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Triangle Group
Triangle Group
Triangle Group
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Triangle Group (Chinese: 三角輪胎; also known as Triangle Tyre) is a Chinese tire company that manufactures a range of tires for vehicles from passenger cars to construction equipment and tires fit for special purposes under the Triangle and DIAMONDBACK brands. As of 2015 it is the 14th largest tire maker in the world according to Tyres & Accessories.[1]

Key Information

History

[edit]

Triangle Group was founded by the Weihai government in 1976. Lacking a car industry in China, the company supplied tiny tires to Indonesian street-sweepers rubbish carts in the following years.[2] The company grew in size but did not make money until in 1993 with the installment of new management, Triangle reworked itself into a competitive enterprise.[2] In making dramatic reforms, the company invested in new, more modern production and implemented strict workforce discipline.[2] The restructuring would continue into the 2000s, when the company considered a public offering and so brought its accounting to developed world standards and continuously invested in more sophisticated manufacturing lines.[2] These efforts to make itself a top league tire maker would be the subject of a profile article in The Economist during June 2008.

In recent years, the company has focused more on research and development, announcing a desire to become a technology leader, through research partnerships with universities.[3] In 2011, it signed an agreement with the University of Akron to work together on polymer research and also opened an office in the same town of Akron, Ohio with plans for 30 employees.[4] It partnered up in 2012 with the Harbin Institute of Technology to carry out research on designing and manufacturing tires for large-bodied aircraft, enabling Triangle to compete with two other companies in China that already produce such tires.

In 2015, Triangle Group announced its first venture into the U.S. market with the opening of their new North American headquarters in Franklin, Tennessee inside the Nashville Metropolitan Area,[5] and in late 2017 selected Edgecombe County, North Carolina as the location for its first manufacturing facility in the United States where it expects to manufacture six million tires annually.[6] In May 2022 the company has scrapped a project due to "a change in investment environment" and other factors such as COVID.

Plants

[edit]

All Triangle Tire factories are located in Weihai prefecture-level city, Shandong province, eastern China:[7]

  • Huasheng Plant specializes in the production of passenger vehicle tires and giant tires, radial tires engineering.
  • Huamao Plant specializes in the production of commercial vehicle tires.
  • Huayang Plant specializes in the production of high-performance tires for passenger cars and SUVs.
  • Huada Plant specializes in tire retreading.
  • Huaxin Plant specializes in bias tire engineering.

Products

[edit]

The company works as a strategic partner and supplier to many overseas companies. It has struck partnerships with Caterpillar, Volvo, and Goodyear.[3] In terms of market share success, the company's most dominant placement is in the off the road (OTR) category, which includes tires for mining, construction, and other industrial uses.[3] It is the 4th largest manufacturer of OTR tires with the company shipping 90% of its OTR stock to export markets.[3]

At the start of 2016, Triangle truck and bus tyres started being distributed by multinational distributor Zenises.[8]

In December 2021 the company has launched a mobile app for Android and iOS smartphones. It includes all products for the European market, from passenger to TBR and earthmoving tires.

