I’ve received a growing number of requests for ideas on school projects and even to respond to “interviews with an expert” which are being required by some teachers for a passing grade on projects.
My hope with this site is to expand knowledge on phi and I appreciate the interest it has generated. Still, there’s only one of me, and it’s becoming increasing difficult to respond to all the requests.
To help meet your needs, I’m including some of the responses I’ve sent to past e-mails. For information on use of site contents in your papers, click HERE.
As to science fair ideas, my primary recommendation would be to make your project interactive, getting other people involved in either making measurements, constructing something, adding their own measurements to your data, voting on what is beautiful or seeing the practical applications for phi. Give them something to touch, feel or experience so they really understand how phi can be used.
Perhaps you could use one of the gauges like a have on my page at http://www.goldennumber.net/gauges.htm, or the do it yourself version on the page athttp://www.goldennumber.net/goldgrid.htm. That page also has some templates for constructing phi-based shapes.
There’s a neat program for creating art based on Penrose tiles on my page at http://www.goldennumber.net/penrose.htm, as well as a source for Penrose tile puzzles.
For other ideas, you might want to:
Take measurements of people’s faces and bodies to see how closely the averages conform to phi and also how it affects the way people are perceived as being beautiful. I recommend using digital photos and doing the measurements by pixel within a photo editing program.
Do a project on the appearance of phi in geometry, showing 2D and 3D figures.
Do a project on its application in art, architecture and design.
Do a project on the unusual properties of Fibonacci numbers.
Do a project on Fibonacci numbers in plants and spirals like pine cones, flower seed patterns, etc.
See also the page on Do-It-Yourself projects.
I hope that helps and I wish you the very best as you pursue all that’s golden in your projects!
As to things that phi relates to, there are dozens of examples on this site to choose from. Just pick some out and talk about how phi plays a role in the proportions of design of life and its processes and has been adopted by man in his creative efforts to create harmony and beauty in the results.
I’m doing a project on the phi for the beauty section. Could you please help me with a question based on phi ??
As to the project, you might want to take measurements of people’s faces and bodies to see how closely the averages conform to phi and also how it affects the way people are perceived as being beautiful. I recommend using digital photos and doing the measurements by pixel within a photo editing program. See http://beautyanalysis.com as well. As a word of explanation, keep in mind that phi appears in humans not as a constant where everyone is identical, but rather as an average of measures taken across large groups, and this average measure is most often perceived as being beautiful.
One aspect of phi that would probably be of most interest to people in general is how it appears in the human body. You could do some experiments in taking measurements of different people’s bodies and then computing how close the measures come to phi. One tip: It may be easiest and most accurate to do this with a digital camera and then to use a graphics program that lets you count the pixels. You can of course just use a measuring tape as well, but the results may not be quite as accurate. You can see my pages under the “Life” section to see where this applies. You can also take surveys to see how people perceive beauty and whether their perceptions of beauty is related to phi proportions, similar to Dr. Marquardt’s work at http://www.beautyanalysis.com.
I am a high school student doing a math project on the topic of Phi. I am a little confused with the data i have got. I have faces of women who have done some plastic surgeries to their faces. I got some measurements of their faces including Pupil Line, Facial Vertical Line, Upper lip to chin, idth of face, Height of face and width/height. With this information, how can I prove that those faces are proportional to phi? Are there any graphs/ charts that i can use to prove this?
Take your measurements and divide the length of the longer one by the shorter one and see how close they come to Phi, which is 1.618. Keep in mind that some measures may be a golden section of a golden section (see my page at http://www.goldennumber.net/life-design/ so some of the relationships may be based on phi squared, phi cubed, etc. or the reciprocals of these numbers:
Exponent Powers of Phi Reciprocal
4 6.854101966 0.145898034
3 4.236067977 0.236067977
2 2.618033989 0.381966011
1 1.618033989 0.618033989
0 1 1
-1 0.618033989 1.618033989
-2 0.381966011 2.618033989
-3 0.236067977 4.236067977
-4 0.145898034 6.854101966
Compare your ratios of measurements to the numbers in the table above. If they match, you’ve shown that the faces are in phi proportion.
Responses to requests for interviews
1. What does phi have to do with the spiral of a pine cone?
The bottom of a pine cone reveals two spirals in opposite directions, both Fibonacci numbers, typically 8 and 13.
2. What does phi have to do with the shape or spiral of leaves or flowers?
Leaves are arrange around the stems of a plant at roughly 137.5 degrees to optimize exposure to the sun. The complement to that angle, 222.5 is 360 degrees divided by phi. Flowers often have a Fibonacci number for the number of petals.
3. What is a Fibonacci number?
Any number in the additive series that was discovered Leonardo Fibonacci, 0 1 1 2 3 5 8 13 etc. – Each one is the sum of the two before.
4. What is a Lucas number?
The difference in the definition is that the Lucas series starts with 2 and 1 rather than 1 and 1.
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, …
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …
5. What is phi and how does it differ from Phi?
Little phi is 0.618… and Big Phi is 1.618. phi + 1 = Phi. 1 / Phi = phi. 1 / Phi = phi.
