Triangle Edges Research Articles

Low-temperature photoluminescence and Raman study of monolayer WSe2 for photocarrier dynamics and thermal conductivity

Low-temperature PL analysis reveals an intriguing temperature-dependent emission pattern in WSe2: excitonic dominance above the 150 K Debye temperature, a balance between excitonic and trionic emissions at 150 K, and trionic dominance below this threshold. At lower temperatures, both excitons and trions display linearly polarized emissions, with polarization increasing from 0% at 300 K to 23% (excitons) and 7% (trions) at 150 K, and 12% for trions at 90 K. Moreover, the synthesized monolayer of WSe2 exhibits high thermal conductivity (246 W m−1 K−1 for A1g and 185 W m−1 K−1 for E2g1 modes). This property is attributed to Se vacancies and defects at triangle edges, which redirect phonons, reducing scattering and enabling efficient heat transport along boundaries. The unveiling of these novel insights within the synthesized 2D WSe2 material holds significant promise for its potential applications in nano-optoelectronics. Its demonstrated efficiency in dissipating heat, coupled with improved thermal stability, suggests the possibility of employing it in future devices. This could facilitate compact designs and the miniaturization of advanced technological tools, showcasing the material's potential for practical implementation.

Crystal Structure of de Novo Designed Coiled-Coil Protein Origami Triangle.

Coiled-coil protein origami (CCPO) uses modular coiled-coil building blocks and topological principles to design polyhedral structures distinct from those of natural globular proteins. While the CCPO strategy has proven successful in designing diverse protein topologies, no high-resolution structural information has been available about these novel protein folds. Here we report the crystal structure of a single-chain CCPO in the shape of a triangle. While neither cyclization nor the addition of nanobodies enabled crystallization, it was ultimately facilitated by the inclusion of a GCN2 homodimer. Triangle edges are formed by the orthogonal parallel coiled-coil dimers P1:P2, P3:P4, and GCN2 connected by short linkers. A triangle has a large central cavity and is additionally stabilized by side-chain interactions between neighboring segments at each vertex. The crystal lattice is densely packed and stabilized by a large number of contacts between triangles. Interestingly, the polypeptide chain folds into a trefoil-type protein knot topology, and AlphaFold2 fails to predict the correct fold. The structure validates the modular CC-based protein design strategy, providing molecular insight underlying CCPO stabilization and new opportunities for the design.

An efficient algorithm for approximate Voronoi diagram construction on triangulated surfaces

Voronoi diagrams on triangulated surfaces based on the geodesic metric play a key role in many applications of computer graphics. Previous methods of constructing such Voronoi diagrams generally depended on having an exact geodesic metric. However, exact geodesic computation is time-consuming and has high memory usage, limiting wider application of geodesic Voronoi diagrams (GVDs). In order to overcome this issue, instead of using exact methods, we reformulate a graph method based on Steiner point insertion, as an effective way to obtain geodesic distances. Further, since a bisector comprises hyperbolic and line segments, we utilize Apollonius diagrams to encode complicated structures, enabling Voronoi diagrams to encode a medial-axis surface for a dense set of boundary samples. Based on these strategies, we present an approximation algorithm for efficient Voronoi diagram construction on triangulated surfaces. We also suggest a measure for evaluating similarity of our results to the exact GVD. Although our GVD results are constructed using approximate geodesic distances, we can get GVD results similar to exact results by inserting Steiner points on triangle edges. Experimental results on many 3D models indicate the improved speed and memory requirements compared to previous leading methods.

Size-controllable synthesis of 2D Mn3O4 triangular-shaped nanosheets by thermal chemical vapor deposition

