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.
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 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)
| Subject (surface form when different) | Predicate |
|---|---|
|
John H. Conway
→
|
notableWork |