octahedron
Let an agent explain octahedra and their kinds, relay properties and formulas from geometric references, describe occurrences in crystals, chemistry and games, and distinguish the regular octahedron from other eight-faced solids.
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 explain octahedra and their kinds, relay properties and formulas from geometric references, describe occurrences in crystals, chemistry and games, and distinguish the regular octahedron from other eight-faced solids.
A polyhedron with eight faces, most commonly the regular octahedron with eight equilateral triangular faces, six vertices and twelve edges, one of the five Platonic solids and the dual of the cube, but also including irregular octahedra such as the hexagonal prism, the tetragonal trapezohedron, the elongated tetragonal disphenoid, the truncated triangular trapezohedron, the augmented triangular prism and the elongated octahedron, and compounds such as the stellated octahedron; octahedra appear in crystals, molecular geometry and dice.
What it is for: Not applicable; a geometric solid.
It can be explain the solid and kinds; relay properties and formulas; describe occurrences; distinguish regular from irregular.
Distinguishing features
Eight faces
Platonic regular form
Cube duality
Many irregular forms
What it looks like
Two square pyramids joined at their bases, in the regular case.
Physical character
regular octahedron faces: 8 count - equilateral triangles
vertices and edges: 6 and 12 count
volume with edge a: (sqrt 2 / 3) a^3 formula
topologically distinct octahedra: 257 count - convex
How it is recognised
Eight-faced polyhedron
Regular octahedron, trapezohedra, disphenoids, prisms, stellated octahedron
The cube is its dual; the hexahedron has six faces; the octagon is a plane figure
Related models
is a kind of - in registry terms
is a kind of - in registry terms
is the dual of - in the regular case
is one of the - set
In practice
Families and kinds
regular octahedron
hexagonal prism
tetragonal trapezohedron
elongated tetragonal disphenoid and elongated octahedron
truncated triangular trapezohedron
augmented triangular prism
stellated octahedron as a compound
octahedra in crystallography and coordination chemistry
Standards and regulation
No regulation; standard geometric definitions
Failure modes and hazards
Assuming all octahedra are regular
Confusing octahedron with octagon
Miscounting faces of compounds
Also called
Where this came from
wikidata · CC0 1.0
Also registered as vr.tr.octahedron
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Understand What an octahedron is.
Mathematics.
Definition
Definition.
Definition
Definition.
- What is an octahedron, and what makes the regular octahedron a Platonic solid? definition
- Is the question about the regular octahedron, an irregular octahedron or an octagon? boundary
Kinds
Kinds.
Kinds
Kinds.
- What other octahedra exist, such as trapezohedra, disphenoids, augmented prisms and the stellated octahedron? definition
- Which entry fits the specific solid? action
Properties Properties.
Mathematics.
Formulas
Formulas.
Formulas
Formulas.
- What are the volume, surface area, angles and symmetry of the regular octahedron? measurement
- Which references are standard? provenance
Duality
Duality and relations.
Duality
Duality.
- How does the octahedron relate to the cube, tetrahedron and other polyhedra? provenance
- Which entry fits dual polyhedron? action
Occur Occurrences.
Science.
Nature
Crystals and molecules.
Nature
Nature.
- Where do octahedra occur in crystals such as diamond and fluorite and in molecular geometry? provenance
- Which sources are cited? provenance
Objects
Dice and design.
Objects
Objects.
- How are octahedra used as eight-sided dice and in design and architecture? provenance
- Which entry fits polyhedral dice? action
Context History and teaching.
Context.
History
History.
History
History.
- How did Plato, Euclid and Kepler treat the octahedron? provenance
- Which entry fits the history of geometry? action
Teaching
Teaching.
Teaching
Teaching.
- How are octahedra built and taught with nets and models? provenance
- Which entry fits mathematics education? action
What the second pass must settle
- Should regular octahedron and stellated octahedron be separate entries?
- How should geometric references be linked?
- The registry entry has merged aliases naming distinct solids; should they be split off?