The ongoing argument over the extent to which the Golden Ratio underlies art, architecture, and nature (described in part on the previous page) provides a rich source of mathematical and artistic inquiry for students and teachers. Before such investigation can occur, however, students need to have a basic understanding of the Golden Ratio and its application to art and nature. The lesson below provides this introduction.
Lesson 1 – Introducing Students to the Golden Ratio
The main materials for this lesson are the collections of images (in PDF format) found below. There are four collections; each one contains four versions of the same image. In each case, the first image conforms to the Golden Ratio. The other three have been distorted to non-golden proportions. Below the downloads, there are images that show where the Golden Ratio may be found in the first three images (Toyota logo, Florence Colgate’s face, and the Stradivarius violin). One was made in Geometer’s Sketchpad. This program provides a good visual of where the Divine Proportion is to be found, but it could do no better than the creation of close approximates. There is no such image for the fourth collection as it is a simple Golden Rectangle. In it, the Golden Ratio is found in the comparison of the length and width of the piece.
The main materials for this lesson are the collections of images (in PDF format) found below. There are four collections; each one contains four versions of the same image. In each case, the first image conforms to the Golden Ratio. The other three have been distorted to non-golden proportions. Below the downloads, there are images that show where the Golden Ratio may be found in the first three images (Toyota logo, Florence Colgate’s face, and the Stradivarius violin). One was made in Geometer’s Sketchpad. This program provides a good visual of where the Divine Proportion is to be found, but it could do no better than the creation of close approximates. There is no such image for the fourth collection as it is a simple Golden Rectangle. In it, the Golden Ratio is found in the comparison of the length and width of the piece.
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- Start the lesson by having students view these collections with the instruction to select from each the version they judge to be most aesthetically pleasing. This can be accomplished by posting the collections along the walls of the classroom and having the students do a gallery walk. Alternately, the images can be delivered to the students electronically. Ideally, most students will choose the images that feature the Golden Ratio (the first one in each collection), but this is not necessary. Regardless of the selection of the majority, the students should be guided to the fact that people tend to be attracted to and develop a preference for images that feature a specific mathematical and artistic property. Indicate the image in each collection that falls into this category without mentioning the Golden Ratio at this time.
- Equip pairs or small groups with a ruler and one of the collections viewed in Step 1. Without providing any more guidance other than the fact that the mathematical aspect of people’s attraction to certain images has something to do with measurement (hence the ruler), instruct students to search their images for the mathematical key to beauty and aesthetics. Some students may already possess background information about the Golden Ratio. This is not a problem. There is much to be gained from students’ search for the Golden Ratio at this point.
- At an appropriate moment (after they have had some time to try out various theories, but before frustration sets in), halt the students’ investigations and lead a discussion about their ideas. If someone mentions the Golden Ratio, use this as a segue to step 4. Otherwise, accept and discuss all theories before teaching a mini-lesson on the Divine Proportion. The video “Golden Ratio #1” from the previous page might be helpful in this endeavour. Alternately, the Geogebra applet “Golden Ratio 1” could be used to provide a visual upon which students can construct their understanding. Note: there are many Geogebra applets that model the Golden Section on a line segment, but this one allows the user to achieve exact equivalence (1.618) for both a/b and 1/a.
- Once students have learned about the Divine Proportion, introduce them to the connection between it and beauty in art, architecture, and nature. Some of the information presented on the previous page could be used to illustrate this connection. Also introduce them to the lack of agreement and ongoing argument about the prevalence of the Golden Ratio in these arenas (again, the information from the previous page could be harnessed for this purpose). Knowing about these things will help provide a purpose for students’ future work with the Golden Ratio.
- Send the students, now armed with new knowledge, back to their image collections. Instruct them to search for the Golden Ratio in the golden versions of their images. At the end of class, consolidate the learning in such a way that allows all students to see the Divine Proportion in all four images. Regardless of students’ initial preferences as stated in Step 1, remind them that, according to popular belief, it is the use and/or presence of the Golden Ratio in these images that makes them more aesthetically appealing to many people.
References
Castillo, A. (2017, April 26). Golden Ratio 1. [Geogebra applet]. Retrieved from https://www.geogebra.org/m/DYWQBEXD
Meisner, G. (2013, September 1). Florence Colgate [Electronic image]. Retrieved from https://www.goldennumber.net/facial-beauty-golden-ratio-florence-colgate/
Nair, P. (2014, July 21). Toyota logo [Electronic image]. Retrieved from http://trancendconsultingtalks.blogspot.com.co/2014/07/the-golden-ratio-brand-identity-design.html
Rubino, T. (2013, October 2). Golden ratio abstract [Electronic image]. Retrieved from https://fineartamerica.com/featured/golden-ratio-abstract-tony-rubino.html
Wu, F. (2015, January 18). The "lady blunt" violin with golden sections marked out [Electronic image]. Retrieved from https://wufengengineering.wordpress.com/2015/01/18/stradivarius-music-of-the-golden-ratio/
Castillo, A. (2017, April 26). Golden Ratio 1. [Geogebra applet]. Retrieved from https://www.geogebra.org/m/DYWQBEXD
Meisner, G. (2013, September 1). Florence Colgate [Electronic image]. Retrieved from https://www.goldennumber.net/facial-beauty-golden-ratio-florence-colgate/
Nair, P. (2014, July 21). Toyota logo [Electronic image]. Retrieved from http://trancendconsultingtalks.blogspot.com.co/2014/07/the-golden-ratio-brand-identity-design.html
Rubino, T. (2013, October 2). Golden ratio abstract [Electronic image]. Retrieved from https://fineartamerica.com/featured/golden-ratio-abstract-tony-rubino.html
Wu, F. (2015, January 18). The "lady blunt" violin with golden sections marked out [Electronic image]. Retrieved from https://wufengengineering.wordpress.com/2015/01/18/stradivarius-music-of-the-golden-ratio/