See Ford circle on Wiktionary
Download JSON data for Ford circle meaning in All languages combined (4.7kB)
{
"etymology_text": "Named after American mathematician Lester Randolph Ford, Sr., who wrote about them in 1938.",
"forms": [
{
"form": "Ford circles",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "Ford circle (plural Ford circles)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
{
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "English entries with language name categories using raw markup",
"parents": [
"Entries with language name categories using raw markup",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "English terms with non-redundant non-automated sortkeys",
"parents": [
"Terms with non-redundant non-automated sortkeys",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Circle",
"orig": "en:Circle",
"parents": [
"Curves",
"Shapes",
"Geometry",
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "other",
"langcode": "en",
"name": "Fractals",
"orig": "en:Fractals",
"parents": [],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Geometry",
"orig": "en:Geometry",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "other",
"langcode": "en",
"name": "Infinity",
"orig": "en:Infinity",
"parents": [],
"source": "w"
}
],
"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": "quotation"
},
{
"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": "quotation"
},
{
"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": "quotation"
}
],
"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",
"links": [
[
"geometry",
"geometry"
],
[
"irreducible",
"irreducible"
],
[
"fraction",
"fraction"
],
[
"coprime",
"coprime"
],
[
"integer",
"integer"
]
],
"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"
],
"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.",
"forms": [
{
"form": "Ford circles",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "Ford circle (plural Ford circles)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English entries with language name categories using raw markup",
"English eponyms",
"English lemmas",
"English multiword terms",
"English nouns",
"English terms with non-redundant non-automated sortkeys",
"English terms with quotations",
"English terms with usage examples",
"Quotation templates to be cleaned",
"en:Circle",
"en:Fractals",
"en:Geometry",
"en:Infinity"
],
"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": "quotation"
},
{
"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": "quotation"
},
{
"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": "quotation"
}
],
"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": [
[
"geometry",
"geometry"
],
[
"irreducible",
"irreducible"
],
[
"fraction",
"fraction"
],
[
"coprime",
"coprime"
],
[
"integer",
"integer"
]
],
"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"
}
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-05-09 from the enwiktionary dump dated 2024-05-02 using wiktextract (4d5d0bb and edd475d). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.