A journey to an alternate universe based on hyperbolic geometry can serve as an entrée to eye-popping wallpaper designs. Frank A. Farris of Santa Clara University made this provocative notion the basis of a colorful, animated public lecture, recently presented at the MAA's Carriage House.
Farris began by describing a universe that has no edge because all matter—including all beings and all measuring devices—shrinks as it nears the apparent edge. In such a universe, a being (dubbed a Poincarite) would have no way of detecting the edge, which would always be infinitely far away.
Farris then introduced a convenient representation of such a universe—a two-dimensional model of hyperbolic geometry known as the Poincaré Upper Halfplane. In such a setting, "how do Poincarites go straight?" Farris asked. "What would the straightest possible path look like for them, with their different concept of measurement and distance?"
To answer these and related questions, Farris explored the nature of hyperbolic geodesics (straightest possible lines), hyperbolic isometries (transformations that move things around without changing their size), and modular functions (hyperbolic wallpaper designs).
In the halfplane representation, for example, the x-axis serves as the unattainable edge of this hyperbolic space. In this setting, the straightest path for a Poincarite heading in a "horizontal" direction is a portion of a Euclidean circle whose center is on the x-axis and that meets the edge at right angles.
In the course of his talk, Farris went back and forth between two points of view: one as an omniscient being looking down upon this hypothetical universe and the other from that of a Poincarite experiencing this curious geometry.
Farris also paid tribute to one of his mathematical heroes, Henri Poincaré (1854-1912). "It is said of Poincaré that he was the last person to know all of mathematics," Farris remarked.
To construct hyperbolic wallpaper, Farris turned to the isometries of the Poincaré Upper Halfplane—the hyperbolic analogs of translations, reflections, and rotations. In the Euclidean realm, these transformations lead to repeating patterns with particular symmetries, as represented by the 17 fundamental wallpaper patterns. You can find analogous transformations for the hyperbolic plane—dilations, inversions in circles, and so forth.
An example of hyperbolic wallpaper.
For example, if you look at enough swatches of Euclidean wallpaper, you see centers of 2-fold, 3-fold, 4-fold, and 6-fold rotation, but not 5-fold centers. They cannot occur because of the Crystallographic Restriction, a fundamental result about wallpaper patterns, which are defined to be invariant under two linearly independent translations. Nonetheless, Farris showed an image that seemed to show 5-fold symmetry, suggesting that this apparent paradox would offer material for another talk.
In the hyperbolic realm, Farris set about explaining one particular type of wallpaper pattern, which enjoys translational symmetry in just one direction, along with one center of 2-fold symmetry. Requiring these symmetries entails bewilderingly many others, and the patterns with these symmetries are dazzling.
By showing what hyperbolic wallpaper would look like, Farris took his audience to the far shores of mathematics, offering an intriguing mathematical model of a hypothetical universe. Physics suggests that such an expansive universe—one in which a random walker could wander forever and never return to its starting point—would be cold and lonely. "There's so much room down near the edge of the universe," Farris insisted. "But we can make beautiful wallpaper for the inhabitants."