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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
14
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.
17
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.
18
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
19
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
20
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
21
Table 3 Vertex Points of the 4v Dome
Point Original Coordinate Magnitude Stretched Coordinate
1
1
1
22
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
23
.
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.
24
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
25
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:
26
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
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
28
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.
29
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)
30
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)
31
.
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):
.
32
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
33
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
34
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
35
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.
36
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.
37
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
38
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
,
39
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
40
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
41
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
42
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.
43
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
44
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
45
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
46
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.
47
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.
48
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
49
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
50
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.
51
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.
52
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
53
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.
54
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.
55
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
56
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.
57
APPENDICES
58
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
59
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
60
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
61
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
62
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
63
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
64
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
65
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
66
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
67
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
68
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
69
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
70
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
71
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
72
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
73
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
74
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
75
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
76
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;
77
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
78
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')
79
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
80
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
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.
81
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
82
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
83
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