Authors: Juma, Shannon, Jimena
Artwork, Artist: 12 Golden E-Chains Spiral, Philippe Leblanc
We met to discuss whether this would be a viable task, what materials would be required, and to outline the production process for the piece itself. We also looked into how to extend this for the classroom. We tried to understand the way the artist calculated the lengths and positions of each chain. There were also practical considerations about the available box sizes and how the bead sizes might relate to this.
The original artwork was recreated with purchases from Michael’s and Amazon. Some aspects of our piece were chosen for practical reasons (for example, Leblanc’s work utilized a square-based rectangular prism with lateral faces in the proportions of the golden ratio). We began with spreadsheets to calculate bead colour amounts. As we were determining the lengths of each chain and the position of each chain, we found that we were trying to honour the golden ratio that appeared in Leblanc’s work as well as the golden spiral that determined each chain’s place. In reality, we were able to mimic the bead count in the five longest chains; however, the remaining chain lengths were more arbitrary. We worked within the constraint that we wanted 700 beads in total and projected possible lengths that seemed plausible for the shorter chain lengths. To determine the placement of the chains, a printout from Leblanc’s information on the Bridges Math site was used. Another alteration of the piece is that we chose our own colour scheme of 10 colours.
We decided to follow Leblanc’s inspiration of irrational numbers. The first thought was to build a project around pi. However, this didn’t play out in a way that we anticipated. So, we pivoted and focused on square roots. Using either a straight edge and a compass or paper folding, we could create irrational lengths. We were also guided by the Grade 11 curriculum, which requested that we order irrational numbers. From this, we decided to ask the students to fold a Theodorus spiral from a single, long strip of paper. The plan was that the spiral served as the base for our class-focused extension to the art project: a square root mobile. The hope was that this would mimic the beads of Leblanc’s chains. (In a real class, we think that we would have also asked students to create them using different colours for each digit of the decimal expansion of each root, provided it didn’t get too confusing visually.) We have them hung from a ring so that the magnitude of each chain is reflected in relation to the bead chain that represented root one. (We felt this provided a visual to the ordering aspect that is required by the Grade 11 Pre-Calculus syllabus.)
Images from work in process:
Art Project lesson ideas
12 Golden E-Chains Spiral, Philippe Leblanc
https://gallery.bridgesmathart.org/exhibitions/bridges-2025-exhibition-of-mathematical-art/philippe-leblanc
Mathematical Concepts in the artwork itself
The artist says he is inspired by irrational numbers: e, phi, root 2, and root 3 are specifically mentioned
e → decimal expansion
links to irrationality
Catenary → equation of shape:
Phi is also present
The box is in the golden ratio
Each catenary (each strand of the 12 chains) is made to the appropriate length so that they are framed in a golden ratio → so this models the self-similarity of e as well as the recursive nature of the golden ratio (involving phi)
Square roots of perfect squares (Grades 7, 8)
Pythagorean theorem (Grade 8)
Exposure to irrational numbers pi, trig ratios (Grades 7, 10)
** Pre-calculus 11: This seems to be where we can make the most of it
Lesson arc (this might be one or multiple lessons in the “real” classroom)
Real numbers
Think- See-Wonder with the artwork that we made
(possible alternative to help visualization)
https://www.weartxl.be/fr/2025/philippe-leblanc-2/#pll_switcher
This one (right) is just called Golden E-Chains
We only have 5 -7 minutes for the interactive element.
Students Make Theodorus Spiral
https://mathworld.wolfram.com/TheodorusSpiral.html
With the help of this video
In our “real” classroom, we would pair the activity with this worksheet (modified).
https://ferrington.weebly.com/uploads/3/5/8/5/3585877/chapter_8_quilt_project.pdf
Later in the lesson or a later lesson
This is where I see the bead construction. In the classroom, this could be the expansion activity – provided it’s not too much busy work – but for this artwork assignment, I would like to give them one that is pre-made to an appropriate length, and they can put the pieces together to make an analogous artwork of square roots.
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