Geometric Principles Research Articles

Not a place for respectable people, but the ends of the earth converge there: insights from Wisad Pools into the nature and context of Jordan’s Black Desert Neolithic

New data, recovered by the Eastern Badia Archaeological Project, from substantial structures at Wisad Pools have provided insights into the nature and context of eastern Jordan’s Black Desert Neolithic cultural complex, dating to between c. 7000 and 5000 cal. BC. Hunter-herder-trapper communities, with economies based partially on ante-mortem exploitation of domestic sheep, were adept at maintaining a balance between local and non-local adaptations in shaping a networked culture that was deeply rooted in its local environment. Evidence for trans-regional connections with the Levant and Mesopotamia raises the possibility that longer-term habitation sites, like Wisad Pools, were instrumental in linking the western and eastern arms of the Fertile Crescent via direct routes across the steppe. At least one structure excavated by the Project displays evidence for sophisticated architectural planning, demonstrating awareness of basic geometric principles. New radiometric assays suggest that adaptive processes were profoundly influenced by episodes of climatic and environmental change, including the longer 8.2 ka event. Emerging data from north-west Saudi Arabia require Wisad Pools to be examined in the context of Arabian Neolithization, considering the possibility that the Black Desert Neolithic was a point of departure for some aspects of that process.

Multi-channel vortex beam based on single layer transmission-type metal metasurface

We design a transmission type flexible metasurface which operates at 0.1 THz. The metasurface consists of a polyimide substrate and a metallic copper film. Using the geometric phase principle, four vortex beams with different channels and different topological charges are generated under the linearly polarized beam incident. Through three-dimensional electromagnetic field simulation and experimental tests, it is proved that the designed metasurface can generate multi-channel vortex beams, where the topological charges of the vortex beam are ± 1 and ± 2. The metasurface has high transmittance in the frequency band of 0.092THz to 0.106THz, 0.202THz to 0.283THz, 0.318THz to 0.345THz, and so on, which can generate multi-channel vortex beams at multiple frequency bands. Metasurfaces designed based on this method can be applied in optical communication.

The geometry uniformity parameters effect on the dynamic behaviour of trapezoidal beam

Understanding the delicate interplay between geometric design principles and dynamic responses is vital for numerous engineering applications because it allows optimization of structural integrity and performance for manufacturing and special use of this structures. In this study, a complete numerical analysis is undertaken to investigate how differences in geometric parameters impact the structural behaviour of a cantilever smart trapezoidal beam, giving useful insights for engineering applications. The beam consists of a host structure made of aluminum couple with two piezoelectric layers on its top and bottom surface. For such purpose, the study a 3D finite element model has been implemented in ANSY APDL and applied by Matlab we used the interfaces between the two software, considering that the beam changes its form in the axial direction following polynomial based function. The function takes two geometric parameters as an input which are the tapering ratios and the degree of non-uniformity. The beam is then subjected to a harmonic based excitation and the effect of changing tapering ratios and the degree of non-uniformity is analyzed., allowing analysis of the effects of altering tapering ratios and the degree of non-uniformity The results led to a conclusion that just by manipulating the mentioned parameters a considerable changing in the fundamental frequency and the amplitude has been noticed. Addition our numerical analysis shows that small changes to the tapering ratios and degree of non-uniformity have a considerable influence on the behavior of cantilever smart trapezoidal beams. These discoveries add to continuing research in smart materials, paving the road for creative engineering solutions.

Permissible domain walls in monoclinic ferroelectrics. Part II. The case of MC phases.

Monoclinic ferroelectric phases are prevalent in various functional materials, most notably mixed-ion perovskite oxides. These phases can manifest as regularly ordered long-range crystallographic structures or as macroscopic averages of the self-assembled tetragonal/rhombohedral nanodomains. The structural and physical properties of monoclinic ferroelectric phases play a pivotal role when exploring the interplay between ferroelectricity, ferroelasticity, giant piezoelectricity and multiferroicity in crystals, ceramics and epitaxial thin films. However, the complex nature of this subject presents challenges, particularly in deciphering the microstructures of monoclinic domains. In Paper I [Biran & Gorfman (2024). Acta Cryst. A80, 112-128] the geometrical principles governing the connection of domain microstructures formed by pairing MAB type monoclinic domains were elucidated. Specifically, a catalog was established of `permissible domain walls', where `permissible', as originally introduced by Fousek & Janovec [J. Appl. Phys. (1969), 40, 135-142], denotes a mismatch-free connection between two monoclinic domains along the corresponding domain wall. The present article continues the prior work by elaborating on the formalisms of permissible domain walls to describe domain microstructures formed by pairing the MC type monoclinic domains. Similarly to Paper I, 84 permissible domain walls are presented for MC type domains. Each permissible domain wall is characterized by Miller indices, the transformation matrix between the crystallographic basis vectors of the domains and, crucially, the expected separation of Bragg peaks diffracted from the matched pair of domains. All these parameters are provided in an analytical form for easy and intuitive interpretation of the results. Additionally, 2D illustrations are provided for selected instances of permissible domain walls. The findings can prove valuable for various domain-related calculations, investigations involving X-ray diffraction for domain analysis and the description of domain-related physical properties.