6. How long have you researched phi and what got you interested in it?
I’ve been researching it since about 1997. I learned about phi while searching for answers on the origins of life and evidence for God’s design in creation. I had a life-changing experience that brought me to faith in God through Jesus Christ. As I’d been an agnostic until I was almost 40, this challenged most of what I had blindly accepted my entire life as scientific “fact” about the origins of life. In my pursuit of knowledge to reconcile my past beliefs with my new faith, I came across a book that mentioned the phi proportion in the human body. I began to do my own research and found it in many places.
7. In easier terms, how can you test and see if Phi is connected to nature, plant spirals, ect..?
Just measure the dimensions of anything and then get out your calculator to see if the ratio of 1.618 to 1 can be found in its proportions. It appears many places, as shown on my site.
8. How come not many people are familiar with the number Phi when it has to do with so many things?
Phi unveils a unity, harmony and beauty in the design of life and the universe. Grasping that can be a spiritual experience, like a window into the mind of God. While in past eras mankind integrated scientific and spiritual pursuits as one, we live in a post-Darwinian age in which academicians and scientists are jeered by their colleagues for even speaking of God. Science should openly investigate all possibilities, but doesn’t because of intellectual pride. People who take pride in their intellect find it hard to accept an intellect so much greater than their own could exist. They don’t like the implications of God’s existence, so they rationalize Him away. (See 1 Corinthians 3:18-19) Phi should be included in our curriculums because it can be applied to understanding and unifying many fields of study. It’s unknown because it’s been removed from the curriculums of schools over the past one hundred years. From what I’ve read, it was once included but fell out of favor as we took God out of education replaced Him with evolution.
9. What have you experimented with or tested with phi in nature?
Many things, which are shown in the photos on my site – Human face and body proportions, dolphins, insects, shells, tigers, penguins, etc.
10. ANY other useful information that will make me knowledgeable about the number Phi?
It appears pervasively in nature, both in instances where it seems “natural” or “logical” for it to exist, but also in instances where it seems to exist for no other reason than design and beauty.
1. How did you first learn about Phi?
I learned about phi while searching for answers on the origins of life and evidence for God’s design in creation. I had a life-changing experience that brought me to faith in God through Jesus Christ. As I’d been an agnostic until I was almost 40, this challenged most of what I had blindly accepted my entire life as scientific “fact” about the origins of life. In my pursuit of knowledge to reconcile my past beliefs with my new faith, I came across a book that mentioned the phi proportion in the human body. I began to do my own research and found it in many places.
2. What aspect of Phi do you find most interesting?
To me, the most interesting aspect of phi is that it appears so pervasively, both in instances where it seems “natural” for it to exist, but also in instances where it seems to exist for no other reason than design and beauty.
3. Phi is not generally taught in school. Why do you think this is? Should it be?
Phi unveils a unity, harmony and beauty in the design of life and the universe. Grasping that can be a spiritual experience, like a window into the mind of God. While in past eras mankind integrated scientific and spiritual pursuits as one, we live in a post-Darwinian age in which academicians and scientists are jeered by their colleagues for even speaking of God. Science should openly investigate all possibilities, but doesn’t because of intellectual pride and academic “standards” that we have imposed upon ourselves. In addition, people who take pride in their intellect find it hard to accept an intellect so much greater than their own could exist. They don’t like the implications of God’s existence, so they rationalize Him away. (See 1 Corinthians 3:18-19) Phi should be included in our curriculums because it can be applied to understanding and unifying many fields of study.
4. What is your occupation?
I spent most of the last twenty years in dual roles of Chief Financial Officer and Chief Information Officer, leading the finance and technology areas for divisions of public companies. My background is as a CPA with an MBA in Finance and Marketing. I was an an independent consultant and entrepreneur for a time and am now employed again as a CFO for a small company.
5. Despite the exposure of your website and the book “The Da Vinci Code,” Phi is still relatively unknown to the average person. (in my experience) Why do you think this is?
It’s unknown because it’s been removed from the curriculums of schools over the past one hundred years. From what I’ve read, it was once included but fell out of favor as we took God out of education replaced Him with evolution.
For another perspective on this, I invite you to visit my site at http://snapshotsofgod.com
1. How do you use phi in your life?
I use phi to get deeper understandings into the beauty and harmony in the design of the universe and the life within it, and through that to come to know our God and Creator better.
2. Does it apply to your job?
My career is as a chief financial and information officer, so phi doesn’t apply that much to my work. Phi is used extensively however by artists, architects and stock market analysts, and more and more these days by plastic surgeons and orthodontists, as illustrated on my site.
3. Why is phi called the golden mean?
The term mean came from Euclid (365 BC – 300 BC), in “Elements,” who referred to dividing a line at the 0.6180399… point as dividing a line in the extreme and mean ratio. This later gave rise to the use of the term mean in the golden mean.