In this study, monolayer up to few-layer of Mn 3 O 4 nanosheets grown on c-Si (111) substrates using a thermal chemical vapor deposition (TCVD) technique were investigated in detail, where the growth reaction pressure was varied from 0.12 to 0.72 mbar. The morphology of the triangular-shaped Mn 3 O 4 nanosheet greatly depends on the atomic ratio of O to Mn, which eventually determines the horizontal growth rate of the triangle's edges that forms the shape of the nanosheet. A low O/Mn ratio results in the formation of a triangle with a longer edge as compared to a high O/Mn ratio which produces a shorter edge. The 2D triangular-shaped Mn 3 O 4 nanosheets prepared at 0.42 mbar exhibited a thickness of ∼1.10–2.29 nm (equivalent to the approximate thickness of monolayer-bilayer) with a maximum edge length of ∼850 nm, whereas ultrathin nanosheets with a thickness of ∼1.93–4.68 nm (equivalent to ∼2–3 layers) with an edge length of <250 nm were found at the highest reaction pressure of 0.72 mbar. The results imply a synthesis of 2D Mn 3 O 4 nanosheets with controllability of thickness and edge length that vary from monolayer to ultrathin nanosheets at different O/Mn ratios. By presuming that the Mn 3 O 4 growth along the [001] direction, we performed first-principles density-functional theory calculations to validate the structure and electronic properties of these 2D Mn 3 O 4 nanosheets. Our calculated lattice parameters for the 2D Mn 3 O 4 were considerably close to the experimental measurements. The band structure calculations predicted that the 2D Mn 3 O 4 possessed metallic characteristics. This paper discusses the seed-assisted growth mechanism of these 2D Mn 3 O 4 nanosheets. • Thermal CVD grown Mn 3 O 4 nanosheet. • The atomic ratio between O and Mn controlled the morphologies of the triangular-shaped Mn 3 O 4 nanosheet. • A low O/Mn ratio produced monolayer 2D triangular shaped with a longer edge. • A high O/Mn ratio produced ultrathin nanosheets with thickness up to 3 layers and shorter edge. • Based on DFT calculations, 2D Mn 3 O 4 nanosheet exhibited metallic characteristics.

Guite, the spinel-structured Co2+Co3+2O4, a new mineral from the Sicomines copper–cobalt mine, Democratic Republic of Congo

AbstractGuite (IMA2017-080), Co3O4, is a new mineral species and an important economic mineral found in the Sicomines copper-cobalt mine, located ~11 km southwest of Kolwezi City, Democratic Republic of Congo. The mineral occurs as a granular agglomerate, 50 to 500 μm in size, and is associated closely with heterogenite in a quartz matrix. Guite is opaque, has a dark grey colour with metallic lustre and a black streak. In reflected light microscopy, it is white with no internal reflections. The reflectance values (in air,Rin %) are: 27.0 (470 nm); 25.6 (546 nm); 25.2 (589 nm), and 24.6 (650 nm). The average of 20 electron-microprobe analyses is Co 71.53, Cu 0.58, Mn 0.67, Si 0.25, O 26.78, total 99.82 wt.%, corresponding to the empirical formula calculated on the basis of 4 O apfu: (Co2+0.92Cu2+0.02Si4+0.02)Σ0.96(Co3+1.98Mn3+0.03)Σ2.01O4.00, with Co2+and Co3+partitioned using charge balance. The ideal formula is Co2+Co3+2O4. Guite is cubic with space groupFd$\bar{3}$m. The unit cell parameters refined from the single crystal X-ray diffraction data are:a= 8.0898(1) Å,V= 529.436(11) Å3andZ= 8. The calculated density of guite is 6.003 g/cm3. The eight strongest observed powder X-ray diffraction lines [din Å (I/I0) (hkl)] are: 4.6714 (16.7) (111), 2.8620 (18.4) (220), 2.4399 (100) (311), 2.3348 (10.4) (222), 2.0230 (24.8) (400), 1.5556 (26.3) (511, 333), 1.4296 (37.7) (440) and 1.0524 (10.1) (731, 553). The crystal structure of guite was determined by single-crystal X-ray diffraction and refined toR= 0.0132 for 3748 (69 unique) reflections. Guite has a typical spinel-type structure with Co2+in tetrahedral coordination with a Co2+–O bonding length of 1.941(1) Å, and Co3+in octahedral coordination with a Co3+–O bonding length of 1.919(1) Å. The structure is composed of cross-linked framework of chains of Co3+–O6octahedra sharing the equilateral triangle edges (2.550 Å) in three directions [0 1 1], [1 1 0], [1 0 1] with Co2+filling the tetrahedral interstices among the chains. Guite is named in honour of Prof. Xiangping Gu (1964–).

