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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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
ii
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
iii
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.
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
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
7
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 1. Construction of a concrete geodesic dome home. Used by permission.
9
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.
10
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.
15
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)
16
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 �2 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 √2. Since this is a 1v
dome, then the number of triangular panels on one
face is 1 � 1.
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, 2 � 4.
Table 1 Vertex Points for the 2v Dome
Point Original Coordinate Magnitude Stretched Coordinate
� �0,0,1� 1 �0,0,1�
� �12 , 0, 1
2� √22 �√2
2 , 0, √22 �
� �0, 12 , 1
2� √22 �0, √2
2 , √22 �
�� �1, 0, 0� 1 �1, 0, 0�
�� �12 , 1
2 , 0� √22 �√2
2 , √22 , 0�
��� �0, 1, 0� 1 �0, 1, 0�
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 �:
������������������ � ��√ � � 0 � �√
� 1� � 0.765367 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 4 � 16 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.
Point Original Coordinate Magnitude Stretched Coordinate
� �0, 0, 1� 1 �0,0,1� � �1
4 , 0, 34� √10
4 �√1010 , 0, 3√10
10 �
� �0, 14 , 3
4� √104 �0, √10
10 , 3√1010 �
�� �12 , 0, 1
2� √22 �√2
2 , 0, √22 �
�� �14 , 1
4 , 12� √6
4 �√66 , √6
6 , √63 �
��� �0, 12 , 1
2� √22 �0, √2
2 , √22 �
�& �14 , 0, 3
4� √104 �3√10
10 , 0, √1010 �
�& �12 , 1
4 , 14� √6
4 �√63 , √6
6 , √66 �
�&� �14 , 1
2 , 14� √6
4 �√66 , √6
3 , √66 �
�&& �0, 14 , 3
4� √104 �0, 3√10
10 , √1010 �
�' �1, 0, 0� 1 �1, 0, 0� �' �1
4 , 34 , 0� √10
4 �√1010 , 3√10
10 , 0�
�'� �12 , 1
2 , 0� √22 �√2
2 , √22 , 0�
�'& �34 , 1
4 , 0� √104 �3√10
10 , √1010 , 0�
�'' �0, 1, 0� 1 �0, 1, 0�
22
Table 4 Strut Lengths for the 4v Dome
Vectors with the same strut length Strut Length
��11�21��������������, ��11�22
��������������, ��51�41��������������, ��44�55
��������������, ��51�52��������������, ��54�55
�������������� 0.3204
��21�22��������������, ��41�52
��������������, ��44�54�������������� 0.4472
��21�31��������������, ��22�33
��������������, ��41�31��������������, ��33�44
��������������, ��52�53��������������, ��53�54
�������������� 0.4595
��21�32��������������, ��22�32
��������������, ��41�42��������������, ��43�44
��������������, ��42�52�������������� 0.4389
��31�32��������������, ��32�33
��������������, ��31�42��������������, ��33�43
��������������, ��42�53��������������, ��43�53
�������������� 0.5176
��32�42��������������, ��32�43
��������������, ��42�43�������������� 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:
(���������������� ) ���������������� ( � (����������������( (����������������( sin -. Geometrically, this is the area of the parallelogram defined by the two vectors. Thus the
area of the triangle �11�21�22 would be 1
2 that value. Algebraically, the cross product is also
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 �x1, y1, z1���1, 0, 0�, �x2, y2, z2���0, 0, 1�, and
�x3, y3, z3�� �0, 1, 0�. Using Eq. (5) the volume under this one panel is
a � � b �1 � 0 � 0��0�0� � 0�0� � 1�0� � 1�1� � 0�0�� � 16 .
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 23 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:
Y � XZ[X\VZ[V\ � d[d d[ � d[ � 0
Y� � X`[X\V`[V\ � [d d[ � [ � �1 Y� � X`[XZV`[VZ � [d d[d � d � undefined slope. The coefficients 4, 5, and 6, of the normal vector to the plane of the triangular face
are calculated for the given points of the 1v dome:
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
Appendix E, continued
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')
74
Appendix E, continued 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 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
75
Appendix E, continued 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 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
76
Appendix E, continued % 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 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;
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
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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.
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