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Trihexagonal tiling

The trihexagonal tiling is an edge-to-edge uniform tiling of the Euclidean plane composed of equilateral triangles and regular hexagons of equal edge length, in which each edge is shared by one triangle and one hexagon, and at each vertex three triangles and three hexagons meet alternately in the configuration (3.6.3.6).^[1] This Archimedean tiling, one of eleven convex uniform tilings by regular polygons, exhibits p6mm symmetry and is bipartite, allowing a two-coloring of its tiles.^[2]^[1] Its dual is the rhombille tiling, formed by rhombi meeting three or six at a vertex.^[3] In any finite portion with N hexagons, there are exactly 2N triangles, reflecting the tiling's balanced yet asymmetric composition of tile types.^[1] The trihexagonal tiling holds significance in dynamical systems research, particularly in models of tiling billiards where light rays refract through transparent triangles and hexagons with equal but opposite indices of refraction, producing periodic, drift-periodic, and dense trajectories that reveal ergodic properties.^[4]^[1] It also appears in studies of quantum graphs and dispersion relations, where its periodic structure enables analysis via Floquet-Bloch theory.^[2]

Definition and Properties

Construction Methods

The trihexagonal tiling consists of equilateral triangles and regular hexagons arranged in an alternating fashion such that each edge is shared by one triangle and one hexagon, resulting in a 2:1 ratio of triangles to hexagons and covering the Euclidean plane without gaps or overlaps.^[5] One method to construct the tiling is through rectification of the regular triangular tiling, where vertices are truncated to the midpoints of the edges; this transforms each original equilateral triangle into a regular hexagon while introducing new equilateral triangles at the original vertices, yielding the desired alternation.^[5] Alternatively, the tiling can be obtained by expanding the regular hexagonal tiling, effectively inserting equilateral triangles between the hexagons by separating the vertices along radial lines from a central point, which spaces the hexagons apart and fills the gaps with triangles at the shared edges.^[5] The vertices of the trihexagonal tiling can be placed on the triangular lattice with coordinates given by (m + n/2, n \sqrt{3}/2) for integers m, n, assuming an edge length of 1; this positions the points such that adjacent vertices are exactly distance 1 apart, forming the framework for connecting edges to outline the triangles and hexagons. The lattice is generated by basis vectors (1, 0) and (1/2, \sqrt{3}/2), ensuring all regular polygons align properly without distortion. A repeating unit cell for the tiling contains two equilateral triangles and one regular hexagon, spanning a parallelogram of area $2\sqrt{3} (for edge length 1); this cell captures the fundamental repeating pattern, with the hexagons surrounding the triangles sharing their edges.^[5]

Vertex Configuration and Metrics

In the trihexagonal tiling, the vertex configuration is denoted as (3.6.3.6), indicating that two equilateral triangles and two regular hexagons alternate around each vertex.^[3] This arrangement ensures a quasiregular structure where the sequence of polygons meeting at every vertex follows this repeating pattern. The dual vertex figure corresponds to a rhombus formed by connecting the centers of the adjacent tiles.^[3] The internal angles contributing to the vertex sum are 60° from each triangular tile and 120° from each hexagonal tile, yielding a total of 360° (calculated as $60^\circ + 120^\circ + 60^\circ + 120^\circ = 360^\circ). To arrive at these angles, note that the internal angle of a regular n-gon with edge length s is given by \left(1 - \frac{2}{n}\right) \times 180^\circ; for n=3, this is $60^\circ, and for n=6, it is $120^\circ. This precise fit allows the tiling to cover the Euclidean plane without gaps or overlaps.^[4] In the uniform realization, all edges have equal length, typically normalized to 1 for computational purposes. The area of each equilateral triangular tile is \frac{\sqrt{3}}{4} s^2, derived by dividing the triangle into two $30^\circ-$60^\circ-$90^\circ right triangles with legs s/2 and (s \sqrt{3})/2, yielding a total area of \frac{1}{2} \times \frac{s}{2} \times s \sqrt{3} \times 2 = \frac{\sqrt{3}}{4} s^2. The area of each regular hexagonal tile is \frac{3 \sqrt{3}}{2} s^2, obtained as six times the area of an equilateral triangle: $6 \times \frac{\sqrt{3}}{4} s^2 = \frac{3 \sqrt{3}}{2} s^2. Given the ratio of two triangles to one hexagon in the tiling's repeating unit, the combined area per such unit is $2 \times \frac{\sqrt{3}}{4} s^2 + \frac{3 \sqrt{3}}{2} s^2 = 2 \sqrt{3} s^2, contributing to full plane coverage with a packing fraction of 1.^[4] The tiling's structure is captured by the Schläfli symbol r\{6,3\}, signifying its origin as the rectification of the regular hexagonal tiling \{6,3\}, where vertices are truncated until adjacent edges meet at midpoints. Alternatively, it is denoted h_2\{6,3\} in some extended notations emphasizing the quasiregular alternation. The rectification process preserves the hexagonal lattice's density while incorporating triangular fills, resulting in an average of four edges meeting at each vertex across the infinite plane.^[6]