Allocation of HV/MV Substations in Iraqi Power Grid Using Delaunay Triangulation

In provincial areas, allocation of the HV/MV substations aims to assign a proper set of numbers and locations of substations to supply loads spatially distributed over a large geographic area with continually emerging towns. This requires identifying the sparse load centers, the existing substations, and the new substations required to cover the potential extension. This paper proposes a method to allocate the main substations to the targeted area based on the lines’ length which are assigned by Delaunay triangulation and geographical coordinates. The proposed configuration is identified by a criterion that is tailored to the Iraqi sub-transmission system in which the standard distances among the substation follow specific patterns. The standard distances that are identified by the proposed methods are represented by the edges of Delaunay triangles. This paper aims to improve the voltage regulation of the distribution grid and avoid problems of overloaded feeders by assigning the proposed locations that reduce the lines’ lengths. The proposed method is verified using simulation tests on Diyala 9-bus system which is a part of the Iraqi 132 kV power system. The results show that the proposed configuration of the substations is reasonably valid and quite close to the real locations of the towns of the system with efficient lengths for the lines.

Regional application of C1 finite element interpolation method in modeling of ionosphere total electron content over Europe

• A novel approach for regional ionosphere grid model with high spatial-temporal resolution is suggested. • The results of the new model is analyzed at different disturb geomagnetic and solar conditions. • A comparative analysis between the results of new model, GIM, NeQuick and GPS-TEC is made. In this paper, a novel approach based on C 1 finite element (FE) interpolation is used to construct a regional ionospheric grid model (RIGM) with high spatio-temporal resolution over Europe. The spatial resolution of the new model is 0.5 0 for a longitude and latitude and a temporal resolution is 15 min. Observations of 38 GPS stations at different days and seasons with different geomagnetic and solar activity conditions are used to evaluate the new method. All stations are used to discretize the whole domain of the problem into separate triangular patches using a Delaunay triangulation algorithm. In this case, GPS stations would be the vertices of the triangular patches. Also, the position of stations in sun-fixed coordinate system with coordinates of latitude and longitude are chosen in order to consider the temporal variations of vertical total electron content (VTEC). To model the behavior of VTEC in each triangular patch, a particular shape function is defined which is fitted to VTEC values at three vertices of each triangle while it satisfies C 1 continuity of VTEC at the common edges of the adjacent triangles. The results of the new model are compared with the VTEC of GPS at 4 control stations as well as global ionosphere map (GIM) and NeQuick models. Statistical parameters, including correlation coefficient, root mean square error (RMSE), standard deviation and dVTEC = |VTEC GPS - VTEC model | have been used to evaluate the error of the models. In low geomagnetic activity (KP < 4), the average RMSE at 4 control stations for FE, GIM and NeQuick models are 0.89, 1.05 and 6.42 TECU, respectively, while in the high geomagnetic activity (KP > 4), they are 1.24, 1.68 and 7.62 TECU, respectively. For further analysis, the accuracy of three models in precise point positioning (PPP) has been also evaluated. In PPP method, there is an improvement of about 1–16 mm compared to GIM and NeQuick models at coordinate components of control stations in high geomagnetic activity. The results of the analysis performed in this paper show that the FE model has a very high capability and accuracy in regional VTEC modeling in high and low geomagnetic conditions. The regional accuracy of this model is higher than GIM and NeQuick models and therefore can be considered as a new model for the ionosphere.

Crystal chemistry criteria of the existence of spin liquids on the kagome lattice

The structural-magnetic models of 25 antiferromagnetic kagome cuprates similar to herbertsmithite (ZnCu3(OH)6Cl2)—a perspective spin liquid—have been calculated and analyzed. Main correlations between the structure and magnetic properties of these compounds were revealed. It has been demonstrated that, in all AFM kagome cuprates, including herbertsmithite, there exists the competition between the exchange interaction and the antisymmetric anisotropic exchange one (the Dzyaloshinskii–Moriya interaction), as magnetic ions are not linked to the center of inversion in the kagome lattice. This competition is strengthened in all the kagome AFM, except herbertsmithite, by one more type of the anisotropy (duality) of the third in length J3 magnetic couplings (strong J3(J12) next-to-nearest-neighbor couplings in linear chains along the triangle edges and very weak FM or AFM J3(J d) couplings along the hexagon diagonals). The above couplings are crystallographically identical, but are divided to two types of different in strength magnetic interactions. The existence of duality of J3 couplings originated from the structure of the kagome lattice itself. Only combined contributions of dual J3 couplings with anisotropic Dzyaloshinskii–Moriya interactions are capable to suppress frustration of kagome antiferromagnetics. It has been demonstrated that the possibility of elimination of such a duality in herbertsmithite, which made it a spin liquid, constitutes a rare lucky event in the kagome system. Three crystal chemistry criteria of the existence of spin liquids on the kagome lattice have been identified: first, the presence of frustrated kagome lattices with strong dominant antiferromagnetic nearest-neighbor J1 couplings competing only with each other in small triangles; second, magnetic isolation of these frustrated kagome lattices; and third, the absence of duality of the third in length J3 magnetic couplings.

