The Golden Ratio, the phi relationship, has long been used by craftsmen and professionals of many fields to understand and achieve beauty, balance and harmony in their work.  Artists, architects, designers, engineers, musicians, photographers, sculptors, surgeons and stock analysts alike use this relationship to achieve excellence in their results.

Golden Mean Gauges make it easy to see and apply Phi

The study and application of the golden section is much easier with the golden mean gauges offered on this page.  Put your ruler and calculator away and let the phi relationships just jump out at you as you work or play with these fine instruments.  Originally invented by a doctor for applications in dentistry, these gauges can bring your work to a new level in any field.

Order a Golden Mean Gauge (Steel)

When these gauges  fully expanded, the outside points of the gauge have a spread of 6-3/16″.  Made of 1.5mm steel.  Rustproof but NOT stainless.  Holds positions tightly, sturdy and is accurate to .25 mm.  Excellent value.

U.S. ORDERS ONLY – USPS PRIORITY DELIVERY

Order directly from Dr. Levin’s Golden Mean Gauge UK site at http://goldenmeangauge.co.uk/product/standard-golden-mean-gauge/.

Orders are shipped from the US via USPS Priority Mail, which usually takes 2 to 3 business days for delivery.

 International Shipments and Gauges of other sizes

For added selections of gauges and other dental equipment please visit the Golden Mean Gauge site of Dr. Eddy Levin.  Those orders will be billed in British Pounds and shipped from England.  Options include:

• Economoically priced introductory level gauge.
• 6.5″ boxed professional gauge just like the one above but made of steel suitable for auto-claving and sterilization.
• Dental Set including boxed Golden Mean 6.5″ Professional Gauge with laminated and tear off book of Golden Mean Grids. Template for optimum height to width ratio of incisors. Articles, instructions and tutorials.

The post Dr. Levin’s Golden Mean Gauges appeared first on The Golden Ratio: Phi, 1.618.

If you have a printer and transparencies, you can print the “Golden Ruler”© grid below and use it to see the golden section and phi in various objects.  Simply hold the transparency in front of you and match the lines up to the object to see if its design is based on the golden section, as shown below.

Here’s a small sample grid or click on the Free Golden Ratio “Golden Ruler” Template below for a larger, printable version.

Golden Section Gauge

Make your own golden section gauge using the template below.  It will work best if you construct it from heavy cardboard stock or plastic.  Drill holes and place a brad at each of the indicated points.  When the gauge is adjusted the middle arm will always show the golden section or phi point between the two outer arms.

Note:  The dimensions listed actually represent the distances between the rivets and the tips, not the lengths of the pieces.  It’s best to think of the gauge conceptually as being made of very fine line segments connecting at golden section points. The actual length of piece AF is thus a little longer than the distance from the rivet at A to the tip at F in order to give the rivet some room to hold.  The tips of the gauge should all come together when it’s closed.

You can also download this in AutoCad format with an accompanying PDF file, courtesy of Bill Cummins.

You might find these videos helpful in making one:

https://www.youtube.com/watch?v=nBa6lpRqfE8 – cardboard construction

https://www.youtube.com/watch?v=s37RP3mVnTg- wood construction

Note that some people call this a Fibonacci gauge, but the Fibonacci series (0, 1, 1, 2, 3, 5, 8, …) is a series whose ratios converge on the golden ratio as the series increases. I think it is more appropriately called a golden mean gauge or golden ratio gauge, as that is the precise ratio that it measures.

6 sided quasi-crystal shape

You can make your own quasi-crystal shape using the template below:

Dodecahedron / Icosahedron structure

You can make your own internal structure for a dodecahedron or icosahedron, both based on phi, using the template below:

Great Pyramid of Giza Template

Here’s a template to create a pyramid with the same golden ratio proportions as the Great Pyramid of Giza.

great-pyramid-golden-triangle-template in PDF

The post Golden Ratio Do It Yourself Projects appeared first on The Golden Ratio: Phi, 1.618.

