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Research draft

tetrahedron

vr.tr.tetrahedron · XCT.QLT

Let an agent explain tetrahedra and their kinds, relay properties and formulas from geometric references, describe occurrences in chemistry and computation, and distinguish the regular tetrahedron from other forms and from pyramids generally.

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 explain tetrahedra and their kinds, relay properties and formulas from geometric references, describe occurrences in chemistry and computation, and distinguish the regular tetrahedron from other forms and from pyramids generally.

A polyhedron with four triangular faces, six edges and four vertices, the simplest convex polyhedron and the three-dimensional simplex, including the regular tetrahedron with equilateral faces, one of the Platonic solids, and special irregular forms such as orthocentric tetrahedra whose altitudes meet, Heronian tetrahedra with integer edges, faces and volume, Hill tetrahedra that tile space, and the Reeve tetrahedron used to show that Pick theorem does not extend to three dimensions; tetrahedra appear in chemistry, crystallography and computational geometry.

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 other forms.

Distinguishing features

Simplest polyhedron

Three-dimensional simplex

Self-dual

Many special forms

What it looks like

A triangular pyramid with four triangular faces.

Physical character

faces, edges, vertices: 4, 6, 4 count

regular volume with edge a: a^3 / (6 sqrt 2) formula

regular dihedral angle: about 70.53 degrees

How it is recognised

Four-faced polyhedron

Regular tetrahedron; orthocentric, Heronian, Hill and Reeve tetrahedra

Square pyramids have five faces; the triangle is the two-dimensional simplex

Related models

is a kind of - in registry terms

simplex

is a kind of - in registry terms

pyramid

is a kind of - in registry terms

convex polyhedron

is one of the - set, in its regular form

Platonic solid

In practice

Families and kinds

regular tetrahedron

orthocentric tetrahedra

Heronian tetrahedra

Hill tetrahedra and space-filling tetrahedra

Reeve tetrahedra

disphenoids

tetrahedra in molecular geometry and finite element meshes

Standards and regulation

No regulation; standard geometric definitions

Failure modes and hazards

Assuming all tetrahedra are regular

Confusing with other pyramids

Poorly shaped tetrahedra in meshes

Also called

regular tetrahedronsquare hosohedronHeronian tetrahedronHill tetrahedronorthocentric tetrahedronReeve tetrahedrontrirectangular tetrahedrondisphenoid

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Also registered as vr.tr.tetrahedron

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 a tetrahedron is.

Mathematics.

Definition

Definition.

Definition

Definition.

  1. What is a tetrahedron, and why is it the three-dimensional simplex? definition
  2. Is the question about the regular tetrahedron, a special tetrahedron or another pyramid? boundary

Kinds

Special tetrahedra.

Kinds

Kinds.

  1. What are orthocentric, Heronian, Hill and Reeve tetrahedra, and why are they notable? definition
  2. Which entry fits the specific kind? action
Properties Properties.

Mathematics.

Formulas

Formulas.

Formulas

Formulas.

  1. How are the volume, surface area, circumsphere and angles of a tetrahedron computed? measurement
  2. Which references are standard? provenance

Relations

Relations to other solids.

Relations

Relations.

  1. How does the tetrahedron relate to the cube, octahedron and stellated octahedron, and why is it self-dual? provenance
  2. Which entry fits Platonic solid? action
Apply Applications.

Science.

Chemistry

Chemistry and crystals.

Chemistry

Chemistry.

  1. Where do tetrahedra occur in molecular geometry such as methane and in silicate crystals? provenance
  2. Which sources are cited? provenance

Computation

Computation.

Computation

Computation.

  1. How are tetrahedral meshes used in finite element analysis and graphics? provenance
  2. Which entry fits tetrahedral mesh? action
Context History and puzzles.

Context.

History

History.

History

History.

  1. How did Plato, Euclid and later mathematicians treat the tetrahedron, including the Hill and Reeve results? provenance
  2. Which entry fits the history of geometry? action

Puzzles

Puzzles and dice.

Puzzles

Puzzles.

  1. How do tetrahedra appear in puzzles, four-sided dice and design? provenance
  2. Which entry fits polyhedral dice? action

What the second pass must settle

  • Should regular tetrahedron and Heronian tetrahedron be separate entries?
  • How should geometric references be linked?
  • The registry entry has merged aliases naming special forms; should they be split off?
Draft as JSON