Modeling of the tension stiffening behavior and the water permeability change of steel bar reinforcing concrete using mesoscopic and macroscopic hydro-mechanical lattice model

Mesoscopic and macroscopic hydromechanical lattice simulations are performed for studying the tension stiffening behavior, the change in permeability, and their linking of a tensioned reinforced normal strength concrete (NSC) member. In the lattice framework, the hydromechanical coupling is carried out by dual Delaunay triangles and Voronoi polygons, where transport elements are placed along the edges of the Voronoi polygons while mechanical elements are edges of Delaunay triangles. At the mesoscale, concrete is considered as a constitution of three phases: aggregate, cement paste and interfacial transition zones (ITZ). Two mesoscopic components cement paste and ITZ, as well as the concrete at the macroscale, are modeled by a damage model including softening strain feature. The crack opening is linked to the damage variable. Fluid flow obeys Darcy’s law in the intact conduit element and Poiseuille’s law in the crack. Validation against experimental results available in the open literature for NSC tie specimens shows that both current macroscopic and mesoscopic lattice hydromechanical models are appropriated to describe the tensile stiffening and damage induced permeability of NSC reinforced by a steel bar. Mesoscopic modeling helps to get insights of the aggregate volume fraction effect.

Adaptive sampling point planning for free-form surface inspection under multi-geometric constraints

Free-form surfaces have been widely used in aerospace, automotive and other fields. Due to its complex geometry, free-form surface inspection is generally conducted by touch-trigger or measuring probe-based Coordinate Measurement Machines or On-machine Measurement. Sampling strategy plays a decisive role in improving both measurement accuracy and efficiency, which is determined by sample size and distribution of sample points. However, it is difficult to simultaneously take the surface curvature, sampling density and approximation error into account, considering the complexity of surface geometry. In this paper, triangle mesh simplification is innovatively adopted in sampling planning to achieve multi-geometric constraints. As triangle mesh has outstanding advantages in representing the surface features, strong stability and is easy to modify its structure, free-form surface is converted to a dense triangle mesh. Triangle mesh simplification is implemented by iteratively contracting triangle edges. An improved quadric error metric is established to decide contraction order and optimal target vertices under discrete curvature constraint. Sampling density is controlled by limiting the triangle edge length. Detailed adaptive sampling algorithm under multi-geometric constraints is then developed. Both simulation and experiment are conducted to validate feasibility and robustness of the proposed method. The results are compared with uniform sampling and existing adaptive sampling strategy to show that the proposed method can prominently reduce sampling error when sample size is small.

Dominating Number on Icosahedral-Hexagonal Network

In this paper, we deal with the dominating set and the domination number on an icosahedral-hexagonal network. We will consider all cases of successive halving of the edges of triangles that are the sides of icosahedrons and thus obtain icosahedral-hexagonal networks.

Tunable Mid-Infrared Optical Properties of Monolayer Black Phosphorus Through Au Nanotriangle Arrays via Electric Field Reflection and Surface Plasmon Polaritons

A metamaterial system composed of monolayer black phosphorus and Au triangle arrays is designed. Absorptivity, transmittivity, and reflectivity are investigated in mid-infrared regime. A low transmissivity and high absorptivity can be obtained via surface plasmon polaritons at the black phosphorus and Au triangle array interface. By changing the geometrical parameters, such as angle magnitude and slit width of Au triangle, we can modulate the transmissivity, reflectivity and absorptivity properties. Different from other previous work, it is found the zigzag direction has a better photoresponse than that armchair by changing the slit width. The electric field of the external radiation field is reflected at the Au triangle edges. Thus, electric field component perpendicular to the polarization direction generates, which can also lead to surface plasmon polaritons and is not researched in others’ work. With increase in the angle of Au triangle, transmittivity (or absorptivity) for armchair direction has a blue shift and for zigzag direction has a red shift. The transmittivity decrease ( or absorptivity increase ) for special wavelength caused by the surface plasmon polaritons may be applied to design filter devices.

