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Strumming the Lute of Pythagoras

The views expressed are those of the author and are not necessarily those of Scientific American.


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When I was at the Joint Math Meetings in January, the evocative name “Lute of Pythagoras” jumped out at me in a talk by Ann Hanson of Columbia College in Chicago. Hanson teaches a course, Math in Art and Nature, that satisfies the general math requirement for Columbia College but comes with a healthy helping of creative arts as well. Students learn about geometric constructions, tessellations, and other mathematical ways of generating patterns and designs, and then they find or create artwork using those ideas. Columbia is an arts and communications college, so the course is particularly suited for the school.

The Lute of Pythagoras is just one of the geometric constructions Hanson uses in her course. Two of Hanson’s students generously shared art they created for her class using the Lute of Pythagoras.

A drawing by Joseph Koch incorporates the Lute of Pythagoras into a portrait of Pythagoras himself. Image copyright Joseph Koch. Used with permission.

Both superimpose the Lute on another picture, highlighting the proportions of the underlying images.

The Lute of Pythagoras superimposed on a photograph of a forest. Image copyright Tanya Nikolic. Used with permission.

The Lute of Pythagoras is based on the “golden” isosceles triangle, a triangle with two equal sides and an apex angle of 36 degrees.

A golden triangle. The ratio a:b is the Golden ratio. The angle θ is 36 degrees, or π/5 radians. Image: Krishnavedala, via Wikimedia commons.

Each of the bottom angles is twice the size of the top angle, and with liberal use of sine and cosine addition formulas, you can check for yourself that the ratio a:b is indeed (1+√5)/2, the famous Golden ratio. Some compass and straightedge steps give you a cool pentagon-y, triangle-y, starry figure, the Lute of Pythagoras.

The Lute of Pythagoras, built from the golden triangle with vertices ABC. Image: Ann Hanson.

The ancient mathematical/musical Pythagorean cult is a bit mysterious, and apparently one of those mysteries is why this construction is called a lute! I don’t even know whether the figure was known to the Pythagoreans. I’d be thrilled if a math history buff educated me about the origin of the name.

In my correspondence with Hanson, I focused on the Lute of Pythagoras, but her students have also created quilts, origami, and tessellations for the class, in addition to learning to recognize mathematical inspiration when it appears in other people’s artwork. “One of their assignments is to go to the Art Institute [of Chicago],” Hanson says. “They say, ‘I never thought about all these painting and artwork having a relationship to mathematics.’”

Hanson, who is herself an artist as well as a math instructor, says that her math-art class is useful for students who are anxious about their math skills. “I’m not saying it’s a cure-all. This is just one approach that seems to help.” she says. “They come away with a different appreciation of math.” If you’d like to shake out the geometry cobwebs and get creative with the Lute of Pythagoras, full instructions for making the figure are here (pdf, kindly provided by Ann Hanson).

Evelyn Lamb About the Author: Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.

The views expressed are those of the author and are not necessarily those of Scientific American.



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  1. 1. M Tucker 12:50 pm 05/31/2013

    I am not a mathematician but I love mathematics and the Golden ratio, phi, is one of my favorite numbers. It appears everywhere but I was amazed at the way Ann Hanson and her students were using the Golden Triangle. And you said they were using geometric constructions to form the lute…geometric constructions are a magical technique to me and I have always been fascinated with them. I really enjoyed the pdf description of how to form the lute given a Golden Triangle but that got me thinking. Does she give her students a GT? How is a GT constructed? I had never constructed one of those before. I spent some time thinking about it and then it came to me after looking at the lute construction again. With all those pentagrams I started thinking about pentagons. Sure enough! To construct a GT you first construct a regular pentagon. If Ann had added the construction of a regular pentagon to her file it wouldn’t have made it too much longer. I looked up the construction of a regular pentagon and it is a world of fun!

    Also, this got me thinking too: “Each of the bottom angles is twice the size of the top angle, and with liberal use of sine and cosine addition formulas, you can check for yourself that the ratio a:b is indeed (1+√5)/2, the famous Golden ratio.” I don’t think Pythagoras or Euclid or Plato (who spent some time thinking about the Golden Ratio) had the use of sine and cosine addition formulas yet they knew that the ratio of the long and short sides of an isosceles triangle constructed in a regular pentagon WAS the Golden Ratio. I thought about this for a couple of days, on and off, while washing dishes, doing laundry, making beds, then it came to me…sort of. It had to be done using construction methods. They might have had use of algebra but when dealing with a geometric figure I felt sure they would use good old geometry. That led me to thinking about similar triangles.

    You can prove the ratio of the long side to the base is the Golden Ratio by bisecting one of the base angles forming a similar triangle. With a few steps it follows pretty quickly that b/a is the Golden Ratio, but you need to use Euclid’s definition of extreme and mean ratio.

    Thanks for this post Evelyn! I had a lot of fun!

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