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Triangle Group

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A triangle group is a Coxeter group of rank three in mathematics, generated by three reflections aa, bb, and cc corresponding to the sides of a triangle, with the standard presentation ⟨a,b,c∣a2=b2=c2=(ab)p=(bc)q=(ca)r=1⟩\langle a, b, c \mid a^2 = b^2 = c^2 = (ab)^p = (bc)^q = (ca)^r = 1 \rangle, where the integers p,q,r≥2p, q, r \geq 2 (or ∞\infty) represent the orders of the products of adjacent reflections, related to the dihedral angles π/p\pi/p, π/q\pi/q, and π/r\pi/r of the fundamental triangle. These groups act as discrete reflection groups on the spherical, Euclidean, or hyperbolic plane, tiling the space with congruent copies of the fundamental triangle via the orbit of reflections, and their geometric realization is classified by the sum 1p+1q+1r\frac{1}{p} + \frac{1}{q} + \frac{1}{r}: greater than 1 yields a finite spherical group acting on S2S^2 (e.g., Δ(2,3,3)\Delta(2,3,3), Δ(2,3,4)\Delta(2,3,4), Δ(2,3,5)\Delta(2,3,5)); equal to 1 produces an infinite Euclidean (affine) group acting on E2E^2 with translational symmetries (e.g., Δ(2,3,6)\Delta(2,3,6), Δ(2,4,4)\Delta(2,4,4), Δ(3,3,3)\Delta(3,3,3)); and less than 1 results in an infinite hyperbolic group acting discretely on H2H^2 (e.g., Δ(2,3,7)\Delta(2,3,7), Δ(3,3,4)\Delta(3,3,4)). Triangle groups play a central role in the study of tilings and tessellations, as their actions produce regular triangulations of the underlying space, and they encompass important examples like the Γ=Δ(2,3,∞)\Gamma = \Delta(2,3,\infty) acting on the hyperbolic plane. The even-length words in the generators form an index-two subgroup known as the von Dyck group or triangle rotation group D(p,q,r)D(p,q,r), which is generated by rotations of orders pp, qq, and rr around the 's vertices and is orientation-preserving. Finite triangle groups correspond precisely to the groups of the Platonic solids (tetrahedral, octahedral, icosahedral), linking them to polyhedral and . In broader contexts, triangle groups appear in arithmetic , such as in the study of Fuchsian groups and modular surfaces, and in through their connections to Coxeter graphs and Hecke algebras. Their classification via the Gauss-Bonnet theorem ties the angle sum directly to the of the ambient space, providing a foundational example of how abstract group presentations encode geometric structures.

Fundamentals

Definition

A triangle group is a generated by three reflections in the lines (or great circles) forming the sides of a in a of constant curvature, where the triangle has interior angles π/p\pi/p, π/q\pi/q, and π/r\pi/r with p,q,rp, q, r integers greater than or equal to 2. These reflections act as orientation-reversing isometries of the underlying . The triangle serves as a fundamental domain for the action of the triangle group on the sphere S2S^2, the Euclidean plane E2\mathbb{E}^2, or the hyperbolic plane H2\mathbb{H}^2, depending on whether the sum of the angles exceeds, equals, or is less than π\pi, respectively. The group action tiles the space by repeated reflections across the triangle's sides, producing a tessellation where copies of the fundamental triangle cover the space without overlap except on boundaries. Basic examples include the (2,3,3) triangle group, generated by reflections in a spherical with angles π/2\pi/2, π/3\pi/3, and π/3\pi/3, which realizes the of the , and the (2,4,4) triangle group, generated by reflections in a Euclidean with angles π/2\pi/2, π/4\pi/4, and π/4\pi/4 (a right-angled ), which tiles the plane with squares. The full triangle group consists of orientation-reversing isometries, but it contains an index-2 generated by the products of pairs of reflections, which consists of orientation-preserving transformations such as rotations.

