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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Ulam spiral canonical | 2 |
| Ulam doodled primes on graph paper during a scientific meeting | 1 |
| prime number spiral | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T718424 — 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: Ulam spiral Context triple: [Stanislaw Ulam, notableWork, Ulam spiral]
-
A.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
-
B.
Look-and-say sequence
The look-and-say sequence is a famous integer sequence where each term is generated by verbally describing the digits of the previous term, studied for its surprising combinatorial and growth properties.
-
C.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
D.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
-
E.
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.
- 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: Ulam spiral Target entity description: 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.
-
A.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
-
B.
Look-and-say sequence
The look-and-say sequence is a famous integer sequence where each term is generated by verbally describing the digits of the previous term, studied for its surprising combinatorial and growth properties.
-
C.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
D.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
-
E.
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.
- F. None of above. chosen
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 ⓘ |
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: Ulam spiral Description of subject: 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.
Referenced by (4)
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
this entity surface form:
prime number spiral