The Art of Parametric Equations

Can mathematical curves be beautiful? Most certainly! Precalculus students glimpse the connection between mathematics and art when they graph roses, cardioids, limaçons, and lemniscates. But these curves give just a taste of the beauty that can be achieved when graphing equations.

In a recent article from the online science magazine Quanta, Pradeep Mutalik reviews a gorgeous new math book, Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, by Frank A. Farris.  Below is a Web Sketchpad model of a parametric equation from the book.

The Quanta article notes that the curve has fivefold symmetry and challenges readers to explain the role of the coefficients 6 and 14 in producing this symmetry. You’ll find readers’ explanations for this behavior in the comments section of the article.

In addition to investigating this particular equation, the article asks readers to explore the effects of changing the coefficients of t to other values . The Web Sketchpad model makes this exploration simple: just drag the red, blue, and green sliders in the upper-right corner. Which curves do you find most interesting?

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