See Ford circle on Wiktionary
{
"etymology_text": "Named after American mathematician Lester Randolph Ford, Sr., who wrote about them in 1938.",
"forms": [
{
"form": "Ford circles",
"tags": [
"plural"
]
}
],
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"expansion": "Ford circle (plural Ford circles)",
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"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
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"kind": "topical",
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"name": "Circle",
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"name": "Infinity",
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"examples": [
{
"text": "Every Ford circle is tangent to the horizontal axis y#x3D;0, and any two Ford circles are either disjoint or meet at a tangent.",
"type": "example"
},
{
"text": "There is a unique Ford circle associated with every rational number. Additionally, the axis y#x3D;0 can be considered a Ford circle with infinite radius, corresponding to the case p#x3D;1,#x5C;q#x3D;0.",
"type": "example"
},
{
"ref": "1949, American Journal of Mathematics, volume 71, Johns Hopkins University Press, page 413:",
"text": "Thus the Ford circle [1], drawn tangent to the real axis at #x5C;textstyle#x7B;p#x5F;n#x7D;#x2F;#x7B;q#x5F;n#x7D;, and having radius #x5C;textstylea#x5F;#x7B;n#x2B;1#x7D;#x7B;-1#x7D;q#x5F;n#x7B;-2#x7D;, must contain in its interior some points belonging to #x5C;textstyle#x7B;p#x5F;n#x7D;#x2F;#x7B;q#x5F;n#x7D;, such as #x5C;textstyle#x5C;theta#x2B;ia#x5F;#x7B;n#x2B;1#x7D;#x7B;-1#x7D;q#x5F;n#x7B;-2#x7D; whose imaginary part lies between #x5C;textstyleq#x5F;n#x7B;-2#x7D; and #x5C;textstyleq#x5F;#x7B;n#x2B;1#x7D;#x7B;-2#x7D;.",
"type": "quote"
},
{
"ref": "2008, Jan Manschot, Partition Functions for Supersymmetric Black Holes, Amsterdam University Press, page 78:",
"text": "A Farey fraction #x5C;textstyle#x5C;frackc defines a Ford circle #x5C;textstyle#x5C;mathcal#x7B;C#x7D;(k,c) in #x5C;textstyle#x5C;mathbbC. Its center is given by #x5C;textstylekc#x2B;i#x5C;frac 1#x7B;2c²#x7D; and its radius is #x5C;textstyle#x5C;frac1#x7B;2c²#x7D;. Two Ford circles #x5C;textstyle#x5C;mathcalC(a,b) and #x5C;textstyle#x5C;mathcalC(c,d) are tangent whenever #x5C;textstylead-bc#x3D;#x5C;pm 1. This is the case for Ford circles related to consecutive Farey fractions in a sequence #x5C;textstyleF#x5F;N.",
"type": "quote"
},
{
"ref": "2016, Ian Short, Mairi Walker, “Even-Integer Continued Fractions and the Farey Tree”, in Jozef Širáň, Robert Jajcay, editors, Symmetries in Graphs, Maps, and Polytopes: 5th SIGMAP Workshop, Springer, page 298:",
"text": "Ford circles are a collection of horocycles in #x5C;mathbbH used by Ford to study continued fractions in papers such as [2, 3].[…]Two Ford circles intersect in at most a single point, and the interiors of the two circles are disjoint. In fact, one can check that the Ford circles C#x5F;#x7B;a#x2F;b#x7D; and C#x5F;#x7B;c#x2F;d#x7D; are tangent if and only if #x5C;vertad-bc#x5C;vert#x3D;1.",
"type": "quote"
}
],
"glosses": [
"Any one of a class of circles with centre at (p/q, 1/(2q²)) and radius 1/(2q²), where p/q is an irreducible fraction (i.e., p and q are coprime integers)."
],
"id": "en-Ford_circle-en-noun-i7w3dlgE",
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"raw_glosses": [
"(geometry) Any one of a class of circles with centre at (p/q, 1/(2q²)) and radius 1/(2q²), where p/q is an irreducible fraction (i.e., p and q are coprime integers)."