Improved Grid Relaxation with Zero-Order Integrators

In this research, we have improved a relaxation method for triangular meshes intended for finite element fluid simulations which contain discrete element particles. The triangle edges are treated as springs which relax their lengths towards a “better” force equilibrium where the triangles are closer to equilateral shape. The actual kernel is an improved zero order integrator which is able to follow reconfigurations of the particles faster than earlier methods. The improved relaxation allows larger timesteps in the flow simulation and leads to more stable, faster mesh reconfigurations for fast moving particles in the flow. Additionally, this demonstrates how integrators of the same order zero can nevertheless have different convergence speeds towards equilibrium

Hierarchical Point Matching Method Based on Triangulation Constraint and Propagation

Reliable image matching is the basis of image-based three-dimensional (3D) reconstruction. This study presents a quasi-dense matching method based on triangulation constraint and propagation as applied to different types of close-range image matching, such as illumination change, large viewpoint, and scale change. The method begins from a set of sparse matched points that are used to construct an initial Delaunay triangulation. Edge-to-edge matching propagation is then conducted for the point matching. Two types of matching primitives from the edges of triangles with areas larger than a given threshold in the reference image, that is, the midpoints of edges and the intersections between the edges and extracted line segments, are used for the matching. A hierarchical matching strategy is adopted for the above-mentioned primitive matching. The points that cannot be matched in the first stage, specifically those that failed in a gradient orientation descriptor similarity constraint, are further matched in the second stage. The second stage combines the descriptor and the Mahalanobis distance constraints, and the optimal matching subpixel is determined according to an overall similarity score defined for the multiple constraints with different weights. Subsequently, the triangulation is updated using the newly matched points, and the aforementioned matching is repeated iteratively until no new matching points are generated. Twelve sets of close-range images are considered for the experiment. Results reveal that the proposed method has high robustness for different images and can obtain reliable matching results.

Circumferential confinement consequence on the magnetic properties of a punctured nanotube in the presence of an axial electric field

In this paper, we present magnetic properties of a finite graphene sheet with a triangle punctured vacancy, and its counterpart single-wall carbon nanotube as a rolled-up graphene sheet in the framework of the Hubbard model in the presence of an axial electric field, in order to form a comparison study between these two graphene samples. We have noticed that the tight-binding part of the Hamiltonian consists of two types of zero-energy states in the case of the graphene sheet, the strict zero-energy states, and the quasi zero-energy states. The first type takes part in a ferromagnetic coupling between the triangle edges and one edge of the rectangle graphene sheet, while the latter one has an antiferromagnetic alignment with the opposite edge of the rectangle graphene sheet. Involving the Coulomb interaction through Hubbard term, we have observed that the slope of the cluster edge states in nanotube is higher than the graphene sheet. Additionally, spin-depolarization happens in single-wall nanotube sooner than the graphene sheet by slightly increasing an axial electric field. Also, the graphene sheet is more robust than the single wall nanotube at low electric fields.

A 3D roaming and collision detection algorithm applicable for massive spatial data.

In this paper, a novel 3D roaming algorithm considering collision detection and interaction is proposed that adopts a triangle mesh to organize and manage massive spatial data and uses a customized bounding box intersector to rapidly obtain the potential collided triangles. The proposed algorithm can satisfy the requirements of timeliness and practicability during complicated large 3D scene collision detection. Moreover, we designed a method to calculate the collision point coordinates according to the spatial position relation and distance change between the virtual collision detection sphere and triangles, with the triangle edges and three vertices being considered. Compared to the methods that use the native intersector of OpenSceneGraph (OSG) to obtain the collision point coordinates, the calculation efficiency of the proposed method is greatly improved. Usually, when there is a big split/pit in the scene, the viewpoints will fly off the scene due to the fall of the collision detection sphere, or the region interior cannot be accessed when the entrance of some local region (e.g., internal grotto) of the scene is too small. These problems are solved in this paper through 3D scene-path training and by self-adaptively adjusting the radius of the virtual collision detection sphere. The proposed 3D roaming and collision detection method applicable for massive spatial data overcomes the limitation that the existing roaming and collision detection methods are only applicable to 3D scenes with a small amount of data and simple models. It provides technical supports for freewill browsing and roaming of indoor/outdoor and overground/underground of the 3D scene in cases of massive spatial data.

