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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

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 tessellation self-linksurface differs ⓘ surface form: Farey graph
hasEdgeType	hyperbolic geodesic ⓘ
hasFaceType	ideal triangle ⓘ
hasFundamentalDomain	ideal triangle with vertices 0,1,∞ ⓘ
hasSymmetryGroup	modular group PSL(2,Z) ⓘ surface form: 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 Sr. ⓘ surface form: John Farey
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 upper half-plane model ⓘ surface form: Poincaré disk model

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Conway’s topograph → relatedTo → Farey tessellation ⓘ

Conway’s topograph → inspiredBy → Farey tessellation ⓘ

this entity surface form: Farey graph

Farey tessellation → hasDualObject → Farey tessellation self-linksurface differs ⓘ