Welcome educators, teachers, parents and students to the Golden Ratio Academy page.
The Golden Ratio appears not only in nature and the arts, but also in the Common Core Curriculum mathematics standards. This page is dedicated to educators, parents and students seeking the best, most accurate information on the golden ratio. You can return to this page by simply going to the easily remembered link of goldenratio.academy in your browser.
Separating Math and Myth: Teaching Golden Ratio Fact, not Fiction.
Much information is available on the golden ratio from a variety of sources, but it’s amazing how much of it is incorrect. It perpetuates like a myth or urban legend. Even content produced by Ph.D’s in mathematics and educational sources such as the Discovery Channel contains errors.
My goal is to provide you with the resources to provide our students with the best information available on this fascinating topic. As background, I’ve been researching and writing on this topic through this site since 1997. I developed golden ratio design and analysis software, available since 2004, which is used for research on this site and by users in over 70 countries. I’ve corresponded with hundreds of people in a variety of disciplines who have contributed content to the over 100 pages of information on the golden ratio on this site.
This page includes a number of Golden Ratio resources for your reference, as listed below. Your comments or suggested resources are welcome.
GoldenNumber.net site content
This site includes the following categories:
- Phi – Basics concepts, history and other interesting backgrounds on phi, 1.618, the golden ratio
- Design – Applications of the golden ratio in art, architecture and all kinds of design
- Beauty – Influences of the golden ratio in perceptions of human facial beauty
- Life – Appearances of the golden ratio in the physical dimensions of humans, animals and plants
- Math – The unique mathematical properties of the golden ratio, with many ways to derive it
- Geometry – Appearances of the golden ratio in geometry, with many ways to construct it
- Markets – Applications of the Fibonacci sequence and golden ratio to stock markets and the foreign currency exchange
- Cosmos – Appearances of the golden ratio in the solar system and universe
- Theology – Appearances of the golden ratio in scripture and observations on its meaning
- Blog – Other general interest articles
- Site Map – A complete listing of the entire content of this site.
Highly recommended articles
- Golden Ratio Myth, Fact and Misunderstandings – Covers the most common points of misunderstanding and confusion on the golden ratio, with links to supporting articles.
- Golden Ratio Overview – Many interesting aspects of the golden ratio covered in a single article.
Golden Ratio topics often covered in the various state Core Curriculum Math Standards
- Art – See the Art applications page, and the more detailed articles on use of the Golden Ratio by Leonardo Da Vinci, Sandro Botticelli and Georges Seurat.
- Architecture – See the Architecture applications page, and the more detailed articles on its appearance in the UN Headquarters Secretariat Building, Parthenon and Great Pyramid of Egypt.
- Pascal’s Triangle – See how its rows create the Fibonacci sequence.
- Human proportions – See the pages on beauty and the face, body, hands/foot and teeth.
- Nature/Animals – See the Nature page for appearances in other animals, and the more detailed article on the Nautilus shell.
- Nature/Plants – See the Plants page for appearance in plants.
Golden Ratio lesson plans, assignments and projects
After reviewing dozens of web sites for good lessons and assignments, we recommend the following resources:
- Illuminations Resources for Teaching Math – The Golden Ratio
- Teach Engineering Resources for K-12 – Discovering Phi: The Golden Ratio – Good content and materials, but note that the plant spiral image shown is likely an equi-angular spiral but not a Fibonacci or Golden spiral
- School Projects and Interviews – Ideas on projects and samples of past interviews with students.
- Do It Yourself Projects – Free templates for grids, tools and constructions.
Recommended golden ratio software, golden ratio gauges, golden mean calipers and other tools
- PhiMatrix – Golden Ratio Design and Analysis software for Windows and Mac. Provides grids that overlay any other program or image on your screen to easily find or apply the golden ratio. Free two week trial with a 50% educational discount on download page.
- Golden Ratio Calipers
- Golden mean gauges – For higher precision use and medical/dental applications
Online Videos:
There are very few online golden ratio videos of high enough quality to recommend them as teaching aids. Some show illustrations of equiangular/logarithmic spirals, which, while common in nature, are not based on golden ratio or Fibonacci spirals. Some seek to debunk the golden ratio, saying no evidence exists, ignoring evidence that does exist and presenting no contrary evidence to back their claims. Some claim it as proof of God, when both natural and supernatural reasons may exist for its many appearances. The following videos are of good quality and accuracy in their content:
- Vi Hart’s engaging series on spirals in nature, with Part 1, Part 2 and Part 3.
- Nature by the Numbers – Very well done animation of mathematics in the design of nature, but not all examples shown involve the golden ratio.
- Pythagoras: How to measure beauty – The Human Face – From a BBC Documentary, with Episode 1, Episode 2, Episode 3 and Episode 4. Episode 3 covers the golden ratio in the human face.
- Beauty and the Golden Ratio – Illustrates the many golden ratios that may be found in the human face. Note that this video shows the analysis of only one face, which is not meant to imply that every face would have the same proportions.
- Leonardo da Vinci, Salvator Mundi and the Divine Proportion – Illustrates da Vinci’s application of the golden ratio in art composition.
Online content to use with caution or avoid:
- Donald Duck in Mathmagic Land – The classic Disney video that includes a section on the golden ratio that starts at the 7:12 minute mark of the video. This video teaches some wonderful principles of mathematics and geometry and their application in nature and the arts. Unfortunately, it also contains some oversimplifications of the application of the golden ratio which have undoubtedly led to some of the confusion and controversy around this topic. These include its content on the Parthenon, Notre Dame, United Nations Headquarters, Mona Lisa and Nautilus shell, where it implies a precise appearance of the golden ratio in cases where the actual dimensions only closely approximate a golden ratio, or use it only in a portion of the instances claimed. See the links to articles in the preceding sentence for more details on each topic.
- Discovery Education – The Golden Ratio – Incorrectly states at 1:24 in the video that a golden rectangle is 5 times as wide as it is high (should be 1.618 times as wide). Incorrect illustration of the golden rectangle in Da Vinci’s “The Annunciation” painting.
- The Golden Ratio & Fibonacci Numbers: Fact versus Fiction by Stanford and Math Encounters — Fibonacci & the Golden Ratio Exposed – A Stanford University production has some good information on the Fibonacci series but incorrectly makes categorical statements that there is no evidence for many of the appearances of the golden ratio in art, architecture, the human body, etc. The evidence is quite obvious if you have the right tools and take the time to study the subject, as is documented on this site.
Background on Common Core Curriculum Math Standards for the Golden Ratio
The Grade 7 Mathematics standards include a section on “Ratios and Proportional Relationships.” The standards require “Recognize and represent proportional relationships between quantities,” and one of these requires “Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.”
In 2009, an effort was launched by leaders in various states to develop the Common Core State Standards. This was done in recognition of the value of consistent, real-world learning goals that would ensure all students graduating high school would be prepared for college, career, and life. By the early 2000s, every state had developed and adopted its own learning standards that specify what students in grades 3-8 and high school should be able to do. This lack of standardization was one reason why states decided to develop the Common Core State Standards.
During the development process, the standards were divided into two categories:
- The college and career-readiness standards, which address what students are expected to know and understand by the time they graduate from high school
- The K-12 standards, which address expectations for elementary school through high school
Standards were developed for English language arts and Mathematics. The Grade 7 Mathematics standards include a section on “Ratios and Proportional Relationships.” The standards require “Recognize and represent proportional relationships between quantities,” and one of these requires “Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.”