  MORE PRECISELY 26-1 Curved Space |
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| Euclidean geometry is the geometry of flat space—the geometry taught in high schools everywhere. Set forth by one of the most famous of the ancient Greek mathematicians, Euclid, who lived around 300 b.c., it is the geometry of everyday experience. Houses are usually built with flat floors. Writing tablets and blackboards are also flat. We work easily with flat, straight objects, because the straight line is the shortest distance between any two points.
When we construct houses or any other straight-walled buildings on the surface of Earth, the other basic axioms of Euclid's geometry also apply: parallel lines never meet even when extended to infinity; the angles of any triangle always sum to 180° the circumference of a circle equals In reality, though, the geometry of Earth's surface is not really flat. It is curved. We live on the surface of a sphere, and on that surface, Euclidean geometry breaks down. Instead, the rules for the surface of a sphere are those of Riemannian geometry, named after the nineteenth-century German mathematician Georg Friedrich Riemann. For example, there are no parallel "straight" lines on a sphere. The analog of a straight line on a sphere's surface is a "great circle" the arc where a plane passing through the center of the sphere intersects the surface. Any two such lines must eventually intersect. The sum of a triangle's angles, when drawn on the surface of a sphere, exceeds 180°—in the 90°—90°—90° triangle shown in the accompanying figure, the sum is actually 270°. And the circumference of a circle is less than We see that the curved surface of a sphere, governed by the spherical geometry of Riemann, differs greatly from the flat-space geometry of Euclid. These two geometries are approximately the same only if we confine ourselves to a small patch on the surface. If the patch is small enough compared with the sphere's radius, the surface looks "flat" nearby, and Euclidean geometry is |
approximately valid. This is why we can draw a usable map of our home, our city, even our state, on a flat sheet of paper, but an accurate map of the entire Earth must be drawn on a globe.
When we work with larger parts of Earth, we must abandon Euclidean geometry. World navigators are fully aware of this. Aircraft do not fly along what might appear on most maps as a straight-line path from one point to another. Instead, they follow a great circle on Earth's surface. On the curved surface of a sphere, such a path is always the shortest distance between two points. For example, a flight from Los Angeles to London does not proceed directly across the United States and the Atlantic Ocean, as you might expect from looking at a flat map. Instead, it goes far to the north, over Canada and Greenland, above the Arctic Circle, finally coming in over Scotland for a landing at London. This is the great circle route—the shortest path between the two cities, as you can easily see if you inspect a globe. The "positively curved" space of Riemann is not the only possible departure from flat space. Another is the "negatively curved" space first studied by Nikolai Ivanovich Lobachevsky, a nineteenth-century Russian mathematician. In this geometry, there are an infinite number of lines through any given point parallel to another line, the sum of a triangle's angles is less than 180° (see the accompanying figure), and the circumference of a circle is greater than Most of the local realm of the three-dimensional universe (including the solar system, the neighboring stars, and even our Milky Way Galaxy) is correctly described by Euclidean geometry. |
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times its diameter. If these rules were not obeyed, walls and roof would never meet to form a house!