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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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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 ...................

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