Questioning Euclid and Understanding Escher

Story posted October 23, 2003

On the screen are two circles with repeating images inside - one in black and white, one in color. It takes most people a few moments to realize what they're seeing, but soon images of angels and devils appear in one and fish appear in the other. One drawing is by M.C. Escher; the other is an Escher-like drawing by another artist.

Jennifer Taback, visiting assistant professor of Mathematics, said that when asked for an opinion about such a drawing, most people reply, " Oh, it's very pretty."

Taback spoke on "Questioning Euclid and Understanding Escher" at a recent faculty seminar. "Today I want to go beyond, 'oh, it's pretty,'" she said. "When I look at this, I might see something slightly different than when you look at this."

Most people see the drawings as containing images that grow increasingly smaller as they approach the edge of the circle, Taback said, while she thinks of all of the devils in the first drawing as being the same size. "So somehow I must be interpreting this slightly differently than you are."

Taback's area of expertise is geometry, which is also an important aspect of Escher's art. To understand the drawings, you first have to understand tessellation. Tessellation is simply a repeating pattern; for example, the tile on a bathroom floor is a tessellation. Viewing Escher's drawings as tessellations can help to explain what he's doing.

Looking at another Escher drawing "Liberation" it's easy to see the geometric component.

Liberation, by Escher
Liberation, by M.C. Escher

In this drawing he starts with repeating triangles that eventually morph into repeating birds. Even when he's tessellating a figure, he starts with a geometric shape.

"The essential underlying geometry is easy to understand," Taback said, "because we know about triangles and we know how they repeat."

The circle drawings are a bit different though, because the same shape can't repeat indefinitely since the circle is a confined space.

"Right now, it should look to you like the devils are getting smaller," Taback said, referring to the first drawing. "At the end of the talk, it shouldn't look to you like the devils are getting smaller."

To begin the transformation of the audience, however, Taback had to start with the definition of geometry. Most geometry happens in a plane and is known as Euclidian geometry because it follows five postulates put forth by Euclid:

  1. Given any two distinct points, there exists a line segment between them.
  2. Any line segment can be extended infinitely in either direction.
  3. Given any line segment, one can draw a circle with the segment as a radius, and one endpoint as center.
  4. All right angles are congruent.
  5. Through a given point not on a given line, there exists at most one line through the point parallel to the given line.

The fifth postulate - known as the Parallel Postulate - was the key to Taback's transformation of the audience.

"This is the one that has inspired controversy," she said. "People tried for a long time to prove the parallel postulate."

Some mathematicians felt that geometry only needed four postulates and that the fifth should follow the other four and should be provable. (No one has been able to prove it, so it has remained a postulate.)

This got a few brilliant minds thinking. They wondered what would happen to geometry if they threw out the parallel postulate and invented a geometry where it wasn't true.

"Back then, it was very daring to think that the parallel postulate could be false," Taback said.

Once you assume that infinitely many lines can be drawn, you need to change everything else about geometry too. This required thinking of parallel lines in a different way. They chose to think of parallel lines simply as non-intersecting lines (and to forget about them being equidistant from each another). They realized that for the parallel postulate to be false it couldn't happen in a Euclidian plane. The new system of geometry in which the parallel postulate is false is called hyperbolic geometry.

"Hyperbolic geometry doesn't happen in a plane; hyperbolic geometry happens in a disc," Taback said. A line in hyperbolic geometry is a piece of a circle perpendicular to the boundary of the disc. "[P]arallel now means non-intersecting. This is why the parallel postulate can fail here, it's because the lines curve the way they do."

hyperbolic lines
Lines in hyperbolic space. Diagram courtesy of Professor Douglas Dunham, University of Minnesota at Duluth

So, with the problem of the parallel postulate solved, the next question to address was the length of the lines. Lines in Euclidian geometry are infinitely long; in hyperbolic geometry, they're shown in a disc, so they look short, but is this really the case?

"I really do want my lines to be infinitely long," Taback said. "The problem is that we're all wearing our Euclidian glasses."

In hyperbolic space, repeating objects of the same size have to be drawn smaller and smaller to get them to fit in the disc - but that's only a drawing.

"They look like they're only eight inches long, but they're really infinitely long," Taback said. "So, if you live in hyperbolic space...they would all look the same to you."

Taback turned her attention back to the Escher drawing. In hyperbolic space, triangles, pentagons and other shapes have curved lines, so they look slightly different than they do in Euclidian space. An Escher circle drawing is happening in hyperbolic space, so Taback views the figures as all being the same size.

"These are all different tessellations of a hyperbolic plane," Taback said.

fish tesselation
Diagram courtesy of Professor Douglas Dunham, University of Minnesota at Duluth

"There are infinitely many triangles here, all of which are the same size if you lived in the hyperbolic plane...But to us those fish look really, really tiny because that's how we have to draw them to reflect the hyperbolic plane."

Hyperbolic geometry was developed independently by three different people: Johann Carl Friedrich Gauss (1777-1855); Nicolai Lobachevsky (1792-1856); and Janos Bolyai (1802-1860).

Gauss was a mathematician who had published many other important theoretical works, and though he wrote about hyperbolic geometry, he didn't feel the need to publish any of this work. Lobachevsky was Russian mathematician, who tried to find a publisher for his work on hyperbolic geometry, but wasn't able to for many years. His work didn't become well known until the 1900s. Bolyai came up with hyperbolic geometry when he was just 18 years old, and he published it as a 20-page appendix to one of his father's textbooks. It was the only thing he ever published, but after his death 20,000 pages of other writings were found.

Escher wasn't trained in mathematics and didn't call what he was doing hyperbolic geometry, but he was able, on his own, to develop a system that works in the same way.

"So, without formal training, he really did derive this idea," Taback said.

"Now, I hope when you see a picture like this," she said, pointing the Escher-like fish tessellation, " you won't just say 'isn't that pretty.' You'll say 'There's a hyperbolic plane, and all those fish are the same size.'"

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