Duality and Topological Properties

The dual of the trihexagonal tiling is the rhombille tiling, a tessellation composed entirely of congruent rhombi with interior angles of 60° and 120°.^[7] In this duality, the vertices of the trihexagonal tiling correspond to the faces of the rhombille tiling, while the triangular and hexagonal faces of the trihexagonal tiling become vertices in the rhombille tiling of degree 3 and 6, respectively.^[8] This reciprocal relationship preserves the combinatorial structure, with each edge of the original tiling mapping to an edge in the dual.^[7] As a periodic tiling of the Euclidean plane, the trihexagonal tiling exhibits topological invariants characteristic of planar embeddings, including an Euler characteristic χ = 0 when considering the quotient by the underlying lattice, which forms a torus of genus 1.^[9] This value arises from the balanced counts of vertices, edges, and faces in the infinite structure, where the average number of edges incident to each vertex is 4, reflecting the uniform vertex configuration (3.6.3.6).^[8] The plane itself, as the ambient space, is topologically equivalent to a sphere minus a point, maintaining genus 0, but the tiling's periodicity ensures the characteristic is 0 in the toroidal quotient.^[10] Although the trihexagonal tiling is not strictly self-dual, its dual rhombille tiling possesses isohedral symmetry, meaning it is face-transitive with all rhombi equivalent under the symmetry group, and the rhombi are oriented at 60° angles relative to one another.^[7] This isohedral property highlights a form of reciprocal uniformity between the primal and dual, where the edge-transitive nature of both tilings is preserved.^[8] From a graph-theoretic perspective, the 1-skeleton of the trihexagonal tiling is a 4-regular infinite graph embedded in the plane, consisting of vertices connected by edges that bound triangular and hexagonal faces.^[8] This graph is planar and 3-connected, with the faces corresponding directly to the tiling's polygons, and its regularity underscores the uniform coordination at each vertex.^[7]

Symmetry and Colorings

Wallpaper Group and Uniformity

The trihexagonal tiling exhibits the symmetry of the wallpaper group p6m (*632 in orbifold notation), which comprises 12 isometries including rotations by 0°, 60°, 120°, 180°, 240°, and 300° centered at vertices, reflections across six lines passing through vertices and midpoints of opposite edges (altitudes), and six glide reflections along the directions perpendicular to those reflection lines.^[11] This group fully captures the geometric invariances of the tiling under the Euclidean plane's isometries, ensuring that the arrangement of equilateral triangles and regular hexagons remains unchanged under these transformations.^[11] As one of the 11 Archimedean tilings, the trihexagonal tiling is uniform, meaning it is an edge-to-edge tiling by congruent regular polygons where the symmetry group acts transitively on the vertices, edges, and faces.^[11] This vertex-transitivity is evident from the repeated vertex configuration (3.6.3.6), where two triangles and two hexagons alternate around each vertex, allowing any vertex to be mapped to any other via the group's actions.^[11] The isometry group p6m is generated by a 60° rotation around a vertex and a reflection over an altitude through that vertex, with compositions yielding the full set of symmetries.^[11] Chiral variants form an enantiomorphic pair with the wallpaper group p6, the index-2 rotational subgroup of p6m that omits reflections and glide reflections, consisting only of the six rotations; these mirror-image forms cannot be superimposed without reflection.^[11]

