Triangular tiling
In geometry, the triangular tiling is a regular tessellation of the Euclidean plane using congruent equilateral triangles, in which six triangles meet at each vertex.[1] It is one of only three regular tilings possible in the plane, alongside the square tiling and the hexagonal tiling.[1] The tiling is denoted by the Schläfli symbol {3,6}, indicating regular triangular faces with six meeting at each vertex.[2] This tessellation exhibits the highest degree of symmetry among plane tilings, belonging to the wallpaper group p6m, which includes rotations by multiples of 60 degrees and reflections across lines through vertices and midpoints of edges.[3] The vertices form a triangular lattice, and its geometric dual is the hexagonal tiling, where the centers of the triangles correspond to the vertices of the hexagons.[4] Each edge is shared by exactly two triangles, ensuring a complete covering without gaps or overlaps.[3] The triangular tiling serves as a foundational structure in various mathematical and applied contexts, including crystallography, computer graphics for mesh generation, and the study of periodic patterns.[5] It can be constructed by dividing regular hexagons into six equilateral triangles or by arranging parallelograms formed by pairs of triangles in a repeating pattern.[3] Variations, such as uniform colorings or alternated forms, maintain the underlying symmetry while introducing additional properties for artistic or scientific applications.[1]Fundamentals
Definition and Description
The triangular tiling is one of the three regular tessellations of the Euclidean plane, constructed using congruent equilateral triangles that cover the plane without gaps or overlaps.[6] In this tiling, each edge is shared by exactly two triangles, and six such triangles meet at every vertex, forming a uniform and edge-to-edge arrangement that extends infinitely.[7] This structure distinguishes it from the other two regular tilings: the square tiling with Schläfli symbol {4,4}, where four squares meet at each vertex, and the hexagonal tiling with {6,3}, where three hexagons meet at each vertex.[6] The tiling is denoted by the Schläfli symbol {3,6}, where the first number 3 indicates that each face is a triangle (a regular 3-gon), and the second number 6 specifies that six faces meet at each vertex.[7] Its vertex configuration is expressed as 3.3.3.3.3.3 or equivalently (3^6), reflecting the sequence of six equilateral triangles surrounding each vertex point.[8] English mathematician John H. Conway named this tiling the "deltille," derived from the Greek letter delta (Δ), symbolizing its triangular components.[9] As the dual of the hexagonal tiling, the triangular tiling's vertices correspond to the centers of the hexagons in its dual counterpart.[7]Historical Context
The triangular tiling emerged implicitly in ancient geometric traditions, with equilateral triangles serving as fundamental building blocks in Islamic art from the late 9th century onward. Artists and mathematicians in the Islamic world, influenced by earlier Greek and Sasanian motifs, constructed intricate tessellations using triangles inscribed in circles to form star patterns and repeating designs that symbolized cosmic order and infinity.[10][11][12] These patterns, seen in architectural elements like mihrabs and wall panels, represented an early practical application of triangular grids, though not yet formalized mathematically.[12] The first systematic mathematical treatment of regular tilings, including the triangular one, appeared in Johannes Kepler's Harmonices Mundi in 1619. In Book II, Kepler classified the three monohedral regular tessellations of the Euclidean plane—triangular, square, and hexagonal—as harmonious divisions reflecting cosmic structures, marking a shift from artistic intuition to rigorous enumeration.[13] This work built on classical geometry but provided the earliest complete catalog of edge-to-edge tilings with regular polygons.[14] In the 20th century, the triangular tiling gained prominence through artistic and geometric explorations. M.C. Escher incorporated it into metamorphic tessellations during the 1930s and 1940s, as in his 1955 woodcut Liberation, where a grid of abstract shapes transitions into birds, demonstrating its versatility for visual narratives.[15] Mathematician H.S.M. Coxeter advanced formal classification in Regular Polytopes (1948, revised 1963), integrating the triangular tiling {3,6} into the study of uniform honeycombs and symmetries across dimensions.[16] The tiling's role as a periodic baseline was underscored in the 1970s amid aperiodic tiling discoveries. John Conway coined the term "deltille" for it in his enumerations of plane coverings, emphasizing its delta-like triangular prototile.[17] Roger Penrose's aperiodic tilings using rhombi, developed around 1974, contrasted with the triangular tiling's simplicity, highlighting its foundational status in tiling theory.[18]Properties
