Conway criterion¶

A sufficient boundary-symmetry test guaranteeing that a topological-disk prototile can tile the plane by translations and half-turns.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3905
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Tessellation Theory, Monohedral Tiling Criteria → Mathematics

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.

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Sure-Fit Puzzle Piece Test

Some puzzle-piece shapes can cover a whole floor with copies, with no gaps and no overlaps. The Conway criterion is a check for this. If the piece's edge can be cut into six parts, where two opposite parts match just by sliding a copy over, and each of the other four parts matches itself when spun halfway around, then copies will fit together to cover the floor. Some shapes that fail the check can still cover a floor.

Slide-and-Spin Tiling Test

Tiling means covering a flat surface with copies of a shape, with no gaps or overlaps. The Conway criterion is a test for a shape with no holes. Split its edge into six parts. If two opposite parts are exact slid copies of each other, and each of the other four parts looks the same when you spin it halfway around its middle, then the shape can tile the plane. Copies fit together by sliding to match the paired parts and by half-turning to match the other four. Passing guarantees tiling, but failing doesn't prove a shape can't tile.

Translation–Half-Turn Tiling Certificate

The Conway criterion is a sufficient condition for a shape to tile the plane. The shape must be a topological disk, a single piece with no holes, whose boundary can be split at six points into six arcs. One pair of opposite arcs must be translates of each other, meaning one is the other slid without turning. Each of the other four arcs must be centrally symmetric, meaning it looks the same after a half-turn about its midpoint. Then copies of the tile fit together: translated copies match the paired arcs, and half-turned copies match each symmetric arc, so copies can surround every vertex and extend periodically without gaps or overlaps. It is only a sufficient test: many tiles that do tile the plane fail it, so it is not a classification of all tiles.

The Conway criterion is a constructive sufficient symmetry certificate for plane tiling. A tile that is a topological disk qualifies if its boundary admits a decomposition at six points into six consecutive arcs such that one pair of opposite arcs are exact translates of each other and each of the remaining four arcs is centrally symmetric, invariant under a half-turn about its midpoint. The certificate is constructive: translated copies match the paired arcs, and 180-degree rotations about arc midpoints match each centrosymmetric arc, so copies can surround each vertex and extend periodically without gaps or overlaps. The congruences must be exact, and the six-point decomposition must be declared; approximate or visual resemblance does not suffice. Some arcs may degenerate in particular cases, depending on the convention adopted. The criterion is sufficient but not necessary, so it does not classify all tilers, and failing it says nothing definitive about tilability.

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.
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) — outgoing typed relations

Conway criterion is a kind of Necessity and Sufficiency Prime

The result is explicitly a sufficient but nonnecessary condition.

Hierarchy path (1) — routes to 1 parentless root

Conway criterion → is a kind of Necessity and Sufficiency → is a kind of Relation

Neighborhood in Abstraction Space¶

Conway criterion sits in a crowded region of the domain-specific corpus (25^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Point-Set Topology Structures (51 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25