Edge-to-Edge Colorings

The graph associated with the vertices and edges of the trihexagonal tiling, known as the kagome lattice, has a chromatic number of 3 for proper vertex colorings. This arises from the presence of triangular faces requiring at least three colors, while the lattice's periodic structure permits a 3-coloring where each color class forms a sublattice and every triangle receives one vertex of each color.^[12] Proper edge colorings of the trihexagonal tiling require four colors, as the graph is 4-regular and belongs to Vizing class I, achieving the chromatic index equal to the maximum degree Δ=4. This follows from Vizing's theorem, which bounds the chromatic index between Δ and Δ+1 for simple planar graphs, with explicit constructions confirming 4-colorability via algorithmic methods like the Misra-Gries procedure adapted for lattices. The faces of the trihexagonal tiling can be properly colored with two colors, as the face adjacency graph is bipartite: every edge is shared exclusively between a triangle and a hexagon, allowing all triangles one color and all hexagons the complementary color without monochromatic adjacencies.^[5] The unit cell graph of the trihexagonal tiling consists of three vertices per primitive cell connected periodically, with the minimal number of colors for proper vertex coloring being 3.

Circle Packing Arrangements

The trihexagonal tiling admits a vertex circle packing where equal circles are placed at each vertex, with radius r = \frac{1}{2} for unit edge lengths. This configuration realizes the maximum planar kissing number of 6, as the circle size corresponds to the tight hexagonal arrangement, though the tiling's 4-regular vertex figure results in only 4 local contacts per circle, leaving space for two additional tangent circles that are absent due to the lattice structure.^[13] Face-centered circle packings in the trihexagonal tiling position circles at the centers of triangular and hexagonal faces, with radii r = \frac{\sqrt{3}}{6} for triangles and r = \frac{\sqrt{3}}{2} for hexagons. These circles are tangent to adjacent circles across shared edges, yielding a packing density of \frac{11\pi}{24\sqrt{3}} \approx 0.831.^[14] By Thue's theorem, which establishes the hexagonal lattice as the optimal equal-circle packing in the Euclidean plane, the vertex packing in the trihexagonal tiling represents the densest possible arrangement for equal circles constrained to its underlying lattice sites, with tangencies occurring precisely along the tiling edges.^[14] While the primary focus remains on Euclidean packings, extensions to hyperbolic geometry yield circle packings that conform to the trihexagonal tiling's combinatorial structure via the circle packing theorem, enabling discrete approximations of hyperbolic metrics with tangent circles respecting the tiling's tangency graph.^[15]

Kagome Pattern and Lattice

Historical and Cultural Origins

The trihexagonal tiling was first described by Johannes Kepler in his 1619 treatise Harmonices Mundi, where he classified it among the eleven uniform tilings of the plane as an example of geometric harmony derived from regular polygons.^[16] Kepler's work represented the earliest systematic enumeration of such semiregular tilings, integrating them into his broader exploration of congruence and proportion in nature.^[17] This pattern, consisting of alternating equilateral triangles and regular hexagons, was presented as a harmonious arrangement without gaps or overlaps, foreshadowing later mathematical formalizations.^[18] The tiling later received formal classification within the framework of Archimedean tilings by mathematician H. S. M. Coxeter, whose 1954 paper on uniform polyhedra extended analogous principles to plane tilings, confirming the trihexagonal configuration as one of the eleven convex uniform tilings by regular polygons.^[19] Coxeter's contributions emphasized the tiling's vertex-transitive symmetry and its role in enumerating all edge-to-edge tessellations achievable with regular polygons.^[20] Known alternatively as the kagome tiling, the pattern draws its name from the Japanese term "kagome," derived from "kago" (basket) and "me" (weave or eye), evoking the interlaced hexagonal openings in traditional bamboo basketry that mimic the tiling's structure. This motif has roots in Japanese craftsmanship, appearing in basketry techniques and textiles as early as the Heian period (794–1185 CE), where advanced weaving methods produced fabrics with protective geometric lattices symbolizing barriers against misfortune.^[21] Similar interlaced patterns resembling the trihexagonal tiling appear in medieval Islamic geometric art as variants of girih strapwork, used in architectural decorations across the Islamic world from the 12th century onward, though not explicitly named as such.^[22] These designs, often based on hexagonal grids, served as foundational constructions for intricate mosque tilings and illustrate early cross-cultural recognition of the pattern's aesthetic and structural properties.

