Fibonacci math art project group reflection

From Fibonacci Curved Kites to Our Own Mathematical Art Piece

Group Member: Medha Sharma, Joshua Kanaganayagam, Helen Chen 

For our project, we chose Gauthier Cerf’s Curved Kite Fibonacci Spiral. We chose this piece because it showed us a different way of visualizing the Fibonacci sequence. We were familiar with the usual Fibonacci squares and the resulting spiral, but Cerf’s work showed us how the Fibonacci sequence could be used to make various other geometric constructions while still creating a beautiful art piece. This combination of math and art made us curious about how it was constructed and how we might use the same underlying ideas to make our own artistic design.

Reverse Engineering the Original Artwork

Before creating our own piece, we first tried to understand the math behind Cerf’s curved kite. Each curved kite was constructed from circular arcs, and an interesting property we discovered was the relationship between its dimensions and area. Understanding how these individual kites could be scaled according to the Fibonacci sequence, and rotationally arranged to form the larger spiral, helped us see that the final artwork was not simply a picture of a Fibonacci spiral.  Additional visual patterns emerged from recursively repeating a relatively simple geometric rule.

Helen computed the area of Cerf’s curved kite using geometric formulas available to a grade 9 student. However, we suspect a derivation of this complexity could only be tackled by stronger grade 9 students who are familiar with contest-level math.

After we reconstructed the original design, we obtained the mathematical intuition necessary for our own extension. In particular, our thoughts about the area of the curved kite became important later when we considered what mathematical quantities could be explored in our own design.

Developing Our Own Design

Our attempts at extending the art piece involved many more iterations than originally expected. Firstly, we considered the familiar geometry of Fibonacci squares and transcribed in them various triangular shapes. We experimented with equilateral triangles, circles, semicircles, and Reuleaux triangles.

Although many of these ideas were aesthetically pleasing, none of them matched the aesthetic we were looking for. The original artwork had a strong relationship between its visual form and its geometry, which was challenging to replicate. As the complexity of our designs increased, it became increasingly difficult to find mathematical properties that were both interesting and attainable. We did not want the resulting mathematics to feel contrived, like something added on to the artwork as an after thought. We desired the mathematical structure to be critical to the design.

This insight shaped our progress. Medha, who was our main designer, suggested that instead of trying to force a rotated triangular shape into each Fibonacci square tile, we could work directly with arcs to produce the desired spiral. This insight helped us achieve our exact desire to attain a spiral shape. To produce a solid spiral curve, circular sectors were subtracted from the tiles (in a set-difference sense), which left behind a diagonal region that could combine to form the spiral.

It was not straightforward to reach the final tile design. Medha experimented with many placements of these removed sectors. She also added a special semicircular sector to (almost) every tile, which functioned to smooth the resulting spiral. Our initial design did not include these smaller semicircles inside each tile. Upon adding them later in the design process, we created smoother transitions between the segments in each tile, which made the overall piece feel more connected and therefore cohesive.

Just when we were close to finalizing the design, Josh noticed a small inaccuracy in one of the drawings. After review, we adjusted the design to maintain consistency, and we eventually arrived at our final version.

The colour scheme also involved careful experimentation. Given that regions would be filled, we discussed how colour choices would affect the positive and negative spaces created by the sectors. For our final art piece, Medha developed a blue-and-silver colour combination. The inevitable alternation of colours highlighted the spiral’s curvature while giving our finished artwork a very different aesthetic from Cerf’s original red and white piece.

Finding Mathematics in Our New Artwork

After establishing our Fibonacci tile design, our next question was: “What interesting mathematics can be found in our new artwork?”

We chose to focus on area, as we had found the area of Cerf’s kites to be of interest. Josh focussed on performing analytic calculations and algebraically formalizing our design’s geometry. Josh made some initial errors in his calculation that resulted in much more cancellation of terms than what actually occurs. After Josh corrected his calculations, he eventually organized the full mathematical argument clearly using LaTeX. Even though there was some partial cancellation still possible (using undergraduate index tricks) we did not include this summation cancellation in this project because it exceeded the scope of the BC curriculum.

This helped us appreciate something important about the relation between math and art: changing slight recursive rules can create an entirely different art piece with distinct mathematical relations. We started with Cerf’s artistic construction, but upon making changes we needed to investigate entirely new geometric relationships rather than retaining mathematics from the original piece.

Our Interactive Activities

We want our fellow classmates to experience some of the same problem-solving that we experienced during this project. Instead of simply showing our final calculations, we developed various opportunities for them to participate during our presentation.

We planned interactive activities in four main stages:

1. We would ask the class to consider how the area of Cerf’s curved kite might be related to its smaller radius before we would reveal the relationship.
2. Since one of Medha’s earlier designs contained a small geometric inaccuracy, we planned to challenge the class to see whether they could identify what was wrong. This would allow us to share our authentic design process rather than presenting our final artwork as though everything had worked perfectly on the first attempt.
3. During Josh’s derivation of the area, we would give our classmates opportunities to predict the next step as though they were high school students. Our plan for participation during parts of the calculation should enhance engagement rather than getting them to simply watch us provide the solution.
4. Our final interactive activity involves getting classmates to match terms in the equations with the correct geometric elements in the art piece, so that they can intuit our formula for area. 

If we had more classroom time available, we could consider further extensions. We could instruct our classmates to design their own art piece in a rectangle that is formed by Fibonacci squares. Alternatively, the artwork’s construction could involve physical tiles with different materials and colours that are assigned different costs. We could get students to determine production costs given the area calculations, and perhaps involve constraints to optimize the design. We could connect this artwork to increasingly sophisticated mathematics.

Helen has worked as our project coordinator. She organized the entire powerpoint, and ensured it looked clear and nice. She also considered ways to make the presentation interactive, so allow our classmates to participate rather than just listening to us talk. Helen also connected our project math in BC, while specifically considering how this art could be used in a classroom.

Grade 9 Geometry to Pre-calc 12 Series

During this project, we realized that a single mathematical artwork could be used in a variety of ways to teach. The Grade 9 syllabus requires students to investigate the area of simple geometric shapes, like what was discussed here. At higher grade levels, our design already utilizes series from pre-calculus 12.

We took on different roles through this project. Helen researched and explained Cerf’s original art piece and explored the Fibonacci foundation. Medha focussed on the artistic design and the physical productions, while Helen and Josh contributed feedback during this iterative process. Josh performed much of the mathematical analysis, which predominantly included the area calculations. We collaboratively developed the various interactive activities.

This process was much more iterative than we could have imagined. Since many of our first ideas did not work, we had to change our design repeatedly. There were so many iterations that we needed to correct a nearly finished version. Despite this, these challenges were a valuable part of this project, which made the project fulfilling.

Despite our distinct roles, we worked as a team throughout. This involved sharing ideas, giving feedback and solving problems together, since many things did not work out as expected. We all contributed in different ways, and by helping each other throughout the process, we created the final result that we wanted.

Very Hard Working Students:

Late-night meeting working on the presentation. 

Happy faces after all the things are mostly done.