Ulam spiral
E86904
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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Ulam doodled primes on graph paper during a scientific meeting | 1 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
graphical representation of integers
ⓘ
mathematical visualization ⓘ prime number visualization ⓘ |
| constructionMethod |
integers arranged on a square lattice
ⓘ
numbers placed in a spiral starting from the origin ⓘ |
| coordinateSystem | integer lattice in the plane ⓘ |
| discoveredBy | Stanislaw Ulam ⓘ |
| discoveryContext |
Ulam spiral
self-linksurface differs
ⓘ
surface form:
Ulam doodled primes on graph paper during a scientific meeting
|
| field |
mathematical art
ⓘ
number theory ⓘ |
| generalization |
alternative starting points and orientations
ⓘ
color-coding composites and primes ⓘ higher-dimensional prime spirals ⓘ |
| hasVariant |
spirals based on different step patterns
ⓘ
spirals marking only primes ⓘ spirals marking values of other multiplicative functions ⓘ |
| inspired | further visualizations of arithmetic functions ⓘ |
| mathematicalObservation |
lines often correspond to polynomials of the form an^2+bn+c
ⓘ
many prime-rich diagonals correspond to quadratic polynomials ⓘ |
| namedAfter | Stanislaw Ulam ⓘ |
| notableFeature |
primes tend to lie on diagonal lines
ⓘ
visible linear patterns of prime-rich diagonals ⓘ |
| openProblem | no complete theoretical explanation of all observed patterns is known ⓘ |
| patternType | square spiral ⓘ |
| popularizedBy | Martin Gardner ⓘ |
| publicationContext | first widely publicized in Scientific American ⓘ |
| relatedConcept |
Eratosthenes spiral
ⓘ
Gaussian integers ⓘ Sacks spiral ⓘ Vogel spiral ⓘ prime constellations ⓘ prime number distribution ⓘ quadratic polynomials generating many primes ⓘ randomness of primes ⓘ |
| representationMedium |
computer graphics
ⓘ
graph paper ⓘ |
| reveals | diagonal alignments of prime numbers ⓘ |
| startPoint | number 1 at the center ⓘ |
| suggests | non-random structure in the distribution of primes ⓘ |
| usedFor |
computer experiments in number theory
ⓘ
educational demonstrations of prime distribution ⓘ exploring visual patterns in primes ⓘ |
| visualizes |
positive integers
ⓘ
prime numbers ⓘ |
| visualProperty |
asymmetry in prime distribution along different diagonals
ⓘ
rotational symmetry of the underlying square spiral ⓘ |
| yearOfDiscovery | 1963 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Ulam doodled primes on graph paper during a scientific meeting