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Research draft

polyhedron

vr.tr.polyhedron · XCT.QLT

Let an agent handle polyhedra by faces, edges, vertices, symmetry and classification, with Euler characteristic checks.

Thing Registry Cross-cutting context

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 polyhedra by faces, edges, vertices, symmetry and classification, with Euler characteristic checks.

A three-dimensional solid bounded by flat polygonal faces, straight edges and vertices, such as cubes, pyramids, prisms, Platonic and Archimedean solids, and their duals.

What it is for: Geometry, crystallography, architecture, chemistry and modelling.

It can be count faces, edges and vertices; construct nets and models; classify by symmetry; find duals.

Distinguishing features

Flat faces, straight edges

Euler characteristic V - E + F = 2 for convex polyhedra

Convex and non-convex

Duals swap faces and vertices

What it looks like

Solid shapes with flat faces and straight edges.

How it is recognised

Face shapes and counts

Names such as dodecahedron

Curved solids such as spheres are not polyhedra

Related models

is a kind of - category

polytope and solid figure

has part - components

polygon (face)

includes - example

cube

is studied in - field

geometry

In practice

Families and kinds

Platonic solids

Archimedean solids

prisms and antiprisms

pyramids

Kepler-Poinsot and other star polyhedra

Johnson solids

Failure modes and hazards

Applying Euler formula to non-simple polyhedra

Naming confusion across conventions

Also called

Boerdijk–Coxeter helixdual polyhedronHemi facetted cubeWenninger's polyhedron modelshendecahedronhexadecahedrontangential polyhedrontriacontadihedrondeltahedronDouble Hessian polyhedronheptahedronicosihenahedronicosidihedronicositrihedronicosipentahedronicosihexahedronicosiheptahedronicosioctahedronicosienneahedrontriacontahedronbiaugmented hexagonal prismdecahedronregular-faced polyhedronicositetrahedronWaterman polyhedronsnub polyhedronComposite polyhedronpolycubeOrthogonal polyhedronChazelle polyhedronLeonardo polyhedronpentadecahedronchamfered tetrahedrontridecahedronheptadecahedronholyhedronpolyhedral terrainscalenohedrondodecahedrongyroelongated birotunda

+237

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Also registered as vr.tr.polyhedron

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Structure Faces, edges, vertices.

Counts describe polyhedra.

Counts

V, E, F.

Counts

Element counts.

  1. How many faces, edges and vertices? measurement
  2. Does Euler formula hold? boundary

Faces

Polygon types.

Faces

Face types.

  1. Which polygons form the faces? definition
  2. Are they regular? boundary
Classification Which family.

Families organise polyhedra.

Family

Platonic, Archimedean.

Family

Family.

  1. Which family does it belong to? definition
  2. Is it convex? boundary

Symmetry

Group.

Symmetry

Symmetry group.

  1. What is its symmetry group? definition
  2. What is its dual? definition
Measures Size.

Measures follow from edges.

Volume

Formula.

Volume

Volume.

  1. What is its volume for a given edge length? measurement
  2. What is its surface area? measurement

Angles

Dihedral.

Angles

Dihedral angles.

  1. What are its dihedral angles? measurement
  2. How are they computed? action
Making Models.

Models aid understanding.

Nets

Paper models.

Nets

Nets.

  1. Which net folds into it? definition
  2. How can a model be built? action

Applications

Uses.

Applications

Applications.

  1. Where does this polyhedron appear in nature or design? definition
  2. For example in crystals or molecules? definition

What the second pass must settle

  • Should families be separate entries?
  • How should duals be linked?
  • How should models such as Wenninger models be linked?
Draft as JSON