Angle Trisection with Growth Rate of The Golden Ratio

In this paper, we explore the triangulation of angles using the golden ratio and geometric compass principles. By dividing angles into three equal parts based on the golden ratio, we demonstrate a method for solving this geometric problem. Furthermore, we extend our exploration into the third dimension, where a geometric compass can effectively divide a two-dimensional angle into three equal parts. Additionally, our investigation delves into the realm of six-dimensional space-time, where we observe the simultaneous movement of two arms of the compasses based on the golden ratio to achieve the division of every angle into three equal parts. We emphasize the significance of the growth rate associated with the golden ratio in resolving complex mathematical and physical problems. This study provides valuable insights into the application of geometric principles and the golden ratio in solving challenging problems within mathematics and physics.

Enhancing images for vision measurement in hazy tunnel construction

To address the negative impacts of image degradation on vision measurement in hazy tunnel construction, this paper presents a unified enhancement method of color and grayscale images in tunnel haze environments. Firstly, a tunnel haze environment light scattering model is introduced utilizing geometric optics principles and the conventional atmospheric scattering model. Secondly, a model for perceiving image haze is established within the image feature space to estimate ambient light and transmission. The efficacy of the method is assessed by constructing a tunnel haze simulation chamber and a platform for measurement of tunnel boring machine (TBM) cutter wear. Comparative experiments demonstrate the method reduces the error in 3D point cloud reconstruction by 77% and improves the measurement precision of TBM cutter wear by 62%. The results demonstrate the potential application of this method in automated tunnel construction. However, further research is needed to enhance its efficacy in dense haze images.

The relationship between ball mass and throw distance: Implications for coaching practice

In training for many sports, balls of different masses are thrown. This study examined how ball mass influences distance thrown. Based on geometric scaling principles a relationship was derived between ball mass and throw distance: throw distance was inversely proportional to ball mass. This relationship was tested in two ways. The first approach was using athlete shot put throw distances with different shot masses, and hammer throws performed with different mass hammers. These experimental data were better represented by the non-linear relationship, derived from geometric scaling principles, than assuming a simple linear relationship. The second approach was to use a simulation model to predict throw distance for balls of different masses. The model contained realistic segmental inertial properties, activation dynamics, moment-joint angle properties, and moment-joint angular velocity properties. The non-linear relationship provided a better fit to the results of the simulations compared with a linear model. The experimental and modeling results support the theoretical relationship derived between distance thrown and ball mass. The relationship has utility for coaches monitoring and planning the training of athletes which incorporates throwing different mass balls.

Multifunctional beam modulation based on gratings combined with additive metasurfaces

A design scheme combining grating and element additive metasurface is proposed in this paper. Multi-dimensional joint control of the phase, amplitude, angle and polarization and multi-functional integrated metasurface applications were realized by using the inherent characteristics of the light wave properties (polarization and wavelength). Firstly, suitable structural parameters are obtained based on the reflection response of the unit structure, and a 2-bit encoded metasurface array is constructed by utilizing the combinatorial operation between the unit structure. Combined with the polarization selectivity of the lower grating, the dual-function integration of asymmetric transmission (the contrast ratio is 98%) and abnormal reflections (operating bandwidth is 300 nm) was realized. Next, two new half-wave plates were obtained by addition and rotating the elemental structure, and the polarization conversion efficiency of 95%–97% was achieved thanks to the Fabry–Perot cavity formed by the upper metasurface and lower grating. Finally, encoded metasurface based on the geometric phase principle to achieve 300 nm bandwidth cross-polarized wave deflection characteristics with polarization conversion efficiency close to 100%. Our work combines grating structures with element additive metasurfaces, which solves the problem that element structure needs to be redesigned with the change of function, and also paves the way for the application of multifunctional integrated metasurfaces in optical sensing, optical imaging and other fields.