],
"topics": [
"geometry",
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],
"translations": [
{
"code": "de",
"lang": "German",
"sense": "any of a class of circles, each unique to a rational number and pairwise either tangent to or disjoint from each other circle",
"tags": [
"masculine"
],
"word": "Ford-Kreis"
}
],
"wikipedia": [
"Ford circle",
"Lester R. Ford"
]
}
],
"word": "Ford circle"
}
{
"etymology_text": "Named after American mathematician Lester Randolph Ford, Sr., who wrote about them in 1938.",
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"expansion": "Ford circle (plural Ford circles)",
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"examples": [
{
"text": "Every Ford circle is tangent to the horizontal axis y#x3D;0, and any two Ford circles are either disjoint or meet at a tangent.",
"type": "example"
},
{
"text": "There is a unique Ford circle associated with every rational number. Additionally, the axis y#x3D;0 can be considered a Ford circle with infinite radius, corresponding to the case p#x3D;1,#x5C;q#x3D;0.",
"type": "example"
},
{
"ref": "1949, American Journal of Mathematics, volume 71, Johns Hopkins University Press, page 413:",
"text": "Thus the Ford circle [1], drawn tangent to the real axis at #x5C;textstyle#x7B;p#x5F;n#x7D;#x2F;#x7B;q#x5F;n#x7D;, and having radius #x5C;textstylea#x5F;#x7B;n#x2B;1#x7D;#x7B;-1#x7D;q#x5F;n#x7B;-2#x7D;, must contain in its interior some points belonging to #x5C;textstyle#x7B;p#x5F;n#x7D;#x2F;#x7B;q#x5F;n#x7D;, such as #x5C;textstyle#x5C;theta#x2B;ia#x5F;#x7B;n#x2B;1#x7D;#x7B;-1#x7D;q#x5F;n#x7B;-2#x7D; whose imaginary part lies between #x5C;textstyleq#x5F;n#x7B;-2#x7D; and #x5C;textstyleq#x5F;#x7B;n#x2B;1#x7D;#x7B;-2#x7D;.",
"type": "quote"
},
{
"ref": "2008, Jan Manschot, Partition Functions for Supersymmetric Black Holes, Amsterdam University Press, page 78:",
"text": "A Farey fraction #x5C;textstyle#x5C;frackc defines a Ford circle #x5C;textstyle#x5C;mathcal#x7B;C#x7D;(k,c) in #x5C;textstyle#x5C;mathbbC. Its center is given by #x5C;textstylekc#x2B;i#x5C;frac 1#x7B;2c²#x7D; and its radius is #x5C;textstyle#x5C;frac1#x7B;2c²#x7D;. Two Ford circles #x5C;textstyle#x5C;mathcalC(a,b) and #x5C;textstyle#x5C;mathcalC(c,d) are tangent whenever #x5C;textstylead-bc#x3D;#x5C;pm 1. This is the case for Ford circles related to consecutive Farey fractions in a sequence #x5C;textstyleF#x5F;N.",
"type": "quote"
},
{
"ref": "2016, Ian Short, Mairi Walker, “Even-Integer Continued Fractions and the Farey Tree”, in Jozef Širáň, Robert Jajcay, editors, Symmetries in Graphs, Maps, and Polytopes: 5th SIGMAP Workshop, Springer, page 298:",
"text": "Ford circles are a collection of horocycles in #x5C;mathbbH used by Ford to study continued fractions in papers such as [2, 3].[…]Two Ford circles intersect in at most a single point, and the interiors of the two circles are disjoint. In fact, one can check that the Ford circles C#x5F;#x7B;a#x2F;b#x7D; and C#x5F;#x7B;c#x2F;d#x7D; are tangent if and only if #x5C;vertad-bc#x5C;vert#x3D;1.",
"type": "quote"
}
],
"glosses": [
"Any one of a class of circles with centre at (p/q, 1/(2q²)) and radius 1/(2q²), where p/q is an irreducible fraction (i.e., p and q are coprime integers)."
],
"links": [
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"fraction"
],
[
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],
[
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],
"raw_glosses": [
"(geometry) Any one of a class of circles with centre at (p/q, 1/(2q²)) and radius 1/(2q²), where p/q is an irreducible fraction (i.e., p and q are coprime integers)."
],
"topics": [
"geometry",
"mathematics",
"sciences"
],
"wikipedia": [
"Ford circle",
"Lester R. Ford"
]
}
],
"translations": [
{
"code": "de",
"lang": "German",
"sense": "any of a class of circles, each unique to a rational number and pairwise either tangent to or disjoint from each other circle",
"tags": [
"masculine"
],
"word": "Ford-Kreis"
}
],
"word": "Ford circle"
}
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