Ford Circle

  • In the news

  • Down Memory's Busy Street
    Village Voice, NY -
    ... Christa and Ford perform two duets—her catlike coiled softness a to-die-for ... Diggs, who's been asking questions as if to herself, begins to circle the dancers ...
  • Police Blotter
    Palm Beach Post, United States -
    ... A 1987 Ford Bronco was burglarized in the 2000 block of North Albatross Road ... and Dell laptop computer were stolen from a house in the 2000 block of Par Circle. ...
  • Man in custody after standoff with police
    Dodge City Daily Globe, KS -
    A 31 year-old black man is in custody in the Ford County Detention Center ... a reported incident of aggravated assault of a male and female at 646B Market Circle. ...
  • Life as an NHRA crewmember Part 2, Road Tripping
    NHRA.com, CA -
    After the winner's circle photos have been taken and the ESPN cameras have been turned off ... perks and things you don't like, just like all jobs," Shawn Ford said ...
  • Stage Door Jonny’s Guide to Great Theater
    Windy city Media Group, United States -
    ... s musical version of the 19th Century romantic novel was a failure on Broadway; but Circle Theatre in ... Monty Python’s Spamalot, Ford Center/Oriental, Dec. ...
In mathematics a Ford circle is a circle with centre at (p/q, 1/2q2) and radius 1/(2q2), where p/q is a fraction in its lowest terms (i.e. p and q are coprime integers).

1 History

2 Properties

3 See also

4 External links

Table of contents

History

Ford circles are named after American mathematician Lester R. Ford, Sr, who described them in an article in American Mathematical Monthly in 1938.

Properties

The Ford circle associated with the fraction p/q is denoted by C[p/q] or C[p,q]. There is a Ford circle associated with every rational number. In addition, the line y=1 is counted as a Ford circle - it can be thought of as the Ford circle associated with infinity, which is the case p = 1, q = 0.

Two different Ford circles are either disjoint or tangent to one another. No two Ford circles intersect - even though there is a Ford circle tangent to the x-axis at each point on it with rational co-ordinates. If p/q is between 0 and 1, the Ford circles that are tangent to C[p/q] are precisely those associated with the fractions that are the neighbours of p/q in some Farey sequence.

Ford circles can also be thought of as curves in the complex plane. The modular group of transformations of the complex plane maps Ford circles to other Ford circles.

By interpreting the upper half of the complex plane as a model of the hyperbolic plane (the Poincaré half-plane model) Ford circles can also be interpreted as a tiling of the hyperbolic plane. Any two Ford circles are congruent in hyperbolic geometry. If C[p/q] and C[r/s] are tangent Ford circles, then the half-circle joining (p/q,0) and (r/s,0) that is perpendicular to the x-axis is a hyperbolic line that also passes through the point where the two circles are tangent to one another.

See also

External links