Angle Trisection with Growth Rate of The Golden Ratio

In this paper, we explore the triangulation of angles using the golden ratio and geometric compass principles. By dividing angles into three equal parts based on the golden ratio, we demonstrate a method for solving this geometric problem. Furthermore, we extend our exploration into the third dimension, where a geometric compass can effectively divide a two-dimensional angle into three equal parts. Additionally, our investigation delves into the realm of six-dimensional space-time, where we observe the simultaneous movement of two arms of the compasses based on the golden ratio to achieve the division of every angle into three equal parts. We emphasize the significance of the growth rate associated with the golden ratio in resolving complex mathematical and physical problems. This study provides valuable insights into the application of geometric principles and the golden ratio in solving challenging problems within mathematics and physics.

Metal-Centered Boron-Wheel Cluster of Y©B112- with Rare D11h Symmetry.

Molecules with high point-group symmetry are interesting prototype species in the textbook. As transition metal-centered boron clusters tend to have highly symmetric structures to fulfill multicenter bonding and high stability, new boron clusters with rare point-group symmetry may be viable. Through in-depth scrutiny over the structures of experimentally already observed transition metal-centered boron-wheel complexes, geometric and electronic design principles are summarized, based on which we studied M©B11k- (M = Y, La; Zr, Hf; k = 1, 2) clusters and found that a Y©B112- boron-wheel complex has an unprecedented D11h point-group symmetry. The remarkable stability of the planar Y©B112- complex is illustrated via extensive global-minimum structural search as well as comprehensive chemical bonding analyses. Similar to other boron-wheel complexes, the Y©B112- complex is shown to possess σ and π double aromaticity, indeed following the electronic design principle previously summarized. This new compound is expected to be experimentally identified, which will extend the currently known largest possible planar molecular symmetry and enrich the metal-centered boron-wheel class.

Marking the nipple-areola complex in mastopexy and reduction mammoplasty: a geometrical method.

Mastopexy and reduction mammoplasty are well-established procedures used to address ptotic and hypertrophic breasts. A variety of methods have been described in the literature for managing the three main aspects of these procedures: skin excess, parenchyma, and the nipple-areola complex (NAC). Among these techniques, several have specifically addressed the markings of the NAC in procedures involving a preoperatively established NAC position. While many of these have proven effective over time, the literature lacks standardized and reproducible methods. In this paper, we introduce, step-by-step, a novel approach based on geometrical principles to address this limitation. This method involves the simple process of drawing two hemi circumferences on the vertical limbs of the skin resection pattern, which are commonly marked using the Bisemberg maneuver or similar techniques. The method we introduce is adaptable to virtually all NAC pedicles. Furthermore, as exemplified in the accompanying video, it can be incorporated to most skin resection patterns with a preoperatively defined NAC position. The clinical application, as evidenced in the case reported, appears to be promising. Although it has some limitations, this procedure shows the potential to reduce inaccuracy by minimizing variability factors. As a result, even inexperienced surgeons can achieve precise markings.

Tunable VO2 metasurface for reflective terahertz linear and circular polarization wavefront manipulation at two frequencies independently

In this work, a tunable VO2 metasurface (MS) is proposed and numerically demonstrated in the terahertz (THz) region, which can independently manipulate reflected linearly polarized (LP) and circularly polarized (CP) wavefronts at two frequencies independently. The unit-cell of the proposed VO2 MS consists of a metallic circular ring (CR) structure, rectangular-shaped-patch (RSP) and circular-double-split-ring (CDSR) structure VO2 adhered on a dielectric substrate by a bottom metallic ground plane. The proposed MS structure enables the reflected cross-polarization conversion of LP waves at lower frequencies (fL = 1.34 THz) and CP waves at higher frequencies (fH = 2.53 THz), respectively. By changing the external temperature, the conductivity of the VO2 RSP and CDSR structures can be adjusted dynamically. By varying the conductivity of the VO2, the proposed MS can achieve a reflected LP conversion with efficiency from 0% to 98.86% at lower frequency of 1.34 THz and the reflected CP conversion with efficiency from 0% to 81.86% at higher frequency of 2.53 THz. Moreover, the corresponding modulation depths of this MS are as high as 98.86% and 81.86% at two different frequencies, respectively. Moreover, a 2π phase shift can be realized by the transmission phase and geometric phase principles at lower and higher frequencies, respectively. Finally, two VO2 MSs have been constructed to achieve the reflection deflection for both CP and LP waves, and reflective CP focusing and LP vortex beam generation, respectively. These findings are favorable for expanding the range of temperature-controlled VO2 MS for wavefront manipulation, thereby opening up new possibilities for the development of advanced, tunable, and multifunctional THz devices that can handle both LP and CP waves.

