Non-Eucildean geometry is the study of spatial relationships in which the Euclidean parallel postulate is altered or replaced to produce a system in which the traditional properties of space are altered.

As a 12-year-old, German-American physicist Albert Einstein was given a small book on Euclidean plane geometry. More than half a century later, Einstein referred to it as "that holy geometry booklet." In 1922, in her poem "The Harp Weaver," American poet Edna St. Vincent Millay wrote that "Euclid alone has looked on beauty bare." That beauty, viewed in the third century B.C. by the most prominent mathematician of antiquity, Euclid of Alexandria, concerns the relationships between points and lines lying in a two-dimensional space, a plane.

Move a point uniformly, and a straight line is produced. Move that straight line in a direction perpendicular to itself to produce a plane. Euclid demonstrated the properties of...