4. How does the construction of the “golden rectangle” show phi?
In the golden rectangle, the ratio of the long side to the short side is phi, or 1.6180339…
5. How did you come across phi?
I came across phi when I began to study evidence for design in creation after coming to faith in God through Jesus Christ.
1. How important is the golden ratio in mathematics? I mean, why the golden ratio as a number is so special?
It’s unique in that its square is one more than itself and its reciprocal is one less than itself. It appears in the geometries of many basic shapes and constructions, as well as in mathematical limits. http://www.goldennumber.net/neophite.htm
2. Is it important for living beings? Do you think the fact the golden ratio appears in living beings is just a coincidence, or is it that the living beings tend to have such proportion?
I do not believe it’s a coincidence because it appears it places where there seems to be no reason other than for beauty alone. I think it is an evidence of the beauty and harmony of creation that was put their by our God and Creator. http://www.goldennumber.net/theology.htm
3. Does the golden ratio have any aesthetic features? If so, is this why some buildings and paintings were created following this proportion?
The sense of beauty and aesthetics is wired into us as humans, and because the golden ratio appears so pervasively in nature mankind has naturally incorporated it into our own creations, both consciously and unconsciously.
4. Do you think that the golden ratio is aesthetically more attractive than other proportions and because of that some products containing it are bought by more people?
Studies have shown that it is more aesthetically appealing, and it is in fact used in product design, so it follows logically that this would aid in sales. I do not have specific studies though on the impact on product sales. http://www.phimatrix.com/examples.htm
5. Do you think that in the future buildings will continue being designed based on the golden ratio?
Definitely. There is a growing awareness, a re-awakening, of the existence and significance of the golden ratio so it will continue to be used.
6. Which is the place or object you find more interesting where the golden ratio appears? Why?
The human face, because it appears in so many different aspects of the face and can help us to understand our own perceptions and desires for beauty.http://www.goldennumber.net/face.htm
7. Could you please tell me why the golden ratio has infinite decimals digits?
Simply because it’s an irrational number in our base 10 and cannot be expressed in integers alone. http://www.goldennumber.net/neophite.htm
8. Some experts say that the golden ratio can be found in buildings like the Parthenon or the Notre Dame cathedral, or in some paintings like the Mona Lisa. However, there are also some experts who say that the golden ratio is found in those places because measurements are presented in a way that facilitates the appearance of the golden ratio. In other words, data is taken in non-natural, somehow fake ways, so that the golden ratio appears.
What is your opinion about this? I have found its proportions in both the Parthenon and Notre Dame, but this is based on photos. Whenever I use photos, however, I always seek photos that are taken from a line directly perpendicular to the object to avoid the distorting impacts of perspective. These are matters of fact thought, not opinion. Dispute should only occur when the condition of the building has deteriorated to make measurement subject to interpretation, as in the case of the Great Pyramids. Anyone can conduct the same analysis themselves, and it’s very easy to do with my software product at www.phimatrix.com which can be used to unveil or apply the golden ratio in a matter of seconds. Test it for yourself and then you will know. http://www.goldennumber.net/architecture.htm
9. When applied to the real world, do you think that the golden ratio is just a curiosity, or is something more important (a property from Nature, a law which determines certain things in the Universe…)?
It’s too pervasive in nature and too unique and interesting mathematically to call it a curiosity. It’s not an absolute or a law though. I think of it as a constant used by God to assure the beauty and harmony of His creation, the signature of the Author and Artist of Life.
1. I understand where Phi occurs in nature, but I’m wondering why it does so. (Not sure if this is known, but I wanted to ask about your thoughts.)
Answer: No one knows for sure. There are definitely instances where the golden ratio produces the most efficient result. I think there are many other instances though where its appearance is for no obvious reason other than beauty and harmony. The human face is one great example. That, in my belief, is an evidence, but not proof, of purpose and design, and of God as our Creator. See http://www.goldennumber.net/about/ and my other site at www.snapshotsofgod.com if you want more on my own philosophy and theology on this.
2. Do you think it would possible, or have you ever heard of, any ways Phi is used in other areas than art, beauty, music, architecture, etc.? For example, could Phi be used in manufacturing processes, car designs, dance choreography, the solving of environmental issues, etc.?
Answer: See my page at http://www.phimatrix.com/applications/ for applications. I’ve heard of some uses of mixing food and formulations in phi proportions, but don’t have much evidence on it.
3. Are you aware of any recent developments or new directions with Phi?
Answer: Apple’s new iOS 7 icons are based on a golden ratio template, use in website typography and auto design:
4. Why do you see as coming in the future of Phi?
Answer: It’s part of nature’s design and will continue to be so. Each generation, each person, can discover it anew in their own lifetime. I think the future of phi will be to continue to raise the awareness of those who will believe of the beauty, harmony, design, purpose and oneness of life. It will be a gateway to understanding the incredible expression of all these same things by God our Creator in the creation of life. Of course, there will be those who will not believe and will continue to explain it all away as purposeless pattern seeking, entirely natural and with no cause other than efficient design. Phi brings us to a fork in the road of deciding how and why we are here, and you must upon which road you will make your life’s journey.