Classical and quantum solutions in scalar field cosmology via the Eisenhart lift and linearization

This study introduces a novel approach for solving the cosmological field equations within scalar field theory by employing the Eisenhart lift. The field equations are reformulated as a system of geodesic equations for the Eisenhart metric. In the case of an exponential potential, the Eisenhart metric is shown to be conformally flat. By applying basic geometric principles, a new set of dynamical variables is identified, allowing for the linearization of the field equations and the derivation of classical cosmological solutions. However, the quantization of the Eisenhart system reveals a distinct set of solutions for the wavefunction, particularly in the presence of symmetry breaking at the quantum level.

Blueprinting extendable nanomaterials with standardized protein blocks

A wooden house frame consists of many different lumber pieces, but because of the regularity of these building blocks, the structure can be designed using straightforward geometrical principles. The design of multicomponent protein assemblies, in comparison, has been much more complex, largely owing to the irregular shapes of protein structures1. Here we describe extendable linear, curved and angled protein building blocks, as well as inter-block interactions, that conform to specified geometric standards; assemblies designed using these blocks inherit their extendability and regular interaction surfaces, enabling them to be expanded or contracted by varying the number of modules, and reinforced with secondary struts. Using X-ray crystallography and electron microscopy, we validate nanomaterial designs ranging from simple polygonal and circular oligomers that can be concentrically nested, up to large polyhedral nanocages and unbounded straight ‘train track’ assemblies with reconfigurable sizes and geometries that can be readily blueprinted. Because of the complexity of protein structures and sequence–structure relationships, it has not previously been possible to build up large protein assemblies by deliberate placement of protein backbones onto a blank three-dimensional canvas; the simplicity and geometric regularity of our design platform now enables construction of protein nanomaterials according to ‘back of an envelope’ architectural blueprints.

A magnesium screw with optimized geometry exhibits improved corrosion resistance and favors bone fracture healing

Stress-induced corrosion impairs the mechanical integrity of magnesium (Mg) and its alloys as potential orthopedic implants. Although there has been extensive work reporting the effects of stress on Mg corrosion in vitro, the geometric design principles of the Mg-based orthopedic devices still remain largely unknown. In this work, a numerical simulation model mimicking fractured bone fixation and surgical animal models were applied to investigate the effects of the geometric design of Mg screws on the stress distribution and the stress-induced degradation behavior. Finite element (FE) analysis was used for calculation of stress concentrations around the Mg screws, with different thread type, thread pitch, and thread width. Afterward, the Mg screws of the pre-optimization and post-optimization groups exhibiting the highest and lowest stress concentrations, respectively, were implanted in the fractured distal femora and back subcutaneous tissue of rabbits. Encouragingly, there was a significant difference between the pre-optimization and the post-optimization groups in the degradation rate of the stressed screw parts located around the fracture line. Interestingly, there was no significant difference between the two groups in the degradation rate of the non-stressed screw parts. Consistently, the Mg screw post-optimization exhibited a significantly lower degradation rate than that pre-optimization in the back subcutaneous implantation model, which generated stress in the whole screw body. The alteration in geometric design did not affect the corrosion rate of the Mg screws in an immersion test without load applied. Importantly, an accelerated new bone formation with less fibrous encapsulation around the screws was observed in the Mg group post-optimization relative to the Mg group pre-optimization and the poly (lactic acid) group. Geometry optimization may be a promising strategy to reduce stress-induced corrosion in Mg-based orthopedic devices. Statement of significanceStress concentrations influence corrosion characteristics of magnesium (Mg)-based implants. The geometric design parameters, including thread type, thread pitch, and thread width of the Mg screws, were optimized through finite element analysis to reduce stress concentrations in a fractured model. The Mg screws with triangular thread type, 2.25 mm pitch, and 0.3 mm thread width, exhibiting the lowest maximum von Mises stress, showed a significant decrease in the volume loss relative to the Mg screws pre-optimization. Compared with the Mg screw pre-optimization and the poly(lactic acid) screw, the Mg screw post-optimization favored new bone formation while inhibiting fibrous encapsulation. Collectively, optimization in the geometric design is a promising approach to reduce stress-induced corrosion in Mg-based implants.

