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Research draft

tessellation

vr.tr.tessellation · XCT.QLT

Let an agent explain tessellations and their classification, identify or construct tilings, describe their appearance in art and nature, and support learning and computational use.

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 tessellations and their classification, identify or construct tilings, describe their appearance in art and nature, and support learning and computational use.

A covering of a plane or other space by shapes called tiles with no gaps or overlaps, including regular and semiregular tilings by polygons, periodic and aperiodic tilings such as Penrose tilings, tilings of the sphere and hyperbolic plane, and higher-dimensional honeycombs; tessellations are studied in geometry, appear in art, architecture and nature, and are used in computing and manufacturing.

What it is for: Covering space with tiles.

It can be explain tessellation types and classification; identify or construct a tiling; describe tessellations in art and nature; support computational use.

Distinguishing features

Gapless covering

Classified by symmetry

Periodic or aperiodic

Generalises to other spaces

What it looks like

Repeating or patterned shapes covering a surface without gaps.

How it is recognised

No gaps or overlaps

Regular, semiregular or irregular

A pattern with gaps is not a tessellation

Related models

is a kind of - category

polytope

is a kind of - category

cover

is related to - tilings by dimension

dimension

is related to - decorative tilings

applied arts

In practice

Families and kinds

regular and semiregular tilings

periodic tilings and wallpaper groups

aperiodic tilings

spherical and hyperbolic tilings

honeycombs in higher dimensions

Standards and regulation

Mathematical naming conventions such as vertex configurations

No regulation

Failure modes and hazards

Confusing tilings with patterns that have gaps

Misnaming uniform tilings

Overlooking aperiodic cases

Also called

ditrigonary trigonal hemiapeirogonal tesselationtrigonal hemiapeirogonal tesselationpetrial triangular tilinghexagonal hemiapeirogonal tesselationsquare hemiapeirogonal tesselationuniform honeycomb dualmonohedral tilingpolyhedral tilingsnub hexagonal tilingsnub square tilingdemiregular tilingAugmented truncated hexagonal tilingdomino tilinguniform tessellationtiling by regular polygonsElongated triangular tilingtrihexagonal tilingpentagon tilingWang tileapeirogonal antiprismGilbert tessellationpinwheel tilingOrder 3-7 kisrhombillePythagorean tilingquasiperiodic tilingrhombille tilingaperiodic tilingSnub trioctagonal tilingsphinx tilingsubstitution tilingtiling with rectanglesTruncated triheptagonal tilingSocolar tilingAmmann–Beenker tilingquaquaversal tiling

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

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

Classify Kinds of tessellation.

Geometry.

Types

Classification.

Types

Classification.

  1. What kind of tessellation is this, and how is it named by vertex configuration or symmetry group? definition
  2. Which regular and semiregular tilings exist? definition

Aperiodic

Aperiodic tilings.

Aperiodic

Aperiodic.

  1. What are aperiodic tilings such as Penrose tilings and the recently found aperiodic monotile? provenance
  2. Why do they matter? provenance
Construct Making tilings.

Practice.

Build

Constructing a tiling.

Build

Construction.

  1. How can a tessellation be constructed from a given shape or symmetry? action
  2. Which shapes tile the plane? definition

Compute

Computational tilings.

Compute

Computation.

  1. How are tessellations used in computer graphics, meshing and manufacturing? provenance
  2. Which algorithms generate them? provenance
Beyond Other spaces.

Generalisation.

Spherical

Sphere and hyperbolic plane.

Spherical

Other geometries.

  1. How do tilings of the sphere and hyperbolic plane differ from planar tilings, and what are hemiapeirogonal tilings? definition
  2. How do they relate to polyhedra? definition

Honeycombs

Higher dimensions.

Honeycombs

Honeycombs.

  1. What are honeycombs in three and more dimensions, and how are their duals defined? definition
  2. Which uniform honeycombs exist? provenance
Culture Art, nature and teaching.

Context.

Art

Tessellations in art and nature.

Art

Art and nature.

  1. How do tessellations appear in Islamic art, Escher work, architecture and nature such as honeycombs? provenance
  2. Which entry fits a specific work? action

Teach

Teaching.

Teach

Teaching.

  1. How can tessellations be taught with hands-on activities? action
  2. Which misconceptions arise? provenance

What the second pass must settle

  • Should wallpaper groups be a separate entry?
  • How should tiling catalogues be linked?
  • How should art examples be linked?
Draft as JSON