To reference this site in APA format, the reference will depend on the page you are citing but will look like this:

Meisner, Gary B. (Year, Month Date Published). Article title. Phi: The Golden Number, Retrieved Month Date, Year, from http://www.goldennumber.net

This website was redesigned in 2012, so many of the articles show a 2012 publish date even though they were published much earlier, many as early as 2001 or on a predecessor site back to 1997.  You can determine the original month or year an article was published by using the Wayback Machine at http://web.archive.org/web/*/goldennumber.net

If you would like to use images from this site for school papers, educational purposes and non-commercial uses, all you have to do is add the reference to your material wherever the images appear:

Image courtesy of http://www.goldennumber.net, Gary B. Meisner, Copyright 2001-20xx (insert current year for 20xx).

As added background, Gary holds an MBA in Finance and Marketing from University of Chicago and BS in Accountancy from University of Illinois.

This is an authorization to use up to ten images from the site in your material.

You may also include portions of the text of the site with the same reference.

Please note that this is NOT an authorization to copy substantial portions of this site and include it in your material or represent it as your own work.

Although not an absolute requirement of use, I appreciate being informed of your use by sending an e-mail (see below) with your name, e-mail address and where the image will be used, including a URL if it’s an Internet site.

Commercial uses  – Permissions and custom image development

Before using any of the images on this site for commercial or for-profit uses, please contact me to obtain authorization via e-mail for each specific use.  Some of the images on this site are original copyrighted images.  Others are in the public domain.  Sources are unknown for some images.

As the images on this site were developed for web usage, their resolution is not generally adequate for print.  If you see images that you would like in higher resolutions, I have developed custom images for text books and will be happy to quote you on your specific requirements.

The post Use of Images and Content from this Site appeared first on The Golden Ratio: Phi, 1.618.

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.

Projects

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.

You might also get some ideas from my web sites and http://www.phimatrix.com and http://www.phidental.com on ways that others have put the golden section into practical applications.

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

INTERVIEW #1

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.
Lucas Series:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, …

Fibonacci Series:

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.

INTERVIEW #2

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

INTERVIEW #3

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.

INTERVIEW #4

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.

INTERVIEW #5

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:

http://www.phimatrix.com/web-site-design-golden-ratio/

http://www.goldennumber.net/aston-martin-golden-ratio/

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.

The post School Projects and Interviews appeared first on The Golden Ratio: Phi, 1.618.

YouTube – Donald Duck in Mathmagic Land – The Disney Classic – Part I / Part II – Order from Amazon in DVD or VHS.

Gordon Plummer presentation on “The Nature of Things” – Google Video / ESnips / ESnips Download

BBC UK, In Our Time – The Fibonacci Sequence

BBC UK, Science – 5 Numbers, Programme 3, The Golden Ratio / Direct link to audio

YouTube – The Golden Mean and Spirals

YouTube – Phi:  The Nature of Growth

YouTube – Fibonacci, Fractals and Financial Markets – Socionomics.net

YouTube – Sequence 5, The Golden Rectangle, Construction

YouTube – Golden Ratio- The Perfect Face & Change the Sex of People.  Applying the Beauty Mask of Dr. Stephen Marquardt.

YouTube – The Golden Rectangle – The Musical

YouTube – How to do the Fibonacci Series – Simple illustration of addition of beginning Fibonacci numbers

YouTube – Fibonacci Numbers – Another Evolutionary Miracle (or maybe God did it)

YouTube – Phi (How to Cornify the Divine Proportion) *Grace* Enhanced (Adding a little music and romance to phi)

YouTube – Divine Proportion – Animation of Spirals

YouTube – Dr Ron Knott Fibonacci Numbers Eden Project Cornwall

YouTube – Fibonacci Gauge – Well actually it’s a golden section gauge if you want to be precise.  Application to woodworking and art.

YouTube – The Fibonacci in Lateralus – Spirals in Space with rock music

YouTube – Fibonacci Tree – Illustration of Fibonacci series in branching of trees

YouTube – El Número de Oro; Phi; la Divina Proporción – Well done overview of Phi and Fibonacci video in Spanish

The post Links to Phi in Video and Multimedia resources appeared first on The Golden Ratio: Phi, 1.618.