Enamel microstructure and morphometric discrimination of sympatric species of Microtus (Rodentia)

Prairie (Microtus ochrogaster) and woodland (Microtus pinetorum) voles, which exhibit distinctly different ecological preferences (grassland versus forest), commonly co-occur in paleontological deposits in eastern North America. Despite their ecological differences, their molar morphology is similar. Assuming that those ecologic differences occurred in the past, differentiation of these two taxa is important for paleoenvironmental reconstruction. A sample of 51 lower first molars from living populations were viewed via scanning electron microscope to qualify and quantify schmelzmuster (enamel microstructure) to species-specific standards applicable to the fossil record. The most obvious differences between schmelzmuster of the two taxa are the relatively thicker bands of radial enamel on the leading edges of triangles of M. ochrogaster, as well as the consistent retention of tangential or primitive tangential enamel on the trailing edges and posterior enamel band of the posterior loop of M. pinetorum. Discriminant analysis of landmark data from the same 51 specimens established morphological boundaries for these taxa and successfully separated the recent m1s of M. ochrogaster from those of M. pinetorum.To test identification confidence from previous work, and to add an independent means of identification for future work (at any site), both techniques were applied to a sample of three-closed triangle (“M. ochrogaster” type) m1s from the late Pleistocene Wapsipinicon Local Fauna of Jones County, Iowa. Identifications of the specimens from the Wapsipinicon l.f. based on schmelzmuster and morphometric analysis are consistent with those reached using traditional morphology. Such methods serve as independent “check” of traditional (qualitative) identification, highlight new species-level characters, and quantify previously described features, for discrimination of these taxa.

A greedy region growing algorithm for anisotropic stretch adaptive triangulation of geometry images

The geometry image representation is a remeshing of an irregular triangle surface mesh onto a rectangular grid of points and facilitates the processing of surface geometry. Its connectivity is commonly generated by splitting each rectangular quad of grid points along the shorter diagonal. However, this simple approach typically yields skinny triangles on the surface when the underlying local surface metric tensor eigenvalues differ significantly. Sequences of such triangles can appear like jaggedness.This paper presents a greedy, region growing algorithm for anisotropic triangulation of geometry images obtained by geometric stretch parametrization. The algorithm compensates for the local stretch anisotropy and variations in the principal directions of the metric tensor by minimizing the total length of the new edges of triangles added to the grown region. The surface reconstructed is more faithful to the original surface than the ones reconstructed by quad-splitting connectivity and recognized triangulation approaches for point clouds.

Elliptic Billiards and Ellipses Associated to the 3-Periodic Orbits

The centers of circumscribed and inscribed circles to the triangles that are the 3-periodic orbits of an elliptic billiard are ellipses. In this article, we obtain the canonical equations of these ellipses. Moreover, we complement the previous results about incenters obtained by Romaskevich. Also the geometric locus defined by barycenters of the triangles of an elliptic billiard are ellipses as established by Schwartz and Tabachnikov and explicit equations of these ellipses are also given. In addition, we obtain that the geometric locus defined by the barycenters of the edges of billiard triangles is an ellipse and the canonical equation is obtained.

Removing Depth-Order Cycles Among Triangles: An Algorithm Generating Triangular Fragments

More than 25 years ago, inspired by applications in computer graphics, Chazelle et al. studied the following question: is it possible to cut any set of n lines or other objects in {mathbb R}^3 into a subquadratic number of fragments such that the resulting fragments admit a depth order? They managed to prove an O(n^{9/5}) bound on the number of fragments, but only for the very special case of bipartite weavings of lines. Since then only little progress was made, until a breakthrough in 2016 by Aronov and Sharir, who showed that O(n^{3/2}{{,mathrm{polylog},}}n) fragments suffice for any set of lines. In a follow-up paper Aronov et al. proved an O(n^{3/2+varepsilon }) bound for triangles, but their method uses high- (albeit constant-) degree algebraic arcs to perform the cuts. Hence, the resulting pieces have curved boundaries. Thus the following natural version of the problem is still wide open: is it possible to cut any collection of n disjoint triangles in {mathbb R}^3 into a subquadratic number of triangular fragments that admit a depth order? And if so, can we compute the cuts efficiently? We answer this question by presenting an algorithm that cuts any set of n disjoint triangles in {mathbb R}^3 into O(n^{7/4}{{,mathrm{polylog},}}n) triangular fragments that admit a depth order. The running time of our algorithm is O(n^{3.69}). We also prove a refined bound that depends on the number, K, of intersections between the projections of the triangle edges onto the xy-plane: we show that O(n^{1+varepsilon } + n^{1/4} K^{3/4}{{,mathrm{polylog},}}n) fragments suffice to obtain a depth order. This result extends to xy-monotone surface patches bounded by a constant number of bounded-degree algebraic arcs in general position, constituting the first subquadratic bound for surface patches.