Topological Properties
The triangular tiling of the Euclidean plane forms an infinite cell complex consisting of vertices, edges, and triangular faces. The Euler characteristic of this complex is given by χ = V - E + F = 0, where V, E, and F denote the infinite cardinalities of vertices, edges, and faces, respectively. This value arises from the structural relations of the tiling: each vertex has degree 6, yielding 2E = 6V or V = E/3, while each face is bounded by 3 edges shared by 2 faces, yielding 2E = 3F or F = 2E/3; substituting these into the formula gives χ = (E/3) - E + (2E/3) = 0.[19] In graph-theoretic terms, the 1-skeleton of the triangular tiling is the infinite triangular lattice graph, a 6-regular planar graph where every vertex has exactly six neighboring vertices connected by edges. This graph is simple, connected, and embeddable in the plane without crossings, with faces corresponding to the triangular tiles. The vertex figure of the tiling, which describes the local configuration around a vertex, is a hexagon formed by connecting the midpoints of the adjacent edges.[19][20] A fundamental domain for generating the triangular tiling can be a single equilateral triangle, serving as the basic unit from which the entire structure is constructed by successive edge identifications and attachments, replicating the tiling across the plane. This approach highlights the combinatorial simplicity of the tiling, where the topology is built modularly from this prototile without introducing singularities.[21] The triangular tiling realizes a planar, orientable surface of infinite extent, equivalent topologically to the Euclidean plane, which has genus 0. As a contractible space, its homology groups are H_0(\mathbb{R}^2; \mathbb{Z}) \cong \mathbb{Z} and H_n(\mathbb{R}^2; \mathbb{Z}) = 0 for all n > 0, reflecting the absence of non-trivial cycles or higher-dimensional holes. These topological properties remain invariant under any homeomorphism of the plane that preserves the tiling's cell structure, mapping vertices to vertices, edges to edges, and faces to faces while maintaining adjacency relations. Such homeomorphisms ensure that the combinatorial and topological features, including the Euler characteristic and homology, are preserved.Metric Properties
The triangular tiling fills the Euclidean plane with congruent equilateral triangles, each having side length s, where s > 0 is an arbitrary unit of length. All interior angles of these triangles measure exactly 60°.[22] In this arrangement, the shortest distance between adjacent vertices is s, corresponding to the edges of the triangles. The next-nearest neighbor distance, connecting vertices separated by two edges along a straight line through a vertex, is s\sqrt{3}.[23] The height of each equilateral triangle, from any vertex perpendicular to the opposite side, is given by \frac{s\sqrt{3}}{2}.[22] The area of one such triangle is \frac{s^2 \sqrt{3}}{4}.[24] As the tiling partitions the plane completely without voids or overlaps, it achieves a total areal density of 1. The vertices of the triangular tiling form a triangular lattice, which can be embedded in the Cartesian plane using coordinates \left( m + \frac{n}{2}, n \frac{\sqrt{3}}{2} \right) for all integers m and n. This positioning ensures that neighboring points are separated by vectors of length s, with the basis vectors typically taken as (s, 0) and \left( \frac{s}{2}, s \frac{\sqrt{3}}{2} \right).[23] For finite approximations of the infinite tiling, such as bounded patches used in computational models or discrete simulations, edges on the boundary are often identified in pairs to simulate periodicity, effectively quotienting the patch into a toroidal or other compact surface while preserving local metric relations.[25] In the Euclidean realization, the dihedral angle across each shared edge—representing the angle between the planes of adjacent triangles—is 180°, maintaining the flat geometry essential for the tiling's planarity.[26] The metric structure of the triangular tiling also provides the foundation for the densest known packing of equal circles in the plane, where circles of radius s/2 centered at the vertices achieve a packing density of \pi / \sqrt{12} \approx 0.9069.[27]Symmetry and Colorings