Generators and Relations

Triangle groups are abstractly defined via a involving three generators corresponding to reflections. These generators, denoted ss, tt, and uu, represent reflections across the sides of a fundamental and satisfy the relations s2=t2=u2=1s^2 = t^2 = u^2 = 1, as each reflection is an involution of order two. The pairwise products of these generators obey additional relations (st)p=(tu)q=(us)r=1(st)^p = (tu)^q = (us)^r = 1, where p,q,rp, q, r are integers greater than or equal to 2 that classify the group type based on the geometry of the . The complete presentation of the triangle group is thus ⟨s,t,u∣s2=t2=u2=(st)p=(tu)q=(us)r=1⟩.\langle s, t, u \mid s^2 = t^2 = u^2 = (st)^p = (tu)^q = (us)^r = 1 \rangle. These parameters p,q,rp, q, r directly correspond to the angles π/p\pi/p, π/q\pi/q, and π/r\pi/r at the vertices of the , with the relation (st)p=1(st)^p = 1 arising because the composition of reflections ss and tt yields a by twice the angle between their lines of reflection, which closes after pp applications when the angle is π/p\pi/p. In the geometric realization, these relations are tied to the triangle's angles through the in the ambient space—Euclidean, spherical, or hyperbolic—which relates the angles to the side lengths and determines the existence of the triangle for given p,q,rp, q, r. For instance, in , the hyperbolic cosh⁡c=cos⁡γ+cos⁡αcos⁡βsin⁡αsin⁡β\cosh c = \frac{\cos \gamma + \cos \alpha \cos \beta}{\sin \alpha \sin \beta} (where α=π/p\alpha = \pi/p, β=π/q\beta = \pi/q, γ=π/r\gamma = \pi/r) allows computation of side lengths, confirming the action when the angle sum is less than π\pi. Similar formulas apply in spherical and Euclidean cases to verify the configuration. A degenerate case occurs when one parameter, say r=∞r = \infty, corresponding to a zero angle, yielding the infinite dihedral group generated by two reflections across parallel lines or a straight angle.

Geometric Classifications

Spherical Triangle Groups

Spherical triangle groups arise as finite Coxeter groups generated by reflections across the sides of a spherical triangle with vertex angles π/p\pi/p, π/q\pi/q, and π/r\pi/r, where p,q,rp, q, r are integers greater than or equal to 2 satisfying 1/p+1/q+1/r>11/p + 1/q + 1/r > 1. This inequality corresponds to the positive of the sphere, ensuring the group's action tiles the sphere discretely with a finite number of triangular fundamental domains, resulting in a finite group order. The classification of such groups includes the Δ(2,2,n)\Delta(2,2,n) of order 4n4n for n≥2n \geq 2, and the three exceptional polyhedral groups Δ(2,3,3)\Delta(2,3,3), Δ(2,3,4)\Delta(2,3,4), and Δ(2,3,5)\Delta(2,3,5). The case Δ(2,4,4)\Delta(2,4,4) achieves equality in the inequality and represents the Euclidean limit, so it is excluded from the spherical classification. The exceptional spherical triangle groups Δ(2,3,3)\Delta(2,3,3), Δ(2,3,4)\Delta(2,3,4), and Δ(2,3,5)\Delta(2,3,5) serve as the full s (including reflections) of the Platonic solids, acting on the circumscribed . Specifically, Δ(2,3,3)\Delta(2,3,3) is the of the , with order 24; Δ(2,3,4)\Delta(2,3,4) is the of the (or dual ), with order 48; and Δ(2,3,5)\Delta(2,3,5) is the of the (or dual ), with order 120. For these groups, the order is given by the formula ∣Δ(p,q,r)∣=4/(1/p+1/q+1/r−1)|\Delta(p,q,r)| = 4 / (1/p + 1/q + 1/r - 1), which derives from the spherical excess of the fundamental triangle: the excess π(1/p+1/q+1/r−1)\pi(1/p + 1/q + 1/r - 1) determines the area of each triangular domain, and the 's total area 4π4\pi implies the number of domains is the reciprocal times 4, yielding the group order via the reflection action. The orientation-preserving subgroups of these exceptional groups are the rotation groups of the Platonic solids, isomorphic to the (order 12) for the , the (order 24) for the , and the (order 60) for the . These rotation groups lift to central extensions in SU(2)\mathrm{SU}(2), known as the binary polyhedral groups: the binary tetrahedral group of order 24, the binary octahedral group of order 48, and the binary icosahedral group of order 120, which play a key role in representations of 3-dimensional symmetries and quaternionic structures.

