Stern–Brocot tree
E656695
The Stern–Brocot tree is an infinite binary tree that systematically lists all positive rational numbers in lowest terms exactly once, ordered by increasing value.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Stern–Brocot tree canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T7338659 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Stern–Brocot tree Context triple: [Farey tessellation, relatedTo, Stern–Brocot tree]
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A.
Sylvester sequence
The Sylvester sequence is an integer sequence defined recursively where each term is one more than the product of all previous terms, yielding rapidly growing, pairwise coprime numbers closely related to Egyptian fraction representations.
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B.
Continued Fractions
Continued Fractions is a classic mathematical monograph by Aleksandr Khinchin that systematically develops the theory and applications of continued fraction expansions in number theory and analysis.
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C.
Farey tessellation
The Farey tessellation is a geometric partition of the hyperbolic plane into ideal triangles whose vertices correspond to rational numbers, closely linked to number theory and modular group actions.
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D.
Zeckendorf
Zeckendorf is a surname most notably associated with American real estate developer William Zeckendorf and his influential role in mid-20th-century urban development.
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E.
Khinchin's constant
Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Stern–Brocot tree Target entity description: The Stern–Brocot tree is an infinite binary tree that systematically lists all positive rational numbers in lowest terms exactly once, ordered by increasing value.
-
A.
Sylvester sequence
The Sylvester sequence is an integer sequence defined recursively where each term is one more than the product of all previous terms, yielding rapidly growing, pairwise coprime numbers closely related to Egyptian fraction representations.
-
B.
Continued Fractions
Continued Fractions is a classic mathematical monograph by Aleksandr Khinchin that systematically develops the theory and applications of continued fraction expansions in number theory and analysis.
-
C.
Farey tessellation
The Farey tessellation is a geometric partition of the hyperbolic plane into ideal triangles whose vertices correspond to rational numbers, closely linked to number theory and modular group actions.
-
D.
Zeckendorf
Zeckendorf is a surname most notably associated with American real estate developer William Zeckendorf and his influential role in mid-20th-century urban development.
-
E.
Khinchin's constant
Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
infinite binary tree
ⓘ
mathematical object ⓘ |
| algorithmicUse |
implementing binary search on rationals
ⓘ
searching for rational approximations ⓘ |
| application |
Diophantine approximation
ⓘ
computing best rational approximations to real numbers ⓘ gear tooth design ⓘ |
| classification | enumeration scheme for rational numbers ⓘ |
| constructionDetail |
children of adjacent fractions a/b and c/d are formed using mediant (a+c)/(b+d)
ⓘ
each internal node has exactly two children ⓘ |
| constructionMethod | mediant operation ⓘ |
| discoveredBy | Moritz Stern ⓘ |
| edgeRepresents | adjacency of fractions via mediant ⓘ |
| encoding |
paths correspond to continued fraction expansions
ⓘ
paths correspond to sequences of left and right moves ⓘ |
| field |
combinatorics
ⓘ
number theory ⓘ |
| independentlyDiscoveredBy | Achille Brocot ⓘ |
| leftBoundary | 0/1 ⓘ |
| lists | positive rational numbers ⓘ |
| mathematicalCategory | object in discrete mathematics ⓘ |
| namedAfter |
Achille Brocot
ⓘ
Moritz Stern ⓘ |
| nodeRepresents | reduced fraction ⓘ |
| ordering | increasing value ⓘ |
| originalMotivation | optimization of gear ratios ⓘ |
| property |
adjacent fractions a/b and c/d satisfy bc - ad = 1
ⓘ
each rational appears in lowest terms ⓘ every positive rational appears at a unique node ⓘ gives a complete enumeration of Q_{>0} ⓘ lists each positive rational number exactly once ⓘ no two distinct nodes represent the same rational number ⓘ tree is ordered by increasing fraction value from left to right ⓘ |
| relatedStructure |
Calkin–Wilf tree
ⓘ
Farey sequence ⓘ continued fractions ⓘ |
| relationToContinuedFractions | each rational’s path encodes its continued fraction ⓘ |
| relationToFareySequences | adjacent fractions are Farey neighbors ⓘ |
| representation | often drawn between sentinel fractions 0/1 and 1/0 ⓘ |
| rightBoundary | 1/0 ⓘ |
| rootNode | 1/1 ⓘ |
| searchProperty | supports exact search for any positive rational via comparisons ⓘ |
| topology | countably infinite set of nodes ⓘ |
| usedIn |
computer arithmetic algorithms
ⓘ
data structures for rational search ⓘ |
| visualization | binary tree on the projective line over Q ⓘ |
| yearIntroduced | 1858 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Stern–Brocot tree Description of subject: The Stern–Brocot tree is an infinite binary tree that systematically lists all positive rational numbers in lowest terms exactly once, ordered by increasing value.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.