Visualizing Hyperbolic Geometry

In preparation for her Distinguished Lecture on September 15, Evelyn Lamb practiced reciting Euclid’s fifth postulate without taking a breath mid-statement.

“If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles,” it reads, “then the two lines inevitably must intersect each other on that side if extended far enough.”

Hyperbolic geometry grew, Lamb explained to a packed Carriage House, from the irksome fact that this mouthful of a parallel postulate is not like the first four foundational statements of the axiomatic system laid out in Euclid’s Elements.

In “Visualizing Hyperbolic Geometry” Lamb helped her audience wrap their minds around—and get their hands on—the strange new world that arises from adopting an alternate version of the infamous fifth.            

To make it to the hyperbolic meat (kale, actually, but more on that later) of her talk, Lamb first had to speed through millennia of mathematical history, abridge 2000 years of mathematical thought.

“‘Ugh, I hate that postulate!’” she paraphrased. “‘It is not as self-evident as the other postulates! Can I prove it from the other postulates?’”  

Mathematicians’ efforts to prove Euclid’s fifth postulate had what Lamb considers a felicitous (if ironic) outcome.

“So many theorems about hyperbolic geometry were proved by people who were trying to disprove hyperbolic geometry,” she said.

Alternatives to the parallel postulate are often phrased as variations on an equivalent but more comprehensible statement: Playfair’s axiom. Playfair states that given a line L and a point P not on L, there is exactly one line through P that does not intersect L. Change “one” to “none” in this formulation (and modify the English accordingly, of course), and you get spherical geometry.

Lamb used a “spherical interlude” to ease her audience out of “Euclid-land.” Many aspects of spherical geometry are familiar to us as residents of a more-or-less spherical planet, she noted. Think of Playfair’s line as the equator and Playfair’s point as the North Pole, for example. All lines—in this case great circles—through the North Pole do in fact intersect the equator.

Lamb also illustrated how edible spheres can aid in understanding spherical geometry. Wrap rubber bands around a grapefruit to form a triangle on its surface, and you’ll be able to convince yourself that, in spherical geometry, the sum of the angles of a triangle exceeds 180°.

In hyperbolic geometry, however, the opposite is true: If, given a line L and a point P not on L, there are infinitely many lines through P that do not intersect L—this is the hyperbolic version of Playfair’s axiom—the angle sum of a triangle is less than 180°. >Food again proves helpful for visualization. Think of a Pringle. Imagine drawing a triangle on the crisp, and you’ll realize that it will be “squished in a little bit on the sides.”

A Pringle provides “a perfect local picture of hyperbolic geometry,” but how can we represent a space that looks like a Pringle everywhere? Enter the Poincaré disk. In this mathematical model, hyperbolic space is a disk and “straight lines” are the arcs of circles that intersect the boundary of the disk at right angles. The model defines distance such that the boundary of the disk is infinitely far away, the going slower the farther you are from the center.

“You can think of it as there’s really thick sand, and the sand gets thicker towards the edges of the Poincaré disk,” Lamb explained.

Those fish in Escher’s “Circle Limit I” may to our Euclidean eyes like they’re getting smaller and smaller, but they’re actually all the same size.

Physical models can not only convince you of this but also offer what Lamb called a “visceral experience of how much space there is in hyperbolic geometry.”

Lamb passed around a “hyperbolic soccer ball” constructed by taping Euclidean hexagons and heptagons together. She showed a photograph of a triangle-and-square construction Berkeley’s Katie Mann decorated with Escher-inspired fish.

Lamb’s slides included images of “a lot of people’s favorite visualization of hyperbolic geometry,” hyperbolic crochet, and she brought along her own first foray into the needlecraft. She raved about the impressive twirl of Andrea Hawksley’s hyperbolic skirt.

The negative curvature characteristic of hyperbolic space appears often in nature, Lamb observed, in creatures from coral to kale.

Lamb ended her lecture on a speculative note. “Is it just an accident that Euclidean geometry feels so real to us?” she asked. What if we were insects that lived on kale? If we were tiny enough to call a curly vegetable home, and yet retained the mental capacity for mathematics, would hyperbolic have been the first geometry we discovered?

“Would it have been our natural view of the world?” Lamb wondered. “I think that’s a fun little thing to think about.” –Katharine Merow

Lamb’s lecture, part of MAA’s NSA-funded Distinguished Lecture series, was live-streamed on Facebook. Visit the Videos Section on MAA’s Facebook page to watch the talk.

Katharine Merow is a freelance writer living in Washington, D.C.