Geometric Properties of the Kagome Lattice

The Kagome lattice is the graph-theoretic skeleton of the trihexagonal tiling, comprising its vertices and connecting edges to form a network of corner-sharing equilateral triangles embedded within larger hexagonal voids, where each vertex exhibits a coordination number of 4.^[23] This structure arises as the line graph of the honeycomb lattice or equivalently as the medial lattice of the trihexagonal tiling, with vertices positioned at the centers of the hexagons and at the midpoints of the triangle edges.^[24] Consequently, the lattice displays fourfold connectivity, distinguishing it from the sixfold coordination of the underlying triangular lattice while preserving the geometric motif of frustrated triangular units.^[25] The primitive unit cell of the Kagome lattice consists of three inequivalent sites arranged in an equilateral triangle, forming a basis on a triangular Bravais lattice. The lattice vectors are typically defined at a 60° angle, such as \vec{a_1} = a(1, 0) and \vec{a_2} = a\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right), where a is the nearest-neighbor distance, yielding a hexagonal Brillouin zone with high-symmetry points at the \Gamma, M, and K corners.^[26] This three-site unit cell reflects the lattice's chiral symmetry and enables the description of its geometric properties through a basis of sites labeled A, B, and C, each connected to four neighbors via bonds of equal length in the ideal case.^[24] In the context of tight-binding models, the Kagome lattice's band structure is characterized by the coexistence of dispersive Dirac cones and flat bands, with the latter emerging due to destructive interference of electron hopping paths around the hexagonal plaquettes, which localizes states and pins their energy across the Brillouin zone.^[27] These Dirac points, appearing at the [K](/page/K) points (corners of the Brillouin zone), exhibit linear dispersion akin to massless fermions, with Fermi velocities on the order of $10^5 m/s in prototypical realizations.^[28] The flat bands, often positioned near the Fermi level, contribute to enhanced electron correlations and topological features inherent to the lattice geometry.^[27] The Kagome lattice manifests geometric frustration in classical and quantum antiferromagnetic models, stemming from the odd-numbered triangular loops that prevent collinear spin alignments and minimize all nearest-neighbor exchange energies simultaneously.^[29] In the classical Heisenberg antiferromagnet, this frustration stabilizes non-collinear ground states featuring 120° spin angles within each triangle, forming a coplanar \sqrt{3} \times \sqrt{3} magnetic ordering pattern that extends across the lattice while preserving the overall degeneracy.^[30] Such configurations highlight the lattice's role as a canonical example of two-dimensional spin frustration, influencing macroscopic properties like zero-point entropy in quantum variants.^[29]

Applications in Physics and Materials Science

The Kagome lattice serves as a foundational structure for investigating geometrically frustrated magnetism in materials such as jarosites and herbertsmithite, where strong quantum fluctuations prevent conventional magnetic ordering and give rise to quantum spin liquid states. In iron jarosites and related variants, the magnetic ions occupy a perfect kagome arrangement, leading to a spin liquid phase characterized by fractionalized excitations and absence of spin freezing down to millikelvin temperatures, as evidenced in neutron scattering experiments conducted after 2005.^[31] Similarly, herbertsmithite, a synthetic copper-based kagome antiferromagnet with spin-1/2 ions, exhibits quantum disorder and gapless spinon excitations, confirmed through muon spin relaxation and inelastic neutron scattering measurements that highlight its role as a prototype for two-dimensional spin liquids.^[32] In superconductivity research, the kagome metals AV₃Sb₅ (A = K, Rb, Cs) have emerged as a key platform since their discovery in 2020, hosting unconventional superconductivity coexisting with charge density wave (CDW) order that distorts the lattice and modulates electronic states.^[33] These materials exhibit CDW transitions around 80–100 K, accompanied by time-reversal symmetry breaking and possible chiral ordering, which may underpin the subsequent superconducting dome peaking at 2–3 K under ambient or applied pressure.^[34] Recent investigations from 2020 to 2024 suggest topological superconductivity in AV₃Sb₅, with evidence from scanning tunneling microscopy revealing Majorana-like edge states and pairing symmetries potentially mediated by spin fluctuations or CDW fluctuations.^[35] Kagome lattices also provide experimental and theoretical analogs to the Sachdev-Ye-Kitaev (SYK) model, capturing non-Fermi liquid behavior through strongly interacting fermions in frustrated geometries. In optical lattice realizations, spinless fermions on a kagome lattice with random all-to-all couplings reproduce the complex SYK Hamiltonian, yielding low-temperature entropy and conformal invariance characteristic of SYK non-Fermi liquids.^[36] Additionally, the flat bands inherent to kagome structures enable strong electron localization, fostering correlated phases with potential applications in quantum computing via topologically protected states that support robust qubit encoding.^[37] Advancements in the 2020s have focused on twisted kagome lattices, where moiré superlattices amplify electron correlations through tunable band flatness and frustration. In heterostructures like twisted ZrS₂ bilayers, small twist angles generate emergent moiré kagome patterns with ultra-strong spin-orbit coupling, enhancing interaction-driven phenomena such as correlated insulators and potential superconductivity.^[38] Experiments on twisted configurations inspired by AV₃Sb₅ compounds further demonstrate how moiré-induced flat bands intensify charge and spin correlations, opening pathways to engineer exotic quantum phases.^[39]

