The truncated order‑4 hexagonal tiling is a uniform tiling of the hyperbolic plane obtained by applying a truncation operation to the regular order‑4 hexagonal tiling {6,4}. In the regular tiling {6,4}, four regular hexagons meet at each vertex. Truncating each vertex replaces it with a regular square and expands each original hexagon into a regular dodecagon. The resulting tiling is composed of two face types—squares and dodecagons—arranged so that each vertex is surrounded by one square and two dodecagons, giving the vertex figure 4·12·12.
Schläfli and Coxeter notation
- Schläfli symbol: t{6,4}
- Coxeter notation: (4 12 12)
Symmetry
The tiling possesses the full symmetry group of the regular order‑4 hexagonal tiling, often denoted by [6,4] in Coxeter’s notation. The fundamental domain is an isosceles triangle with angles π/4, π/6, and 0 (ideal vertex), reflecting the hyperbolic nature of the tiling.
Geometric properties
- Face types: regular squares (4‑gons) and regular dodecagons (12‑gons).
- Vertex configuration: each vertex is incident to one square and two dodecagons (4·12·12).
- Edge transitivity: all edges are equivalent under the symmetry group.
- It is a member of the family of truncated regular tilings {p,q} → t{p,q}, where p = 6 and q = 4.
Related tilings
- The regular order‑4 hexagonal tiling {6,4}, from which it is derived.
- The truncated order‑3 hexagonal tiling t{6,3}, a Euclidean counterpart with vertex figure 4·8·8.
- Other truncated hyperbolic tilings such as t{5,4} and t{7,3}.
Construction
The tiling can be constructed by:
- Beginning with the regular {6,4} tiling of the hyperbolic plane.
- Replacing each vertex by a square whose edges are perpendicular bisectors of the original meeting edges.
- Expanding each original hexagon into a dodecagon by inserting new edges at the points where the original edges were truncated.
Applications and significance
Uniform hyperbolic tilings such as the truncated order‑4 hexagonal tiling are studied in the fields of geometric group theory, hyperbolic geometry, and the theory of wallpaper groups. They illustrate how regularity and symmetry can be preserved under truncation even in non‑Euclidean settings, and they serve as examples in visualizations of hyperbolic space.