Euclidean Triangle Groups

Euclidean triangle groups are infinite discrete groups of isometries of the E2\mathbb{E}^2 generated by reflections across the sides of a triangle with angles π/p\pi/p, π/q\pi/q, and π/r\pi/r, where p,q,rp, q, r are integers greater than or equal to 2 satisfying the condition 1p+1q+1r=1\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 1. This condition ensures that copies of the triangle the plane without gaps or overlaps, yielding a crystallographic action with translational symmetries. Unlike finite spherical groups or infinite hyperbolic ones, these groups are affine and correspond to the symmetry groups of the three regular tilings of the plane. Up to , the solutions to the condition are the (2,3,6)(2,3,6), (2,4,4)(2,4,4), and (3,3,3)(3,3,3). These correspond to the triangular lattice (for (2,3,6)(2,3,6)), the (for (2,4,4)(2,4,4)), and the (for (3,3,3)(3,3,3)). Specifically:
  • The (2,3,6)(2,3,6) group acts as the full of the triangular tiling {3,6}\{3,6\}, where tiles are equilateral triangles meeting six at each vertex, with rotation orders 2, 3, and 6 at the triangle's vertices.
  • The (2,4,4)(2,4,4) group symmetries the square tiling {4,4}\{4,4\}, with squares meeting four at each vertex and rotation orders 2, 4, and 4.
  • The (3,3,3)(3,3,3) group symmetries the hexagonal tiling {6,3}\{6,3\}, where regular hexagons meet three at each vertex, with all rotation orders 3, though it also relates to the dual triangular tiling in the .
These groups are among the 17 wallpaper groups, specifically the reflection groups denoted in Conway notation as ∗632*632 (for (2,3,6)(2,3,6)), ∗442*442 (for (2,4,4)(2,4,4)), and ∗333*333 (for (3,3,3)(3,3,3)), which include both orientation-preserving rotations and orientation-reversing reflections. They generate the full crystallographic symmetries of their respective lattices, extending the orientation-preserving subgroups by reflections across the tiling's edges. The fundamental domain for each group is the generating triangle itself, which tiles the plane by repeated reflections. The area of this domain satisfies π(1−1p−1q−1r)=0\pi \left(1 - \frac{1}{p} - \frac{1}{q} - \frac{1}{r}\right) = 0, reflecting the zero curvature of the and the infinite order of the group. This area condition distinguishes them from spherical cases (positive area, finite groups) and underscores their role in periodic plane tilings.
Triple (p,q,r)(p,q,r)TilingLatticeWallpaper Group (Conway)Angles (π/p,π/q,π/r\pi/p, \pi/q, \pi/r)
(2,3,6)Triangular {3,6}\{3,6\}Triangular*632π/2,π/3,π/6\pi/2, \pi/3, \pi/6 (90°, 60°, 30°)
(2,4,4)Square {4,4}\{4,4\}Square*442π/2,π/4,π/4\pi/2, \pi/4, \pi/4 (90°, 45°, 45°)
(3,3,3)Hexagonal {6,3}\{6,3\}Hexagonal*333π/3,π/3,π/3\pi/3, \pi/3, \pi/3 (60°, 60°, 60°)

