![]() ![]() 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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