At most two edges can meet at any one point. A hyperbolic polygon is a region of the hyperbolic plane whose boundary is decomposed into finitely many generalized line segments (recall that this includes circle segments), called edges, meeting only at their endpoints, called vertices. Since lines in hyperbolic space differ from our intuition about lines in Euclidean space, we must adjust our understanding of polygons as well. A geodesic between points A and B (created with GeoGebra). This disc model is precisely what Escher saw in Coxeter’s book, and is what he used to create his art.įigure 6. Thus, the idea of “lines” in Euclidean space is generalized when in hyperbolic space to include circles. In Figure 6, the shortest distance, called a geodesic, between A and B is the arc length of the given circle. It turns out that the shortest distance between two points lies along the arc of a circle that is perpendicular to the boundary. Thus, the shortest path between two points may be curved to take advantage of the “less dense” area towards the center of the disc. We think of the space in the disk getting infinitely more “dense” as we approach the boundary of the disc, so the distance of a straight line between two points get longer as we approach the boundary. We can imagine hyperbolic space as an open disk in the complex plane $\C$. To understand them, we will explore an important model of hyperbolic space: the Poincaré disc model. Hyperbolic geometry has many interesting properties that counter our ingrained Euclidean intuition. This is because, while a Euclidean surface has curvature equal to zero everywhere, a hyperbolic surface has constant negative curvature (for comparison, a sphere has constant positive curvature).įigure 4. The wavy structure is the tip-off that their surfaces exhibit hyperbolic geometry. Two common examples are sea slugs (Figure 4) and lettuce (Figure 5). Although it at first seems unnatural to think about parallel lines performing in “new” ways, hyperbolic surfaces can be found in nature. Proving that the postulate need not hold led to the discovery of an important “non-Euclidean” geometry called hyperbolic geometry. However, the Parallel Postulate need not hold true in all cases, such as on the surface of a sphere. Proving that triangles have 180˚ angle sums is an application of this postulate. The “Parallel Postulate,” which states that if one straight line crosses two other straight lines to make both angles on one side less than 90˚, then the two lines meet. However, one of them was a great source of debate between mathematicians. This assumes Euclid’s axioms, which he intended to be the basis of all geometry. So what is hyperbolic space? Grade school mathematics is taught using Euclidean geometry. To fully understand the beauty of his works, it is helpful to have a basic understanding of hyperbolic geometry. This image sparked a new area of Escher’s exploration of infinity. Coxeter sent Escher a copy of the talk, which included an illustration depicting a tessellation of the hyperbolic plane (Figure 3). Coxeter and Escher struck up a correspondence when Coxeter hoped to use Escher’s unique depictions of symmetry in a presentation for the Royal Society of Canada. Escher’s ideas about structure, pattern, and infinity were suddenly enhanced when he came across the work of geometer H.
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