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Spectre tile

A strictly chiral aperiodic monotile, also known as Tile(1,1), discovered in 2023.

Overview

The Spectre is a 13-sided polygon (Tile(1,1)) that tiles the plane aperiodically using only orientation-preserving copies — no reflected tiles are needed. It was introduced in A chiral aperiodic monotile as the solution to the strictly chiral einstein problem.[1]

Tile(1,1) and Spectre edge variants: straight, jagged, wavy, stepped, scalloped, and rounded silhouettes
Tile(1,1) / Spectre variants. One aperiodic monotile footprint with many equivalent edge silhouettes — straight polygon, jagged, wavy, stepped, scalloped, and rounded forms. All tile the same way; only the boundary decoration changes.
Zoomed aperiodic tiling patch by Tile(1,1) with odd tiles shaded
Spectre / Tile(1,1) patch. A zoomed substitution patch with alternating tile handedness highlighted. Sample image from Kaplan et al., CC BY 4.0.

Substitution structure

Like other modern aperiodic tiles, Spectre patches are generated by substitution: a finite set of metatiles refines into smaller copies until a target region is filled. Public tooling — including Kaplan's Spectre explorer and community ports — implements these rules for interactive exploration and export.[1]

The Aperiodic Monotile Generator API packages this mathematics for production workflows: clipped patches, stable tile transforms, and exporters (SVG, STL, GLB, CSV, JSON).

Relationship to the Hat

The Spectre construction refines the Hat discovery by removing the need for mirrored tiles. Researchers have also studied conversions between Tile(1,1) tilings and other aperiodic layouts.[7] Kaplan's historical survey traces the full path from Penrose tiles to modern monotiles.[3]

See also

Aperiodic monotile, Hat tile, Substitution tiling

Categories: Mathematics · Concepts