Topologically Equivalent Variants

Topologically equivalent variants of the trihexagonal tiling maintain the same combinatorial structure of alternating equilateral triangles and regular hexagons but allow for geometric distortions that reduce symmetry while preserving the overall topology. These variants are homeomorphic to the original tiling, ensuring the same connectivity and incidence relations among tiles, vertices, and edges. Such distortions can be achieved through affine transformations, which map the plane to itself as homeomorphisms, thereby preserving the Euler characteristic χ = 0 characteristic of infinite planar tilings where the density of vertices, edges, and faces satisfies V - E + F = 0 in the limit. Variants exemplify this, featuring irregular hexagons and triangles that preserve the (3.6)_2 vertex configuration—where each vertex meets a triangle, hexagon, triangle, and hexagon in sequence—but with non-uniform edge lengths, leading to tilings that are not tile-transitive. These can be constructed using convex polygons capable of tiling the plane via translations in three directions separated by 60 degrees, allowing for irregular shapes while maintaining the trihexagonal-like arrangement.^[40]^[41] Further topological tilings include isohedral but non-isogonal forms, where the tiling is transitive on tiles (isohedral) but not on vertices (non-isogonal), such as elongated deformations that stretch the lattice along one direction. For instance, a deformed trihexagonal tiling can be obtained by scaling the edges of downward-pointing triangles to half the length of upward-pointing ones, resulting in a periodic structure with reduced rotational symmetry while retaining the genus-0 planar covering topology.

Quasiregular and Archimedean Connections

The trihexagonal tiling holds a prominent position among Archimedean tilings as the unique uniform tiling with vertex configuration (3.6)<sub>2</sub>, where triangles and hexagons alternate around each vertex in a quasiregular manner.^[42] This configuration distinguishes it as one of the 11 uniform plane tilings enumerated by Conway, sharing the p6m wallpaper group symmetry with other Archimedean tilings.^[43] Its quasiregular nature arises from the alternation of regular triangular and hexagonal tiles, making it edge-transitive and vertex-transitive while incorporating two distinct regular polygon types in a symmetric pattern.^[42] The tiling emerges as the rectification of the hexagonal tiling {6,3}, where vertices are truncated to the midpoints of edges, yielding the (3.6)<sub>2</sub> arrangement; equivalently, it rectifies the dual triangular tiling {3,6}.^[8] This operation connects it to broader families of uniform tilings, including the truncated hexagonal tiling t{6,3}, which further modifies the hexagonal base by cutting vertices to edge midpoints but introducing dodecagons. In three dimensions, the trihexagonal tiling serves as a cell in the trihexagonal prismatic honeycomb, extending its quasiregular properties into paracompact structures.^[42] As a quasiregular tiling, it is represented by the Wythoff symbol | 3 6 2, indicating its construction from the {3,6} and {6,3} dual pair via rectification in the Coxeter group framework.^[43] It relates to the snub hexagonal tiling through chirality, where the snub operation introduces left- or right-handed twists to the alternating triangle-hexagon pattern, producing a semiregular variant with four triangles and one hexagon per vertex while preserving the underlying uniform symmetry.^[8]