(036) The Comparison of an Efficacy and Safety of Geometrical Principles Based Corporoplasty Combined with Intraoperative Intracavernosal Prostaglandin E1 Injections and the Non-Geometrical Surgical Techniques for Peyronie's Curvature Correction

Abstract Introduction There is a lack of research evidence and properly conducted comparative long-term studies in regards to whether the intra and post-operative safety and efficacy of geometrical principle of plaque incision and grafting technique, especially if combined with an intraoperative intracavernosal prostaglandin E1 (PGE1) injection instead of saline, is better than the non-geometrical techniques. Objective The aim of this retrospective study was to present our 8-year single-center experience of surgical management of Peyronie's disease in the means of efficacy and safety by comparing geometrical principle of plaque incision and grafting technique combined with the intraoperative intracavernosal PGE1 injection with the other non-geometrical surgical techniques used at our centre for the treatment of penile curvature. Methods Our andrology centre approved this retrospective study and commenced an audit in order to obtain anonymised data. In total 70 patients underwent non-geometrical surgical techniques based penile curvature correction between 2014 and 2017. All the surgeries were carried out performing the intraoperative intracavernosal PGE1 injection instead of obtaining saline-induced erection. The demographic data was retrieved and a comparative analysis was performed. Results In total 216 (N=216) patients, aged 42–73 years old (average age 59 years) suffering from Peyronie's disease had surgical treatment of the curvature at our centre from 2014 to 2023. Group 1 (G1) patients accounted for 32.41% (N=70) of total patients - they were 46–78 years old (average age 64 years) and underwent non-geometrical principals based surgery for Peyronie's disease in between 2014 and 2017. Group 2 (G2) patients accounted for 68.52% (N=148) of total patients - they underwent the surgery using plaque's incision and grafting technique based on geometrical principles in between 2018 and 2023. 31 patients (44.3%) out of G1 required additional curvature corrections after an initial surgery, but only 3 patients (2.0%) out of G2 required that. Preoperative dorsal penile curvature degree among G1 was 50° - 100° (the mean degree 70°) and 60° - 110° (the mean degree 78°) among G2. 11 patients (15.7%) out of G1 and 15 patients (10.1%) out of G2 had the combined dorsolateral curvature. 3 patients (4.3%) out of G1 and 6 patients (4.1%) out of G2 had ventral curvature. 9 patients (12.9%) out G1 and 2 patients (1.4%) out of G2 have reported a subjective statement of their penile shaft appearing shorter after the curvature correcting surgery. 2 patients (2.9%) out G1 and 1 patient (0.7%) out of G2 have reported a subjective statement of erectile dysfunction post the surgery. There were no intraoperative or postoperative complications associated with the intracavernosal PGE1 injection. Conclusions In summary, we found that plaque's incision and grafting technique based on geometrical principles was advantageous against the other non-geometrical penile curvature surgical techniques due to better efficiency achieved by the higher initial success rates that require an additional correction almost twice less often. We also revealed that the intracavernosal PGE1 injection was a safe and sufficient method for inducing erection intraoperatively, that had not been associated with any complications and maintained the good outcomes of the surgical Peyornie's disease treatment. Disclosure No.

Digital measuring the ocular morphological parameters of guinea pig eye in vivo with Python.

To quantitatively measure ocular morphological parameters of guinea pig with Python technology. Thirty-six eyeballs of eighteen 3-week-old guinea pigs were measured with keratometer and photographed to obtain the horizontal, coronal, and sagittal planes respectively. The corresponding photo pixels-actual length ratio was acquired by a proportional scale. The edge coordinates were identified artificially by ginput function. Circle and conic curve fitting were applied to fit the contour of the eyeball in the sagittal, coronal and horizontal view. The curvature, curvature radius, eccentricity, tilt angle, corneal diameter, and binocular separation angle were calculated according to the geometric principles. Next, the eyeballs were removed, canny edge detection was applied to identify the contour of eyeball in vitro. The results were compared between in vivo and in vitro. Regarding the corneal curvature and curvature radius on the horizontal and sagittal planes, no significant differences were observed among results in vivo, in vitro, and the keratometer. The horizontal and vertical binocular separation angles were 130.6°±6.39° and 129.8°±9.58° respectively. For the corneal curvature radius and eccentricity in vivo, significant differences were observed between horizontal and vertical planes. The Graphical interface window of Python makes up the deficiency of edge detection, which requires too much definition in Matlab. There are significant differences between guinea pig and human beings, such as exotropic eye position, oblique oval eyeball, and obvious discrepancy of binoculus. This study helps evaluate objectively the ocular morphological parameters of small experimental animals in emmetropization research.

