Farey tessellation
E169192
geometric tessellation
hyperbolic tessellation
ideal triangulation
object in hyperbolic geometry
object in number theory
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Farey graph | 2 |
| Farey tessellation canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1484055 — 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: Farey tessellation Context triple: [Conway’s topograph, relatedTo, Farey tessellation]
-
A.
Conway’s topograph
Conway’s topograph is a geometric visualization tool introduced by mathematician John H. Conway to study binary quadratic forms and their arithmetic properties using a planar graph of curves and regions.
-
B.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
C.
Ulam spiral
The Ulam spiral is a graphical arrangement of the positive integers in a spiral pattern that reveals striking diagonal alignments of prime numbers, suggesting unexpected structure in their distribution.
-
D.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
-
E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- 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: Farey tessellation Target entity description: 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.
-
A.
Conway’s topograph
Conway’s topograph is a geometric visualization tool introduced by mathematician John H. Conway to study binary quadratic forms and their arithmetic properties using a planar graph of curves and regions.
-
B.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
C.
Ulam spiral
The Ulam spiral is a graphical arrangement of the positive integers in a spiral pattern that reveals striking diagonal alignments of prime numbers, suggesting unexpected structure in their distribution.
-
D.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
-
E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
geometric tessellation
ⓘ
hyperbolic tessellation ⓘ ideal triangulation ⓘ object in hyperbolic geometry ⓘ object in number theory ⓘ |
| appearsIn |
Teichmüller theory
ⓘ
hyperbolic 2-orbifolds ⓘ study of mapping class groups of surfaces ⓘ |
| boundaryIdentifiedWith | projective line over Q ⓘ |
| constructedFrom | geodesics between pairs of rational points on the real line and infinity ⓘ |
| definedOn | hyperbolic plane ⓘ |
| edgeConnects | fractions a/c and b/d with |ad − bc| = 1 ⓘ |
| embeddedIn | Poincaré upper half-plane model ⓘ |
| encodes |
adjacency of rationals in Farey sequences
ⓘ
mediant operation on fractions ⓘ |
| generalizedBy | tessellations associated to other Fuchsian groups ⓘ |
| hasCombinatorialStructure | infinite planar triangulation ⓘ |
| hasCurvatureContext | constant negative curvature ⓘ |
| hasDualObject |
Farey graph
ⓘ
linked to:
Farey tessellation
|
| hasEdgeType | hyperbolic geodesic ⓘ |
| hasFaceType | ideal triangle ⓘ |
| hasFundamentalDomain | ideal triangle with vertices 0,1,∞ ⓘ |
| hasSymmetryGroup |
PSL(2,Z)
ⓘ
linked to:
modular group PSL(2,Z)
|
| hasVertex |
0
ⓘ
1 ⓘ ∞ ⓘ |
| hasVertexSet |
extended rational numbers
ⓘ
rational numbers union infinity ⓘ |
| induces | triangulation of the boundary circle by rationals ⓘ |
| isInvariantUnder |
group SL(2,Z) acting projectively
ⓘ
modular group PSL(2,Z) ⓘ |
| isLocallyFinite | false ⓘ |
| mathematicalDomain |
hyperbolic geometry
ⓘ
number theory ⓘ |
| namedAfter |
John Farey
ⓘ
linked to:
John Farey Sr.
|
| relatedTo |
Farey sequence
ⓘ
Ford circles ⓘ Stern–Brocot tree ⓘ continued fractions ⓘ geodesics in the modular surface ⓘ modular group ⓘ modular surface ⓘ rational approximations ⓘ |
| usedIn |
Diophantine approximation
ⓘ
coding of geodesic flows ⓘ study of Fuchsian groups ⓘ study of modular forms ⓘ symbolic dynamics on the modular surface ⓘ |
| visualizedIn |
Poincaré disk model
ⓘ
linked to:
Poincaré upper half-plane model
|
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: Farey tessellation Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Farey tessellation
linked to: Farey tessellation