Hyperbolic Triangle Groups

Hyperbolic triangle groups are discrete groups generated by reflections across the sides of a with interior angles π/p\pi/p, π/q\pi/q, and π/r\pi/r, where p,q,r≥2p, q, r \geq 2 are integers satisfying 1/p+1/q+1/r<11/p + 1/q + 1/r < 1. This condition ensures that the angle sum is less than π\pi, characteristic of hyperbolic geometry, resulting in an infinite group of isometries acting on the hyperbolic plane without fixed points at infinity in a compact manner. Unlike finite spherical or periodic Euclidean cases, these groups produce non-compact tessellations that cover the entire hyperbolic plane. These groups are typically realized in conformal models of the hyperbolic plane, such as the Poincaré disk model, where the hyperbolic plane is the interior of the unit disk with Möbius transformations preserving the boundary circle, or the upper half-plane model, consisting of points {z∈C∣ℑ(z)>0}\{z \in \mathbb{C} \mid \Im(z) > 0\} with the group of transformations PSL(2,R)\mathrm{PSL}(2, \mathbb{R}). The serves as a fundamental domain for the , tiling the plane through repeated reflections, and its hyperbolic area is determined by the Gauss-Bonnet theorem: π[1−(1/p+1/q+1/r)]\pi \left[1 - \left(1/p + 1/q + 1/r\right)\right]. This area deficit reflects the negative curvature and governs the group's covolume in the space of isometries. Prominent examples include the (2,3,7)(2,3,7) triangle group, whose normal torsion-free subgroup of index 168 quotients the hyperbolic plane to yield the , a -3 with 168 automorphisms, maximizing the order for its genus. Another is the (2,3,8)(2,3,8) group, which generates hyperbolic tessellations where eight triangles meet at each vertex, corresponding to order-3 rotational symmetries in certain infinite tilings. The orientation-preserving subgroups, generated by rotations rather than reflections, form Fuchsian groups, which are index-2 subgroups isomorphic to Δ+(p,q,r)\Delta^+(p,q,r) and consist of discrete faithful representations in PSL(2,R)\mathrm{PSL}(2, \mathbb{R}). Hyperbolic triangle groups comprise infinitely many distinct examples, enumerated across infinite families parameterized by integer triples (p,q,r)(p,q,r) with p≤q≤rp \leq q \leq r and 1/p+1/q+1/r<11/p + 1/q + 1/r < 1, such as fixing p=2,q=3p=2, q=3 and letting r≥7r \geq 7. These families densely populate the parameter space of possible angle configurations, contributing significantly to the moduli space of hyperbolic structures, where their rigid actions influence the topology of quotients and the distribution of Fuchsian groups among all discrete subgroups of PSL(2,R)\mathrm{PSL}(2, \mathbb{R}).

Algebraic Structure

Von Dyck Groups

Von Dyck groups, also known as ordinary triangle groups, are the index-two subgroups of triangle groups consisting of the orientation-preserving isometries. They can be defined abstractly using Von Dyck's theorem, which guarantees that the group generated by elements aa, bb, and cc satisfying the relations ap=1a^p = 1, bq=1b^q = 1, cr=1c^r = 1, and abc=1abc = 1—where p,q,r≥2p, q, r \geq 2 are integers—is isomorphic to the subgroup of rotations in the corresponding triangle group acting on the sphere, plane, or hyperbolic plane depending on the value of 1p+1q+1r\frac{1}{p} + \frac{1}{q} + \frac{1}{r}. This construction realizes the abstract presentation as a concrete geometric group, emphasizing the free product structure amalgamated along the cyclic relations. The standard presentation of a Von Dyck group Δ0(p,q,r)\Delta_0(p, q, r) is ⟨a,b,c∣ap=bq=cr=abc=1⟩,\langle a, b, c \mid a^p = b^q = c^r = abc = 1 \rangle, where aa, bb, and cc correspond to rotations by 2π/p2\pi/p, 2π/q2\pi/q, and 2π/r2\pi/r around the vertices of a fundamental triangle. This presentation captures the group's structure as a quotient of the free product of three cyclic groups by the normal closure of the relation abcabc. An equivalent two-generator presentation is ⟨x,y∣xp=yq=(xy)r=1⟩\langle x, y \mid x^p = y^q = (xy)^r = 1 \rangle, obtained by setting c=(xy)−1c = (xy)^{-1}. In relation to the full triangle group Δ(p,q,r)\Delta(p, q, r), generated by reflections s1,s2,s3s_1, s_2, s_3 with relations si2=1s_i^2 = 1 and (sisj)mij=1(s_i s_j)^{m_{ij}} = 1 (where m12=pm_{12} = p, etc.), the Von Dyck group is the kernel of the orientation homomorphism to Z/2Z\mathbb{Z}/2\mathbb{Z}. The rotations are given by a=s2s3a = s_2 s_3, b=s3s1b = s_3 s_1, c=s1s2c = s_1 s_2, satisfying abc=1abc = 1, and the reflections can be recovered as products of two rotations, such as s1=bcs_1 = bc. This index-two covering structure ensures that the Von Dyck group double-covers the full triangle group topologically and algebraically. A prominent example is the binary tetrahedral group, which is the Von Dyck group Δ0(2,3,3)\Delta_0(2, 3, 3) of order 24, with presentation ⟨a,b∣a2=b3=(ab)3=1⟩\langle a, b \mid a^2 = b^3 = (ab)^3 = 1 \rangle. This group arises as the preimage of the A4A_4 (the rotation group of the tetrahedron) under the double cover SU(2)→SO(3)SU(2) \to SO(3), illustrating its role in the classification of finite subgroups of SL(2,C)SL(2, \mathbb{C}). The concept of Von Dyck groups stems from Walther von Dyck's foundational 1882 work on abstract group presentations, where he first systematically used generators and relations to define groups combinatorially, applying it to polygonal and polyhedral symmetry groups including those derived from triangles. This contribution marked a pivotal advancement in combinatorial group theory, bridging geometric realizations with algebraic abstractions.

