More info on your write-ups for your math art project (for September 22)

Our group members are: Eric, David, and Henry. For this project, we choose Caroline Bowen's 2026 artwork <y, x>. This artwork is a 30 x 45.5 cm drawing made with gesso, coloured pencil, and graphite pencil. Bowen describes it has a revisiting of a series of vector field drawing that she made in 2019. What interested us most about the origin of the artwork was Bowen's decision to create the vector field manually. She calculated the direction of each vector using a handheld calculator, and then drew each vector individually on a grid of circles made from a compass. She describes this process as creating a sense of "mathematical intimacy" with mathematics that we would normally let computers calculate for us.

To remake the artwork, we approached the process in the opposite direction. Instead of calculating and drawing every vector by hand due to a lack of time, we used MATLAB to generate the vector field using Main.m and Regenerate_graph.m. Main.m allows the user to choose either differential or implicit equations and generate their corresponding fields. Main.m also allows initial conditions to be entered so that solution curves can be displayed. The program generates vectors over a grid and uses numerical integration to determine how the field moves from each starting point. We then created Regenerate_graph.m which was adjusted specifically to reproduce the behaviour of Bowen’s <y, x> artwork by making sure that the arrows pointed in the correct direction in different regions of the plane rather than simply producing lines with the correct slope. We also had to deal with the behaviour near the origin, where the usual slope-based method becomes problematic, so the program uses nearby starting points and checks how they behave within the grid.

Regenerate_graph.m

One difficulty occurred around the origin, where the vector behaviour could not be handled in exactly the same way as the rest of the grid. Working through these problems helped us see the difference between simply graphing a differential equation and representing an actual vector field. It also gave us a new appreciation for Bowen’s original process. MATLAB can calculate hundreds of vectors almost instantly, while Bowen deliberately performed this repetitive calculation and drawing by hand. Her artwork therefore makes the process of doing the mathematics itself part of the artwork, whereas our recreation explores what happens when that same process is automated.

We also slightly changed the piece to make it our own. Rather than trying to create an exact digital copy of Bowen’s physical drawing, we developed a MATLAB “Vector Field Studio” application that can generate other vector fields using the same basic visual idea. The program uses a dark background, a coloured gradient across the vectors, and allows the user to enter different equations and initial conditions. This change allowed the project to take in different user inputs and see how the graph is altered. 

Our interactive activity with the class is still being designed. Our current idea is to connect differential equation vector fields to functions that students have already learned, then allow them to enter their own functions and observe the resulting vector field patterns. However, we think this activity may be a little too difficult, especially for students who are unfamiliar with differential equations, so we may simplify or change the activity later.