Conway’s topograph
E29423
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Conway’s topograph canonical | 1 |
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
geometric tool
ⓘ
graphical representation of binary quadratic forms ⓘ mathematical visualization ⓘ |
| appliesTo |
definite binary quadratic forms
ⓘ
indefinite binary quadratic forms ⓘ |
| associatedWith | binary quadratic form ax^2+bxy+cy^2 ⓘ |
| basedOn | planar graph ⓘ |
| category |
mathematical diagrams
ⓘ
visual tools in number theory ⓘ |
| creator | John H. Conway ⓘ |
| encodes | values of a quadratic form ax^2+bxy+cy^2 on primitive integer pairs (x,y) ⓘ |
| field |
algebra
ⓘ
arithmetic of binary quadratic forms ⓘ geometry ⓘ number theory ⓘ |
| hasAlternativeName | topograph of a quadratic form ⓘ |
| hasConcept |
rivers representing sequences of reduced forms
ⓘ
wells and peaks corresponding to minima and maxima of the form ⓘ |
| hasPart |
edges corresponding to primitive integer vectors
ⓘ
faces corresponding to values of the form on pairs of integers ⓘ regions labeled by integer values of a quadratic form ⓘ |
| hasProperty |
encodes arithmetic information in a planar picture
ⓘ
equivariant under the action of SL(2,Z) ⓘ organizes integer solutions of quadratic equations ⓘ |
| hasRepresentation | infinite planar tree-like graph ⓘ |
| hasSymmetry | action of the modular group on the upper half-plane ⓘ |
| helpsWith |
computing class numbers of binary quadratic forms in some cases
ⓘ
visualizing equivalence of forms under SL(2,Z) ⓘ |
| inspiredBy |
Farey tessellation
ⓘ
surface form:
Farey graph
classical reduction theory of Gauss ⓘ |
| introducedInContextOf | study of quadratic forms over the integers ⓘ |
| language | integer lattice Z^2 ⓘ |
| notablePublication | Conway’s work on the sensual quadratic form ⓘ |
| relatedTo |
Farey tessellation
ⓘ
continued fraction expansions of real numbers ⓘ modular group PSL(2,Z) ⓘ reduction of indefinite binary quadratic forms ⓘ |
| teaches | geometric intuition for algebraic properties of quadratic forms ⓘ |
| usedFor |
studying binary quadratic forms
ⓘ
understanding reduction theory of binary quadratic forms ⓘ visualizing arithmetic properties of binary quadratic forms ⓘ visualizing continued fractions ⓘ visualizing geodesics on the modular surface ⓘ |
| usedIn | expository work on quadratic forms and modular groups ⓘ |
| visualizes |
flow of values of a quadratic form along edges
ⓘ
level sets of a binary quadratic form ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.