Conway criterion¶
A sufficient boundary-symmetry test guaranteeing that a topological-disk prototile can tile the plane by translations and half-turns.
Core Idea¶
The Conway criterion requires one pair of opposite boundary arcs to be translates and four remaining arcs to be centrally symmetric, which is sufficient for planar tiling. Translated copies match the paired arcs while half-turns match each centrosymmetric arc, allowing copies to surround vertices and extend periodically without gaps or overlaps. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Conway criterion belongs to tessellation theory and is useful where the analyst can specify a closed disk-like tile, six cyclic boundary points, paired boundary arcs, translations, half-turn symmetries, and an edge-to-edge placement construction, then evaluate the tile is a topological disk and its boundary admits the declared six-point decomposition with exact translation and half-turn congruences. The scope is broad within that domain but bounded by the need for the tile is a topological disk and its boundary admits the declared six-point decomposition with exact translation and half-turn congruences. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the tile is a topological disk and its boundary admits the declared six-point decomposition with exact translation and half-turn congruences the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Conway criterion can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Conway criterion. Conway criterion compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: a closed disk-like tile, six cyclic boundary points, paired boundary arcs, translations, half-turn symmetries, and an edge-to-edge placement construction. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the tile is a topological disk and its boundary admits the declared six-point decomposition with exact translation and half-turn congruences independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of tessellation theory because they reuse a closed disk-like tile, six cyclic boundary points, paired boundary arcs, translations, half-turn symmetries, and an edge-to-edge placement construction, Translated copies match the paired arcs while half-turns match each centrosymmetric arc, allowing copies to surround vertices and extend periodically without gaps or overlaps., and type the carrier, state every parameter and convention in the definition, test that the tile is a topological disk and its boundary admits the declared six-point decomposition with exact translation and half-turn congruences, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Conway criterion Domain-specific
Parents (1) — more general patterns this builds on
-
Conway criterion is a kind of Necessity and Sufficiency Prime
The proposed strict upward parent is
prime:necessity_and_sufficiency.
Hierarchy path (1) — routes to 1 parentless root
- Conway criterion → Necessity and Sufficiency → Relation
Neighborhood in Abstraction Space¶
Conway criterion sits in a moderately populated region (41st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Manifold & Simplicial Constructions (8 abstractions)
Nearest neighbors
- Triangulation (topology) — 0.90
- Perfect rectangle — 0.90
- Simply connected at infinity — 0.89
- Tetrahedron packing — 0.89
- Fort space — 0.89
Computed from structural-signature embeddings · 2026-09-08