regular polygon
Let an agent define regular polygons, relay formulas for angles, area and circumradius, explain constructibility and star polygons, and identify specific regular n-gons.
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 define regular polygons, relay formulas for angles, area and circumradius, explain constructibility and star polygons, and identify specific regular n-gons.
A polygon that is both equiangular and equilateral, with all sides and all interior angles equal, including convex forms such as the equilateral triangle, square, regular pentagon and hexagon and star forms such as the pentagram and henicosagram, with n-gons for any n of 3 or more such as the 257-gon and 65537-gon, and degenerate cases such as the digon and the zerogon in some conventions; regular polygons are central to geometry, tilings, symmetry groups and constructibility with compass and straightedge.
What it is for: Not applicable; a geometric object.
It can be define and classify; relay formulas; explain constructibility; identify specific n-gons.
Distinguishing features
Equal sides and angles
Dihedral symmetry
Schlafli symbol notation
Gauss-Wantzel constructibility
What it looks like
A closed figure with equal sides and equal angles, convex or star-shaped.
Physical character
interior angle: (n-2)*180/n degrees - convex
area: (1/4)*n*s^2*cot(pi/n) formula - side s
How it is recognised
Equilateral and equiangular polygon
Convex n-gons and star polygons
Equilateral or equiangular alone is not regular; regular polytopes generalise to higher dimensions
Related models
is a kind of - in registry terms
is a kind of - in registry terms
is a kind of - in two dimensions
is described by - the notation
In practice
Families and kinds
convex regular polygons from the triangle upward
star polygons such as the pentagram
constructible polygons including the 17-gon, 257-gon and 65537-gon
degenerate cases such as the digon and zerogon
improper regular polygons in some treatments
Identifiers
Schlafli symbol {n} or {n/m}
Standards and regulation
No regulation; mathematical conventions
Failure modes and hazards
Confusing equilateral with regular
Convention disputes on degenerate cases
Errors in formulas
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.
Define Definition.
Mathematics.
Definition
Definition.
Definition
Definition.
- What is a regular polygon, and what does the Schlafli symbol encode? definition
- Is the figure regular, or only equilateral or equiangular? boundary
Kinds
Convex and star forms.
Kinds
Kinds.
- How do convex and star regular polygons differ, and what are the degenerate cases? definition
- Which entry fits the specific n-gon? action
Compute Formulas.
Mathematics.
Angles
Angles and radii.
Angles
Angles.
- What are the formulas for interior angles, circumradius and inradius? measurement
- Which entry fits polygon geometry? action
Area
Area and perimeter.
Area
Area.
- How are area and perimeter computed, and how do they approach a circle as n grows? measurement
- Which references are standard? provenance
Construct Constructibility and symmetry.
Mathematics.
Constructibility
Constructibility.
Constructibility
Constructibility.
- Which regular polygons are constructible, and what does the Gauss-Wantzel theorem say? provenance
- Which entry fits constructible polygon? action
Symmetry
Symmetry and tilings.
Symmetry
Symmetry.
- What symmetry groups do regular polygons have, and which tile the plane? provenance
- Which entry fits tessellation? action
Context History and applications.
Context.
History
History.
History
History.
- How were regular polygons studied from Euclid to Gauss? provenance
- Which sources are cited? provenance
Applications
Applications.
Applications
Applications.
- Where do regular polygons appear in design, nature and engineering? provenance
- Which entry fits the specific application? action
What the second pass must settle
- Should star polygons be a separate entry?
- How should reference formulas be linked?
- The registry entry has merged aliases naming specific n-gons and edge cases; should they be split off?