People often wonder whether artists really use geometry. I am convinced that they use geometry all the time—and that they use every kind of geometry. Sometimes artists have studied geometry; Salvador Dalí learned mathematics from Thomas Banchoff and René Thom. M.C. Escher used ideas from H.S.M. Coxeter and Roger Penrose; Leonardo da Vinci was himself a mathematician. In other cases, it seems like artists get their geometric knowledge and inspiration the same way we mathematicians do; one can imagine ancient artisans developing symmetric designs by trying out different ideas to see what works. 

I have recently completed a textbook, Geometry for the Artist (Gorini, 2023), that shows how art can enrich the teaching of geometry. The main topics are symmetry, perspective, fractals, lines and curves, surfaces, Euclidean and non-Euclidean geometry, and topology. In this note, I will share some ways to incorporate art into a geometry course.

Symmetry. We find symmetric designs and patterns everywhere. Artists use symmetry to convey a feeling of balance and orderliness and even, in the case of band ornaments or tilings, the idea of infinity. The mathematics of symmetry explains the structure of designs and patterns and gives artists ways to create new designs and patterns. Interestingly, artists found two-dimensional patterns well before mathematicians could prove their properties. To analyze and create symmetric designs, we use symmetry transformations, which are simply the transformations—rotation, reflection, translation, or glide reflection—that leave a pattern or design unchanged. 

There are many ways to use symmetric designs while studying geometric transformations:

• Find the symmetry transformations (by giving rotocenters and mirror lines) of finite designs.
• Find the symmetry transformations (by giving rotocenters, mirror lines, translations, and glide reflections) of band ornaments and tilings.
• Find the symmetry transformations of patterns and designs from a variety of world cultures.
• Find finite designs, band ornaments, and tilings in the environment and describe their symmetry transformations. 
• Use dynamic geometry software to re-create designs found in the environment and then create new designs and patterns. 

Perspective. Accurate perspective depends on geometry. (See Gorini, 2023.) Some ways to challenge students with the geometry of perspective include:

• Show that a circle in perspective is an ellipse.
• Show that a sphere in perspective is an ellipse. This is unexpected, and artists usually use a circle when drawing a sphere in perspective.
• Draw a row of identical columns in perspective. This gives columns that are narrower in the middle of the row and wider as they extend out. This also is unexpected, and artists do not draw columns this way.
• Draw a checkerboard floor in perspective and explain the geometry of the construction.
• Determine how M.C. Escher uses perspective in his “impossible worlds” pictures.
• Look at René Magritte’s series The Human Condition and explain his use of perspective.

Ratio and Proportion. When artists draw people or other characters at different sizes, they make them similar—larger if they are closer, smaller if they are farther away. For similar triangles, we use the ratio of two sides. For people and cartoon characters, artists use the head-to-body ratio (head height from tip of chin to top of head divided by total height) or its reciprocal, the number of heads tall (total height divided by head height) to ensure that figures are similar. Children and child-like cartoon characters have large head-to-body ratios (1/3 or 1/4), while adults have smaller head-to-body ratios (1/7), and action heroes and high-fashion models have even smaller head-to-body ratios (1/9 or 1/10). Some ways students can work with head-to-body ratios include the following:

• Measure the head-to-body of several figures. Determine if the figures are meant to be child-like, adult, or heroic.
• Measure their own head-to-body ratio and compare it to the ratios of a child and an adult.
• Create cartoon characters that are child-like, adult, or heroic based on their head-to-body ratios.

Fractals. Fractals are intricate self-similar shapes like a fern, lightning, the branches of trees, or clouds. They have been called the “geometry of nature.” The key to a shape being a fractal is that there are similar shapes found in ever increasing or decreasing scales. There are many ways you can have students work with fractals:

• Draw the Cantor set, Koch snowflake, and Sierpinski triangle. Determine the length of the Cantor set and the perimeter and area of the other two.
• Photograph fractals in nature. Explain the fractal structure.
• Find fractals in works of art. Typically, these are waves, clouds, and mountains, but some abstract work, like that of Jackson Pollock, have a fractal appearance.

Non-Euclidean geometry. Non-Euclidean geometry is the geometry of curved surfaces, such as a sphere or a saddle, in contrast with Euclidean geometry, the geometry of a flat plane. We see non-Euclidean geometries on all the surfaces around us. The human face shows different geometries—the flatness of the brow, the roundness of the tip of the nose, and the saddle of the bridge of the nose. Non-Euclidean geometries are fun to work with. Here are some suggestions:

• Determine whether surfaces are flat, elliptic, or hyperbolic using “flappy triangles” (Casey, 1994). 
• Find regions on one’s faces or hands and classify them as flat, elliptic, or hyperbolic.
• Find different geometries within one painting.

Topology. Topology is a fascinating enrichment topic for a geometry course. The properties studied by topology, like the number of holes in a donut or pretzel, do not depend on measurements of length or angle. These properties do not change even when a shape is bent, stretched, or twisted, as if it were made of infinitely stretchy rubber (which is why topology is called “rubber-sheet geometry”). Exploring the properties of the Möbius band forces students to think in new ways. M.C. Escher has instructive pictures of the Möbius band (Ernst, 1985). Möbius Strip I shows what happens when a Möbius strip is cut down the middle, and Möbius II shows that a Möbius strip has only one side. Works by El Greco, Modigliani, Botero, Picasso, and Dalí show a variety of effects that result from stretching and shrinking. Pictures by Rob Gonsalves show dramatic topological transformations and give students ideas for creating their own.

Conclusion. I hope that you can use these ideas to incorporate art into your geometry classes and engage students who might otherwise not be interested in mathematics. When students start seeing how geometry is used in art, they love finding their own examples and bringing them into class. With a little help, they can start using geometry to create art on their own. There are many more examples everywhere in art not discussed here for you and your students to explore.

References

Casey, J. (1994). Using a Surface Triangle to Explore Curvature. The Mathematics Teacher, 87(2), 69–77. 

Gorini, C.A. (2023) Geometry for the Artist. CRC Press.

Ernst, B. (1985). The Magic Mirror of M.C. Escher. Tarquin Publications. 

You can request an examination copy of Geometry for the Artist or purchase with discount code  AEVV23 at https://www.routledge.com/Geometry-for-the-Artist/Gorini/p/book/9780367628253.