Mrs. Miniver's problem
Encyclopedia
Mrs. Miniver's problem is a geometry
problem about circle
s. Given a circle A, find a circle B such that the area of the intersection of A and B is equal to the area of the symmetric difference
of A and B (the sum of the area of A − B and the area of B − A).
The problem derives from "A Country House Visit", one of Jan Struther
's newspaper articles featuring her character Mrs. Miniver
. According to the story:
Alan Wachtel writes of the problem:
In the case of two circles of equal size, the ratio of the distance between their centers and their radius is often quoted as approximately 0.807946. However, that actually describes the case when the three areas each are of equal size. The solution for the problem as stated in the story ("when the area of the two outer crescents, added together, is exactly equal to that of the leaf-shaped piece in the middle") is approximately 0.529864.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
problem about circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s. Given a circle A, find a circle B such that the area of the intersection of A and B is equal to the area of the symmetric difference
Symmetric difference
In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....
of A and B (the sum of the area of A − B and the area of B − A).
The problem derives from "A Country House Visit", one of Jan Struther
Jan Struther
Jan Struther was the pen name of Joyce Anstruther, later Joyce Maxtone Graham and finally Joyce Placzek , an English writer remembered for her character Mrs...
's newspaper articles featuring her character Mrs. Miniver
Mrs. Miniver
Mrs. Miniver is a fictional character created by Jan Struther in 1937 for a series of newspaper columns for The Times, later adapted into a movie of the same name.-Origin:...
. According to the story:
She saw every relationship as a pair of intersecting circles. It would seem at first glance that the more they overlapped the better the relationship; but this is not so. Beyond a certain point the law of diminishing returns sets in, and there are not enough private resources left on either side to enrich the life that is shared. Probably perfection is reached when the area of the two outer crescents, added together, is exactly equal to that of the leaf-shaped piece in the middle. On paper there must be some neat mathematical formula for arriving at this; in life, none.
Alan Wachtel writes of the problem:
It seems that certain mathematicians took this literary challenge literally, and Fadiman follows it with an excerpt from "Ingenious Mathematical Problems and Methods," by L. A. Graham, who had evidently posed the problem in a mathematics journal. Graham gives a solution by William W. Johnson of Cleveland for the general case of unequal circles. The analysis isn't difficult, but the resulting transcendental equation is messy and can't be solved exactly. When the circles are of equal size, the equation is much simpler, but it still can be solved only approximately.
In the case of two circles of equal size, the ratio of the distance between their centers and their radius is often quoted as approximately 0.807946. However, that actually describes the case when the three areas each are of equal size. The solution for the problem as stated in the story ("when the area of the two outer crescents, added together, is exactly equal to that of the leaf-shaped piece in the middle") is approximately 0.529864.