Great Circles and Angles
The circular paths formed by the rubber bands around a ball (see "Lines on a Sphere") are called great circles. If you were to slice a ball exactly in half, the rim would be a great circle.
On Earth, one geographic example of a great circle would be the equator. The lines of longitude are great circles that intersect one another at the North Pole and at the South Pole.
Great circles are the largest of all circles that can be drawn on the surface of a sphere.
These great circles divide up the surface of a sphere into various regions, such as the curved triangle shown.
The shortest distance between two points on a sphere is along the arc of a great circle joining the two points. On any three-dimensional surface, including a sphere, the shortest distance between two points is called a geodesic.
To find the geodesic between any two points marked on a baseball or a tennis ball, just stretch a rubber band around the ball to form a great circle that passes over both points.
A baseball diamond on a small, spherical asteroid would have baselines that are arcs of great circles. That's why the baselines end up being curved (see "Alien Baseball Field").
Four great circles intersect to form a "square."
Curved baselines are not the only weird things about an asteroid baseball diamond. If you were to measure the angles at the four corners of the diamond, you would get a surprising result.
On a flat surface, the angles are each 90 degrees, but on a spherical surface, they are not. Furthermore, the sum of the four angles is 360 degrees on a flat baseball diamond, but not on a sphere. How large are the angles?
The baseball diamond is a square shape, with all four angles the same size. On a sphere, each angle of a square is greater than 90 degrees. So, the sum of the angles of a four-sided figure on a sphere is greater than 360 degrees. The larger the diamond (square), the larger the angles on any given sphere.
The sum of the angles of a triangle on a sphere is greater than 180 degrees.
Great Soccer Circles
Study the edges of the pentagons and hexagons on a classic soccer ball. Try covering the edges with large rubber bands. Are all of the "edge" lines on a soccer ball arcs of great circles?
Pentagons and hexagons happen to fit together so their edges all meet on great arcs on a sphere. Because these shapes are rounded on a sphere, their angles are larger than the angles of regular pentagons and hexagons that are flat.
Minipuzzler
A pilot flies due south 100 kilometers, then goes east 100 kilometers, then north 100 kilometers. She ends up right back where she started. Where was her starting point? (Hint: Study a world globe.)
Answer: The simplest answer is that the pilot started at the North Pole. It is also possible, however, that the pilot started somewhere on a great circle that is 116 kilometers from the South Pole. After flying south 100 kilometers, then east 100 kilometers, she would have completed a circle around the South Pole. Then, when she goes north 100 kilometers, she ends up right back where she started.
See also "Touring the Poles."
Sphereland
Pilots and airline route planners covering large distances across the globe have to consider Earth's shape in their calculations. The shortest route from New York to Tokyo, for example, doesn't go directly from east to west along a line of latitude, but actually follows a great circle that passes near the North Pole. Drawn on a flat map, such a route may look curved, but it's really a shortcut for both pilot and passengers.
In fact, spherical geometry plays an important role not just in navigation but also in many other fields. In mathematics it appears in trigonometry, topology, calculus, and other areas.
Spherical geometry also has applications in physics, chemistry, crystallography, earth sciences, astronomy, art, technical drawing, industrial design, and engineering. You couldn't put a satellite into orbit around Earth or send a spacecraft to Mars without understanding spherical geometry.
NEXT: Why a Baseball Could Orbit
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