We were drawn to the idea of connecting math and art for our project because we wanted to highlight the similarities between the two disciplines. Math and art are often regarded as unrelated, but we wanted to find some point in which they intersect. To do this, we came up with the idea of recreating a well-known piece of art with mathematics as our medium. We considered many sculptures as options to recreate, but we were ultimately drawn to Koons’s Balloon Dog because of its bold colors and striking simplicity. It was a piece of art that made a statement about life, yet was also reasonable to recreate with graphs. Also, since Balloon Dog is so widely recognized, we thought it would translate well when we presented our final project to a wider audience.
Recreating Balloon Dog was a fairly simple process; it mainly concerned the manipulation of the general sphere equation (x^2 + y^2 + z^2 = r2) to fit the overall design for the sculpture’s shape. Transformations such as multiplying the x, y, and z terms by a multiplier of 1/a, 1/b, and 1/c stretched the sphere along an axis to create an ellipsoid. Increasing the a, b, and c terms increased the horizontal or vertical length of the ellipsoid. This technique helped differentiate the different sections of the balloon dog. For instance, we wanted the body of the dog to be thicker and longer than its ears. Thus, we ensured that the r and a values for the ellipsoid that modeled the dog’s body were larger than the r and a values for the ellipsoid that modeled the dog’s ears. The final transformation we utilized during this process was using addition to horizontally and vertically shift the ellipsoid. Adding to or subtracting from the x, y, and z terms caused the center of the ellipsoid to shift. This was advantageous when positioning the parts of the dog in space. For example, the equation ( 1/6 𝑥^2 + 𝑦^2 + (𝑧 − 1)^2 = 1/1. 5) was used to model the body of the dog. The “(𝑧 − 1)^2” term caused the center to be shifted up by one unit. Similar transformations were performed on the x and y terms that varied based on where we wanted the ellipsoid to be positioned in space. Ultimately, the use of mathematical transformations to the general equation of a sphere was the key technique we used to achieve our goal of recreating Koons’s masterpiece.
