Implications of Theorem in Real World
The theorem implies that on any great circle around the world, the temperature, pressure, elevation, carbon dioxide concentration, or any other similar scalar quantity which varies continuously, there will always exist two antipodal points that share the same value for that variable.
Proof: Take f to be any continuous function on a circle. Draw a line through the center of the circle, intersecting it at two opposite points A and B. Let d be defined by the difference f(A) − f(B). If the line is rotated 180 degrees, the value −d will be obtained instead. Due to the intermediate value theorem there must be some intermediate rotation angle for which d = 0, and as a consequence f(A) = f(B) at this angle.
This is a special case of a more general result called the Borsuk–Ulam theorem.
Another generalization for which this holds is for any closed convex n (n>1) dimensional shape. Specifically, for any continuous function whose domain is the given shape, and any point inside the shape (not necessarily its center), there exist two antipodal points with respect to the given point whose functional value is the same. The proof is identical to the one given above.
The theorem also underpins the explanation of why rotating a wobbly table will bring it to stability (subject to certain easily-met constraints).
Read more about this topic: Intermediate Value Theorem
Famous quotes containing the words real world, implications of, implications, theorem, real and/or world:
“...one of my motivating forces has been to recreate the world I know into a world I wish I could be in. Hence my optimism and happy endings. But I’ve never dreamed I could actually reshape the real world.”
—Kristin Hunter (b. 1931)
“The power to guess the unseen from the seen, to trace the implications of things, to judge the whole piece by the pattern, the condition of feeling life in general so completely that you are well on your way to knowing any particular corner of it—this cluster of gifts may almost be said to constitute experience.”
—Henry James (1843–1916)
“The power to guess the unseen from the seen, to trace the implications of things, to judge the whole piece by the pattern, the condition of feeling life in general so completely that you are well on your way to knowing any particular corner of it—this cluster of gifts may almost be said to constitute experience.”
—Henry James (1843–1916)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)
“I love you in my dreams, but not in real life.”
—Mason Cooley (b. 1927)
“One of the most significant effects of age-segregation in our society has been the isolation of children from the world of work. Whereas in the past children not only saw what their parents did for a living but even shared substantially in the task, many children nowadays have only a vague notion of the nature of the parent’s job, and have had little or no opportunity to observe the parent, or for that matter any other adult, when he is fully engaged in his work.”
—Urie Bronfenbrenner (b. 1917)