Following are sources of jewelry related to phi, the golden section and the Fibonacci series.  Click on the image to see the jewelry and the description below to see background information.

Ka-Gold Jewelry

Nautilus Spiral

Golden Mean

 Icosahedron

Phi and The Golden Section

Fibonacci Series

Other Ideas

 

 

Below are sources from which you can order merchandise that bears the symbol  for phi or that spells out PHI.  Some items are offered only for nationally recognized college fraternities and sororities.  Most items, however, allow you to choose from one to three Greek letters on a variety of jewelry, wearables and a variety of other items in colors of your choice.

Order any of the combinations below:

Sources:

 

Vendor		Types of items offered
Something Greek		A great source for clothing, accessories, keychains, gift items, glassware, jackets, jerseys, jewelry, paddles and wood accessories, pants, hats, shorts, flip flops, shirts, baby t’s, tank tops, sweatshirts, sweatpants and more
Amazon		Charms.  Also search under “phi” once on the site for other items.

 

Get creative:

Make up your own unique expression using one, two or three of the following Greek letters.

The post Phi and Fibonacci Jewelry appeared first on The Golden Ratio: Phi, 1.618.

A lot of effort has gone into making this site one of the best available on phi, and all at no charge to site viewers.  Support from site visitors allows me to devote more time yet to exploring and expanding on this topic.  If you would like to help meet the financial needs of this site, you can help in one of two ways.   Purchasing anything from our affiliates generates a commission or by simply making a donation below.

Whether this site has just helped you with your homework or changed your view of life and the universe, you can make a donation for any amount you want, from a minimum of 50 cents on up!

I sincerely appreciate any and everything you do to show your support, even if it’s just a kind word of thanks!

After you click, you will be given a choice to donate via credit card or to donate from a PayPal account. I sincerely appreciate your support!

 

 

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In addition, GoldenNumber.net uses products and services from the following companies and as is pleased to offer them to you for your consideration through our affiliation with them.  By purchasing goods and services from these companies, you help us to provide this site to you for free.

PhiMatrix – Phi-based graphical analysis and design software

Dr. Levin’s Phi Dental Grid Software – Phi-based software for cosmetic dentistry

Dr. Levin’s Golden Mean Gauges – Steel phi gauges for dentistry, art, stock market analysis and more

Ka-Gold Jewelry – Golden spirals and Nautilus spirals in silver and gold jewelry

PhiSource Merchandise – T-Shirts, hats, mouse pads and much more with phi-based designs and logos

Greek Merchandise – A place for phi stuff from Something Greek.

Amazon.com – Providing book, CD’s, software, toys and much more at great prices, with quality services and fast delivery.

Elliott Wave International – One of the premier market analysis services applying the principles of phi, the Fibonacci series and the Golden Section to technical analysis of the stock markets and other investment instruments.

Phi Chi Sticks – A novel and enjoyable combination of Phi and Tai Chi that uses sticks based on the curve of a golden spiral.

PayPal – Sign up for PayPal and start accepting credit card payments instantly.  Refer New Merchants to PayPal and Earn up to $1,000.00 USD for Each Referral.

And, whether this site has just helped you with your homework or changed your view of life and the universe, you can show your support with a donation for any amount you want.

I sincerely appreciate your support!

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PhiMatrix golden ratio software is a design and analysis software application for Windows and Mac. The PhiMatrix grid system quickly overlays any image in any software on your screen to identify or apply phi proportions in various grid styles, as well as custom user-defined ratios. It’s not a plug-in but a universal overlay, so you can use it with PhotoShop, Illustrator, InDesign, Picassa and any other software you use.

Investing: Elliott Wave International – Fibonacci trading techniques for stocks and other markets

Learn how to apply the golden ratio and Fibonacci patterns that appear in nature to timing and price movements in the stock markets at Elliott Wave International.  See overview at the Stocks page on this site.