Geometrical Structure of Bifurcations during Spatial Decision-Making

Animals must constantly make decisions on the move, such as when choosing among multiple options, or “targets,” in space. Recent evidence suggests that this results from a recursive feedback between the (vectorial) neural representation of the targets and the resulting motion defined by this consensus, which then changes the egocentric neural representation of the options, and so on. Here we employ a simple model of this process both to explore how its dynamics accounts for the experimentally observed abruptly branching trajectories exhibited by animals during spatial decision-making, and to provide new insights into spatiotemporal computation. Essential neural dynamics, notably local excitation and long-range inhibition, are captured in our model via spin-system dynamics, with groups of Ising-spins representing neural “activity bumps” corresponding to target directions (as in a neural ring-attractor network, for example). Analysis, employing a novel “mean-field trajectory” approach, reveals the nature of the spontaneous symmetry breaking bifurcations in the model that result in literal bifurcations in trajectory space—and how it results in new geometric principles for spatiotemporal decision-making. We find that all bifurcation points, beyond the very first, fall on a small number of “bifurcation curves.” It is the spatial organization of these curves that is shown to be key to determining the shape of the trajectories, such as self-similar or space filling, exhibited during decision-making, irrespective of the trajectory's starting point. Furthermore, we find that a non-Euclidean (neural) representation of space (effectively an elliptic geometry) considerably reduces the number of bifurcation points in many geometrical configurations (including from an infinite number to only three), preventing endless indecision and promoting effective spatial decision-making. This suggests that a non-Euclidean neural representation of space may be expected to have evolved across species in order to facilitate spatial decision-making. Published by the American Physical Society 2024

The efficient data-driven solution to estimate the nonlinear bending of sandwich doubly curved panel subjected to transient loading

This paper investigates nonlinear transient bending characteristics of sandwich doubly curved panels under transient loading which is extensively used in aerospace structures. Due to the lack of a dataset for the nonlinear transient bending of the sandwich doubly curved panel for the data-driven solution, mathematical modeling is used. The sandwich doubly curved panel is made of two graphene nanoplatelets (GPLs) reinforced face sheets, and a polymer core. In the mathematical modeling section, the effective Young's modulus is obtained via modified Halpin-Tsai methodology, and the role of the mixture is used to estimate other material properties of the composite face sheets. The presented structure is analyzed using Reddy's third-order shear deformation theory (RTSDT), which incorporates von Kármán nonlinear geometric assumptions and Hamilton's principle. The governing equations and associated boundary conditions are derived accurately by imposing shear stress-free conditions on the upper and lower surfaces of the panel. The meshfree radial point interpolation method (RPIM) with interpolation functions is utilized to determine the unknown degrees of freedom to numerically simulate the mid-plane of the sandwich doubly-curved panel. The Newton-Raphson approach is used to solve the nonlinear ordinary differential using the constant-average acceleration strategy from Newmark's time integration method. After obtaining the dataset, the data is collected, and using an efficient data-driven approach, the results are tested, validated, and trained. In future studies, the presented efficient data-driven approach can be used with low computational cost to simulate nonlinear transient bending characteristics of sandwich doubly curved panels under transient loading for related aerospace industries.

A modified geometric technique to increase peri-implant keratinized mucosa.

The free gingival graft (FGG) has been identified as the most effective method for increasing keratinized mucosa width (KMW). However, the challenge emerges in cases of extensive keratinized mucosa deficiency, where efficient utilization of the patient's limited keratinized tissue to achieve optimal results is crucial. This article introduces a modified geometric technique to address this clinical issue. Utilizing geometric principles, the modified technique involves dividing the rectangular graft into two triangular or trapezoidal sections, which are then reassembled to form an approximate diamond shape. Through strategic cut and splice, the graft is reshaped to suit the recipient site. Preliminary observations in cases employing the modified geometric technique have increased the KMW around implants. This method enhances graft utilization and offers a viable clinical option for surgical plans aimed at widening keratinized mucosa in instances of large-area KMW deficiency. This article proposed a modified method to increase KMW, which may be an optimal choice for patients with insufficient KMW in large area, avoiding the waste of limited graft, decreasing patient morbidity, and effectively widening keratinized mucosa.