Presentations and Coxeter Diagrams

Triangle groups are realized algebraically as rank-3 Coxeter groups, generated by three reflections s,t,us, t, u satisfying the relations s2=t2=u2=1s^2 = t^2 = u^2 = 1 and (st)p=(tu)q=(us)r=1(st)^p = (tu)^q = (us)^r = 1, where p,q,r≥2p, q, r \geq 2 are integers specifying the orders of the products of adjacent generators. This presentation captures the full symmetry group including reflections, distinguishing it from the rotation subgroup discussed in von Dyck presentations. The Coxeter matrix associated with this system is a 3×3 symmetric matrix with 1's on the diagonal and off-diagonal entries m12=pm_{12} = p, m23=qm_{23} = q, m13=rm_{13} = r. The corresponding Coxeter diagram consists of three vertices representing the generators s,t,us, t, u, with an edge between vertices ii and jj labeled mijm_{ij} if mij>3m_{ij} > 3, unlabeled if mij=3m_{ij} = 3, no edge if mij=2m_{ij} = 2, and labeled ∞\infty if mij=∞m_{ij} = \infty. For spherical groups, one of p,q,rp, q, r is typically 2, resulting in a linear without the closing edge. In contrast, Euclidean and hyperbolic cases with all p,q,r≥3p, q, r \geq 3 yield a , as in the [3,3,3][3,3,3] with three unlabeled edges forming a . Infinite arise when at least one label is ∞\infty, such as [3,3,∞][3,3,\infty] for certain hyperbolic cases. All irreducible Coxeter groups of rank 3 are isomorphic to triangle groups of this form, encompassing the finite types A3,B3,H3A_3, B_3, H_3 (spherical), affine types like A~2,C~2,G~2\tilde{A}_2, \tilde{C}_2, \tilde{G}_2 (Euclidean), and infinite hyperbolic families. This classification follows from the geometric realization via reflections in spherical, Euclidean, or , with the group's finiteness determined by whether 1/p+1/q+1/r>11/p + 1/q + 1/r > 1, =1=1, or <1<1, respectively. These presentations facilitate computational studies in software such as GAP, via the CHEVIE package, which implements algorithms for Coxeter groups including character tables, representations, and subgroup computations for both finite and infinite cases. For instance, the finite group [3,3,3][3,3,3] (affine A~2\tilde{A}_2 quotient, but finite presentation) can be constructed and analyzed for its order and structure, while infinite groups like [3,7,3][3,7,3] allow enumeration of cosets or growth rates. The associated Artin group, sharing the same Coxeter diagram, replaces the Coxeter relations with braid relations of length mijm_{ij}, such as st⋯⏟p=ts⋯⏟p\underbrace{st \cdots}_{p} = \underbrace{ts \cdots}_{p}
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