Dutch graphic artist M.C. Escher (1898–1972) devised many highly original schemes in his attempts to capture the concept of infinity visually. One strategy he often employed was to create repeating patterns of interlocking figures. However, although he could imagine how such arrays extended to infinity, the actual pattern he drew represented only a fragment of an infinite expanse.
Another approach Escher tried was to draw replicas of a figure, such as a fish, that diminish in size as they recede from a point in the middle of a circular frame. In this case, he took advantage of the peculiarities of hyperbolic geometry to create an illusion of infinity.
If you draw any triangle on a sheet of paper and add up its three angles, the result is always 180 degrees. When you draw a triangle on a saddle-shaped surface, however, the angles add up to fewer than 180 degrees.
Just as a flat surface is a piece of the infinite mathematical surface known as the Euclidean plane, so a saddle-shaped surface can be thought of as a small piece of the hyperbolic plane. Picturing what the hyperbolic plane looks like on a larger scale, however, requires some mind-bending ingenuity.
To get a feel for what the hyperbolic plane is like, you could try to sew together pieces of cloth in the shape of pentagons. Mathematician and sculptor Helaman Ferguson has done that to fashion a persistently wrinkly hyperbolic quilt by sewing the fabric so that four pentagons meet at each corner. He describes his creation as an unruly quilt that refuses to lie flat.
Helaman Ferguson’s unruly hyperbolic quilt, sewn together from pentagons of cloth, refuses to lie flat.
Such constructions are not the only way to visualize the hyperbolic plane. More than a century ago, Henri Poincaré (1854–1912) introduced a method for representing the entire hyperbolic plane on a flat, disk-shaped surface. In Poincaré’s model, the hyperbolic plane is compressed to fit within a circle. The circle’s circumference represents points at infinity. In this context, a straight line, meaning the shortest distance between two points, is a segment of a circular arc that meets the Poincaré disk’s circular boundary at right angles.
Poincaré representation of a pentagonal tiling of the hyperbolic plane in which four pentagons meet at each vertex.
Although this model distorts distances, it represents angles faithfully. The hyperbolic measure of an angle is equal to that measured in the disk representation of the hyperbolic plane. A repeating pattern made up of identical geometric shapes in the hyperbolic plane, when represented in a Poincaré model, transforms into an array of shapes that diminish in size as they get closer to the disk’s bounding circle.
In his artwork Hyperbolic 5, Ferguson has created a particularly striking representation of the hyperbolic plane. It shows necklaces of pentagonal tiles going around a central pentagon in alternating gold fives and mirror images of gold fives. The gold “5” is a reference to a famous painting by Charles Demuth (1883–1935) called The Figure 5 in Gold. Demuth’s painting was itself inspired by an imagist poem, “The Great Figure,” by his friend and colleague William Carlos Williams (1883–1963).
“My particular inspiration for Hyperbolic 5 was both Williams and Demuth, especially Williams,” Ferguson says. “Somehow I felt Williams’ verbal images were even more powerful than the painting.”
A QuickTime video clip presenting Williams’ 1928 poem “The Great Figure” is available at http://www.learner.org/catalog/extras/vvspot/video/williams.html.
Hyperbolic 5 by Helaman Ferguson.
“We have a checkerboard of red and blue right-angled pentagons,” Ferguson explains. “This is possible because the hyperbolic plane has uncountably many more triangles and consequently more pentagons than the Euclidean plane has.”
In this representation, each necklace has five times a Fibonacci number of pentagons. The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, and so on, where each successive number is the sum of the previous two numbers.
An infinite checkerboard like the one depicted in Hyperbolic 5 is a wonderful playground for the mind. Anyone interested in a game of hyperbolic chess or Go?