Design Gauges:  Dr. Levin’s Golden Mean Gauge – Design and Analyze

See the Golden Mean Gauge page for information on Dr. Levin’s gauges.  Constructed of stainless steel, these precision gauges were designed for use in cosmetic dentistry and can be applied to any application in arts, design, medical practices or engineering.

Wearables and Home/Office Merchandise:  Cafe Press  – Express yourself

Share your interest in phi through a variety of cool merchandise items.  Order dozens of different Phi items through CafePress, such as the ones below:

Phi “Golden” T-Shirt	Messenger Bag
Phi Mug	Phi Collage T-Shirt

Jewelry: Ka-Gold Jewelry and Something Greek Jewelry

See also the golden ratio jewelry page for a variety of jewelry and other items, everything from simple charms to beautifully designed gold and silver jewelry inspired by the golden ratio.

Books: Amazon.com – Learn

Learn, apply and excel with phi the golden ratio from this list of golden ratio books at Amazon.com

 

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Dr. Ron Knott’s Fibonacci Numbers and the Golden section Site – One of the most thorough sites on the Internet on this topic, and a winner of many awards.

The Museum of Harmony and Golden Section

Fibonacci Quarterly Home Page – Research Magazine

The life and numbers of Leonardo Fibonacci

Fascinating Fibonaccis – Mystery and Magic in Numbers

The Golden Mean

The Fib-Phi Link Page – A great page provided by Dawson Merrill with links to a variety of phi and Fibonacci related sites, categorized by topic with brief descriptions of each site.

Mystery of Fibonacci Numbers – A brief overview with links to other good sites.

Design in Nature

The Evolution of Truth

Art and Architecture

Fibonacci numbers and the Golden Section in Art, Architecture and Music

Golden Section in Art, Architecture and Music

The School of Pythagoras

Secrets of the Great Pyramid

Museum of the Golden Ratio – A virtual museum dedicated to collecting and preserving individual expressions in art of the myriad aspects of the Golden Ratio

Secrets of the Great Pyramid

Beauty

A Model’s Secrets

Beauty Analysis – The Beauty Mask of Dr. Stephen Marquardt

Design

Timeless by Design – The use of orthogons as a system of design

Golden Section in Furniture Design

Ka Gold Jewelry – Sacred geometry jewelry and various related Spiritual resources

Sketchlist – Design software for furniture and cabinets that enables use of the golden section

Geometry

Sacred Geometry Home Page – The unique numbers in math

Life

Golden Mean Gauge – Dr. Eddy Levin’s site on application of the golden section to Dentistry, along with many other golden section topics.

Facial Beauty – Dr. Yosh Jefferson’s site on the application of the golden section to health and the treatment of facial and dental abnormalities.

Beauty Analysis – Dr. Stephen Marquardt’s site on the impact of phi on human beauty.

Beauty Rank – Analyzes beauty with phi as one of its attributes.

Fibonacci Number in Human Growth and Development

Mathematics

Fibonacci – The Golden Mean – Using trigonometry, limits, etc.

Online Calculator with Phi, Pi and e constants

Music

Cardas Audio Insights – Golden Section Stranding

Phi-Music Microtone-Brainwave Generator

JodyJazz DV (Da Vinci) Saxophone Mouthpieces – Using phi proportions in design to achieve superior acoustics and performance.

16 Tones and Phi in the Well Tempered Clavier – by Remi Schulz

Population

Fibonacci – A Study in Population Growth

Solar System

Solar Geometry – Phi and many other relationships in the Solar System (A sister site to this site)

SPIRA SOLARIS ARCHYTAS-MIRABLIS III – Application to the Solar System

SPIRA SOLARIS ARCHYTAS-MIRABLIS IV – Application to the Solar System

Spirals

Xah Special Plane Curves Equiangular Spiral

Fibonacci Spiral Art

Phi Chi Sticks – Combining the beauty of golden spirals with the fitness of Tai Chi

Stock Market

Ermanometry Research:  The Perfectly Patterned Stock Market

Elliott Wave International

International Sites

Presencia del Número de Oro – 1.61803398 by Carlos Calvimontes

Presencia del Número de Oro by Carlos Calvimontes

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