I like to see complexity in the things that seem mundane. While spending many weekends with my grandmother, I learned how to knit plastic mats out of a makeshift yarn she likes to call “plarn”. While struggling to perfect my crocheting skills, not only did I realize it’s harder than it looks, but I noticed that it involves dynamic and intricate shapes. While creating my first knot, intensified by the hovering eyes of my grandmother, I was lost in a sea of fascination and curiosity as I appreciated the changing shape of the plane’s curve. First, it was a loose, elongated spiral, then it became a tighter spiral, then a shorter spiral, then, a super compressed and slightly tighter spiral, until I finally pushed the upper loop through the base, eliminating spirals altogether to get a single, beautiful knot! With the mathematics of parameterization and multivariable calculus fresh in my mind, I began to connect the dots, and decided this had to be my final project.
The main concept from multivariable calculus that is being highlighted in my project are parametrized curves. As we saw in Chapter 17.1: Parameterized Curves, the parameterization for helices or spirals in three space can be written using trigonometric functions sine and cosine and z = t. The concept of Uniform Circular Motion from Chapter 17.2: Motion, Velocity, and Acceleration describes the parameterization for movement in a circular direction where x,y, and z correspond to cos(t), sin(t), and t respectively. Using the properties of Uniform Circular Motion, I was also able to designate a general equation for r(t) with appropriate variables, R, w, and t, where R is the radius, w corresponds to the period 2pi/w, and t is time and our boundary for the parameterization. Concepts involving vector fields were also relevant in this project as each spiral is assigned a direction – in the case of crocheting it is counterclockwise. All of these concepts helped me create my parameterized equation r(t) with the appropriate adjustments for the radius, period, and direction.
