,'Color','Black') plot3([x1 x9],[y1 y9],[z1 z9],'-','Linewidth',3,'Color','Black') plot3([x7 x4],[y7 y4],[z7 z4],'-','Linewidth',3,'Color','Black') plot3([x10 x3],[y10 y3],[z10 z3],'-','Linewidth',3,'Color','Black') L96=sqrt((x9-x6)^2+(y9-y6)^2+(z9-z6)^2) L62=sqrt((x6-x2)^2+(y6-y2)^2+(z6-z2)^2) L63=sqrt((x6-x3)^2+(y6-y3)^2+(z6-z3)^2) L65=sqrt((x6-x5)^2+(y6-y5)^2+(z6-z5)^2) PAreaRoof=L96*10 Onehalfroof=.5*PAreaRoof; AreaFrontSideLeft=L96*L65; AreaFrontSideRight=L65*L63; RoofHeight=10 RtTriangleRoof=.5*L63*RoofHeight; SurfaceAreaView1=((L63+L96)*cos(pi/4))*L65+(1/2*(2*L96*cos(pi/4)+L63*cos(pi/4)))*RoofHeight SurfaceAreaView2=AreaFrontSideRight+RtTriangleRoof SurfaceAreaView3=AreaFrontSideLeft+PAreaRoof

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Appendix F: Email permission to use photographs

American Ingenuity Domes, Inc.

Taralyn,

American Ingenuity gives you permission to use the pictures in your thesis.

Glenda Busick

-------- Original Message --------

Subject: AI Domes: Geodesic Dome Pictures

From: Taralyn Fender <[email protected]>

Date: Mon, April 05, 2010 7:31 am

To: [email protected]

This is an enquiry e-mail via http://www.aidomes.com from:

Taralyn Fender <[email protected]>

Good morning,

I am using the informaton received from you in my mathematical thesis on geodesic domes.

I would like to get permission to use the pictures from your cd and website in my paper.

The paper will be published and I need written permission to include them. While credit is

sited in the paper, the pictures add so much reader appeal and I would like to keep them in

the paper when it is published. Thank you for your permission to use these pictures and for

your immediate attention concerning this. Have a great and beautiful day.

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Appendix F, continued

Natural Spaces Domes, Inc.

Hi Tara,

You may use our pictures for your paper, please note the source of course. Thank you for asking first.

Tim

Natural Spaces Domes

From: [email protected] [mailto:[email protected]]

Sent: Monday, April 05, 2010 9:38 AM

To: [email protected] Subject: Geodesic dome pictures

Good morning,

I currently have included a few of your pictures in my mathematical thesis on geodesic domes. Your pictures add so much reader appeal and knowledge of the homes to my paper. I would also like to include these when my paper is published, but I need your written permission. Thank you for your immediate reply. Have great and beautiful day.

Tara Fender

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Appendix F, continued

FEMA

Dear Ms. Fender:

Thank you for your e-mail dated April 5, 2010, to the Federal Emergency Management

Agency (FEMA) inquiring about the use of FEMA photographs.

U.S. Government materials are not copyright protected. Conditions for use of FEMA

materials are explained on our Web site at http://www.fema.gov/help/usage.shtm.

I hope this is helpful and wish you success.

Sincerely,

Janice Sosebee

FEMA Disaster Assistance Directorate

From: [email protected] [mailto:[email protected]]

Sent: Monday, April 05, 2010 1:02 PM

To: AskFEMA,

Subject: Pictures taken by Mark Wolfe of Hurricane Ivan disaster, September 2004

Good afternoon,

I would like permission to use a few photos taken by Mark Wolfe of the Hurricane

Ivan disaster in my mathematical thesis on geodesic dome homes and how they fare

during a hurricane. The picture numbers are 11737, 11725, and 11724. This paper will

be published, so I need written permission to use them in my paper. Thank you so

much for your immediate attention concerning this. Have a beautiful day.

Tara Fender

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REFERENCES

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