polygonal number
Let an agent handle polygonal numbers by polygon, index, formula and properties.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent handle polygonal numbers by polygon, index, formula and properties.
A number that can be represented by dots arranged in a regular polygon, such as triangular, square, pentagonal and hexagonal numbers, given by a formula for each polygon.
What it is for: Recreational and classical number theory, and teaching patterns.
It can be compute them by formula; test whether a number is polygonal; draw dot patterns; study relations between types.
Distinguishing features
Figurate numbers on polygons
Formula depends on number of sides
Triangular numbers are the simplest
Related to centred polygonal numbers
What it looks like
Dot arrangements forming polygons; sequences of numbers.
How it is recognised
Dot patterns
Names such as hexagonal or dodecagonal number
OEIS sequences
Related models
is a kind of - category
is a kind of - values
forms - series of values
is studied by - number theory
In practice
Families and kinds
triangular numbers
square numbers
pentagonal and hexagonal numbers
higher polygonal numbers such as dodecagonal
centred polygonal numbers
Identifiers
OEIS A-number per polygon type sequences, such as A000217 triangular
Failure modes and hazards
Off-by-one indexing
Mixing ordinary and centred polygonal numbers
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Definition Which polygonal number.
Polygon and index define it.
Type
Number of sides.
Type
Polygon type.
- For which polygon (number of sides)? definition
- Which index? definition
Formula
General term.
Formula
Formula.
- What formula gives the nth number? definition
- Does indexing start at 0 or 1? boundary
Properties Patterns.
Patterns relate types.
Relations
Between types.
Relations
Relations.
- How do polygonal numbers relate to each other? definition
- Which theorems apply? provenance
Test
Is it polygonal.
Test
Membership test.
- Is a given number polygonal of this type? measurement
- How is that tested? definition
Representation Dots and sequences.
Visuals help learning.
Dot patterns
Figures.
Dots
Dot representation.
- How is it drawn as dots? action
- Is the figure regular? boundary
Sequence
OEIS.
Sequence
Sequence reference.
- Which OEIS sequence lists it? provenance
- What are the first terms? measurement
History Classical study.
Polygonal numbers are ancient.
Origins
Greek mathematics.
Origins
History.
- Who studied them historically? provenance
- Which sources? provenance
Teaching
Classroom use.
Teaching
Teaching use.
- How are they used in teaching? definition
- What activities work well? action
What the second pass must settle
- Should each polygonal type be a separate entry?
- How should indexing conventions be stated?
- The registry entry includes Platonic number as an alias; is that a separate concept?