Escher’s Geometry Explained

Mathematics professor Susan Goldstine demonstrates the orientation of spherical triangles on a positively-curved object. (Photo by Rowan Copley)
Mathematics professor Susan Goldstine demonstrates the orientation of spherical triangles on a positively-curved object. (Photo by Rowan Copley)

As part of the Natural Science and Mathematics Colloquia Series, St. Mary’s mathematics professor Susan Goldstine presented The Geometries of Escher on Wednesday, Feb. 3. Her presentation discussed the link between the math and art behind M. C. Escher’s work.

“Professor Goldstine is the reason I am at St. Mary’s College,” said mathematics professor Alex Meadows, who introduced the talk on Wednesday.  “I am especially excited for what she has in store for us today.”

Goldstine, in turn, began the lecture with an introduction of her own, showing the audience a gallery of Escher’s work and including the works of Scott Kim, a letterform-focused artist who follows Escher’s style of reflection and illusion in his work.
As Escher’s work scrolled across the screen, the audience, composed of St. Mary’s college students, professors, and community members, could see the many different kinds of pieces Escher completed, from simple mathematical oddities like the Mobius Strip (a circular band with only one surface) to illusion works like the misplaced ladder of the Belvedere (an impossible cube piece) and the depiction of three-dimensional objects on a flat surface.

From there, Goldstine presented Escher’s works on the regular division of the plane, mainly in forms of animals, including lizards, swans, and fish.  “It’s a tiling of the plane, or if you want to sound all fancy and mathematical, it’s a tessellation,” said Goldstine.  “They were partly mathematical and artistic experiments, and partly tools to base more interesting artwork on.”
Escher’s work with tessellations artistically focused on alternations of foregrounds, but mathematically focused on the reflection of shapes across the plane in such a way that a pattern of polygons, from triangles to hexagons, could be seen around several vertices in the work.

After a mathematical inspiration from a 1924 paper by George Pólya that explained the seventeen possible mathematical structures of divisions of the plane, Escher explored this idea in his works with reflection tiling.  His resulting artwork depicted reflecting triangles with specific angles in a way that would create complex patterns around a center point.
Goldstine continued by discussing Escher’s work with spherical shapes, using spherical triangles (triangles with curved edges and an angle sum greater than the traditional 180 degrees) alternating in similar reflection patterns as his previous work to create complex symmetries on the surface of a sphere.

The patterns created with these triangles were explained in a proof designed by mathematician Leonhard Euler. “There’s this thing where you’re really not supposed to give a math talk without any proofs…and the cool thing about this one is that it uses very simple geometry,” said Goldstine.

In his work, Escher was concerned with showing the concept of infinity in a finite space, in his own mind not feeling successful in doing this until he discovered a paper published in 1957 by H. S. M. Coxeter, who worked with the idea of negative curvature (in mathematics terms, hyperbolic space).

The triangles and other shapes placed on this kind of surface followed different mathematical rules that allowed for different patterns of reflection and translation, as shown by Escher’s Circle Limit 3 and Circle Limit 4, which depicted tessellated angels and demons that decreased in size to infinity, theoretically, on the edge of the piece.

Goldstine concluded with this reflection style, also mentioning how University of Minnesota Duluth professor Douglas Dunham and others have used computers to analyze and experiment with the patterns in Escher’s work, before asking questions from the audience.

“I was amazed that [Escher’s] images were so mathematical, but now that seems rather obvious,” said Dr. Robert Paul, a St. Mary’s Biology professor who attended the lecture. “Dr. Goldstine made all this quite clear and accessible in her talk…to a general audience of mathphobic nincompoops as well as the mathematicians in the audience.”

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