Symmetry Group
The symmetry group of the triangular tiling is the wallpaper group p6m in crystallographic notation or *632 in orbifold notation, which fully describes the isometries preserving the tiling. This group acts on the Euclidean plane through translations by vectors in the underlying lattice, rotations by multiples of 60° around tiling vertices (6-fold centers), 120° around triangle centers (3-fold centers), and 180° around edge midpoints (2-fold centers), as well as reflections across lines in three directions spaced 60° apart and corresponding glide reflections. The point group of p6m, fixing a reference point, is the dihedral group D_6 of order 12, corresponding to the 12 distinct orientations per fundamental domain of the tiling.[28][29] The group is generated by a 60° rotation r around a vertex and a reflection f across a line through that vertex and the midpoint of an opposite edge (an altitude), with presentation relations r^6 = 1, f^2 = 1, and frf^{-1} = r^{-1}; translations are generated by two linearly independent lattice vectors at 60° to each other. Reflections occur over altitudes (lines joining vertices to opposite edge midpoints) and over midlines (lines joining midpoints of non-adjacent edges, perpendicular to the altitudes in the three symmetry directions). This structure ensures all elements map triangles to triangles while preserving adjacency and orientation where applicable.[29][28] As a wallpaper group, p6m exhibits 6-fold rotational symmetry at vertices, classifying it among the hexagonal lattice groups with the highest symmetry order. Subgroups include p6 (rotational symmetries only, with 6-fold and 3-fold rotations), p3m1 (3-fold rotations with reflections in lines through vertices), p31m (3-fold rotations with reflections offset from vertices), and cmm (2-fold rotations with reflections and glide reflections in rectangular arrangements), each corresponding to symmetries of derived or reduced tilings.[29] The reflection subgroup of p6m corresponds to the Coxeter group [3,6], whose Coxeter-Dynkin diagram consists of two nodes connected by edges labeled 3 and 6, reflecting the local arrangement of six equilateral triangles around a vertex; the affine extension incorporates the lattice translations for the full plane group.Uniform Colorings
Uniform colorings of the triangular tiling refer to face colorings that preserve the tiling's uniformity, meaning the symmetry group acts transitively on the faces of each color class while leaving the coloring invariant. These colorings are specified by cyclic sequences of colors around each vertex, following Conway's notation for the six triangles meeting at a vertex in the {3,6} tiling. According to Conway et al., there are exactly 9 distinct uniform colorings using 1 to 3 colors, enumerated based on criteria ensuring rotational and reflective symmetry compatibility.[30] The monochromatic coloring, denoted111111, uses a single color for all faces and maintains the full p6m symmetry of the tiling. Bichromatic examples include the alternating pattern 121212, where colors switch every triangle around a vertex, also preserving p6m symmetry, and 112112, which introduces a different periodicity while reducing to p3 symmetry. Trichromatic colorings, such as 123123, achieve a minimal 3-coloring where no two adjacent faces share the same color, corresponding to a proper graph coloring of the dual hexagonal lattice; this pattern likewise retains full p6m symmetry. Other patterns like 111222 and 121314 exhibit partial symmetries down to p2 in some cases, reflecting reductions from the full group via color class orbits.[30]
Among these, the Archimedean coloring 111112 is 2-uniform, employing two colors where five consecutive triangles around most vertices are one color and the sixth differs, forming alternate rows with every third triangle offset. This configuration relates to the geodesic polyhedron denoted {3,6+}_{2,0}, a convex approximation with icosahedral symmetry derived from subdividing the tiling. The enumeration adheres to Conway's criteria, prioritizing vertex-transitive color partitions under wallpaper group actions, with p6m for fully symmetric cases and subgroups like p3 or p2 for those with broken reflections or rotations.[30]