Regular Complex Apeirogons

The trihexagonal tiling can be extended to the complex plane through representations involving regular complex apeirogons, infinite-sided polygons whose vertices and edges are interpreted in complex coordinates. Specifically, it corresponds to the Schläfli symbol {3,6|3}, where three triangular faces meet at each vertex along a 6-gonal path, realized within the ring of Eisenstein integers \mathbb{Z}[\omega], with \omega = e^{2\pi i / 3} the primitive cube root of unity. The dual form {6,3|6} interchanges the roles of faces and vertex figures, maintaining the same symmetry group derived from reflections in the complex plane. This construction allows the tiling to be viewed as a limit of finite polygons in the Eisenstein lattice, where edges connect points separated by units or associates in the ring, enabling analytic study via modular forms and automorphic functions associated with the lattice.^[44] Petrie paths in the trihexagonal tiling form infinite skewed polygons that zigzag across the edges, traversing alternately between adjacent triangles and hexagons in a manner analogous to a helical winding when the plane is rolled into a cylinder. These paths exhibit a density of 2, meaning each infinite edge of the apeirogon effectively covers two parallel layers of the tiling's structure before repeating the pattern in the limit, a property arising from the quasiregular vertex configuration (3.6).2. This density distinguishes the trihexagonal case from purely regular tilings, reflecting its rectified nature and facilitating connections to higher-dimensional skew polyhedra.^[45] In hyperbolic geometry, analogs of the trihexagonal tiling emerge within the rectified hexagonal tiling honeycomb, obtained by rectifying the regular {6,3,3} honeycomb in three-dimensional hyperbolic space. Here, the cells consist of trihexagonal tilings inscribed on horospheres—flat Euclidean surfaces asymptotic to ideal points at infinity—where two triangular prisms and six such tilings meet at each edge, preserving the Euclidean metric locally on each horosphere. This paracompact structure highlights the tiling's role in filling hyperbolic volume, with vertex figures forming infinite skew polyhedra.^[46] The vertices of the trihexagonal tiling in its complex representation can be coordinatized using roots of unity within the Eisenstein integers, modulated by Gaussian periods to capture the periodic lattice structure. Gaussian periods, sums of primitive roots of unity over subgroups of the Galois group of cyclotomic fields, generate subfields that align with the minimal polynomial of \omega, allowing explicit enumeration of lattice points as \sum k_j \zeta^j for suitable integers k_j and roots \zeta. This algebraic formulation underscores the tiling's connection to number-theoretic constructions in quadratic fields.^[47]

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Feb 14, 2025 · Research on kagome systems has made significant progress, revealing various novel phenomena, including superconductivity, charge density waves, ...2 Kagome Lattices And... · 3.1. 2 Vdw-Bonding Kagome... · 5 Moiré Kagome Few-Layers
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Apr 11, 2014 · In the Euclidean plane there are 11 combinatorial types of tilings, called Shubnikov–Laves tilings (see Fig.). However, the symmetry group for a ...Missing: irregular | Show results with:irregular
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Introduction
For each of the regular tilings by equilateral triangles and regular hexagons with unit edge lengths, and p a vertex of each tiling, determine how many tiles ...<|control11|><|separator|>
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[PDF] Convex polytopes and tilings with few flag orbits
Apr 14, 2015 · Conway, Burgiel, and Goodman-Strauss (2008, p. 273) introduce ... the trihexagonal tiling from the regular hexagonal tiling. Each edge ...
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In Figure 1 we present the trihexagonal tiling in one of its usual representations. It is called hexadeltille by Conway and Burgiel (2008) , combining ...
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[2412.00048] The Hexagonal Tiling Honeycomb - arXiv
Nov 23, 2024 · The hexagonal tiling honeycomb is a beautiful structure in 3-dimensional hyperbolic space. It is called {6,3,3} because each hexagon has 6 edges.Missing: trihexagonal | Show results with:trihexagonal
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[PDF] The graphic nature of Gaussian periods - UCLA Mathematics
Our aim here is to initiate the study of. Gaussian periods from this novel perspective. ... Let ωq be a primitive dth root of unity modulo q, so that A “ xωqy in.Missing: trihexagonal tiling