Point Group Research Articles

Topological interfaces crossed by defects and textures of continuous and discrete point group symmetries in spin-2 Bose-Einstein condensates

Topological interfaces crossed by defects and textures of continuous and discrete point group symmetries in spin-2 Bose-Einstein condensates

The effect of multiple triage points on the outcomes (time and accuracy) of hospital triage during mass casualty incidents

IntroductionDuring mass casualty incidents (MCIs), the accuracy and timing of the triage of patients by the emergency department (ED) triage officers are essential. The primary triage is performed at the event's location by paramedics and intends a quick evaluation of the victims. Secondary triage may be used when the transfer of the victim is delayed. In this study, we aimed to investigate the effectiveness of two-point triage in a simulated environment of an MCI in the hospital setting. Materials and methodsIn this case-control study, we used an online test module to assess single triage points (Group 1, n = 41) and two triage points (Group 2, n = 40). 60 vignettes for Group 1 and 55 vignettes (5 deceased cases removed) for Group 2 were used. The assessment utilized clinical MCI scenarios in a scheduled online meeting by using the Simple Triage and Rapid Treatment (START) system. Triage time and accuracy of the triage, along with the experience, and previous training of the participants, were assessed. ResultsA total of 81 triage officers participated in this study. The participants were divided into two independent groups homogenously according to their profession and experience. Groups were comparable primarily without any statistically significant difference in terms of the profession (p = 0.101), sex (p = 0.923), and MCI experience (p = 0.785). The difference between the two groups was not significant with regard to having received practical or theoretical triage training (p = 0.099). The mean time of a single vignette triage was 19.2 (SD 6.5) seconds and mean percentage of correct triage score was 65.0 (SD 12.6). The participants had a statistically significantly better performance in the single-point triage group regarding the median triage time (p < 0.001) and median percentages of under triage (p = 0.001), but a worse median percentage of over triage (p < 0.001). However, there was no significant difference between the two groups in terms of the percentage of accurate triage. Emergency residents performed better in accuracy and triage time than their non-physician colleagues (p = 0.019). ConclusionsTwo-point triage does not demonstrate better outcomes in terms of accuracy and timing. Triage officers should be trained frequently with the preferred training methodology to prevent improper triage accuracy and timing. Well-defined medical disaster planning should include frequent training of the triage officers with case scenarios.

Scene Text Recognition via Dual-path Network with Shape-driven Attention Alignment

Scene text recognition (STR), one typical sequence-to-sequence problem, has drawn much attention recently in multimedia applications. To guarantee good performance, it is essential for STR to obtain aligned character-wise features from the whole-image feature maps. While most present works adopt fully data-driven attention-based alignment, such practice ignores specific character geometric information. In this article, built upon a group of learnable geometric points, we propose a novel shape-driven attention alignment method that is able to obtain character-wise features. Concretely, we first design a corner detector to generate a shape map to guide the attention alignments explicitly, where a series of points can be learned to represent character-wise features flexibly. We then propose a dual-path network with a mutual learning and cooperating strategy that successfully combines CNN with a ViT-based model, leading to further accuracy improvement. We conduct extensive experiments to evaluate the proposed method on various scene text benchmarks, including six popular regular and irregular datasets, two more challenging datasets (i.e., WordArt and OST), and three Chinese datasets. Experimental results indicate that our method can achieve superior performance with a comparable model size against many state-of-the-art models.

Eight Domains of Pediatrician Wellness: A Stakeholder Informed Model

BackgroundPhysician wellness is important to health care systems and quality patient care. There has been limited research clarifying the physician wellness construct. We aimed to develop a stakeholder-informed model of pediatrician wellness. MethodsWe performed a group concept mapping (GCM) study to create a model of pediatrician wellness. We followed the four main steps of GCM and recruited pediatricians at multiple sites and on social media. During brainstorming, pediatricians individually responded to a prompt to generate ideas describing the concept of pediatrician wellness. Second, pediatricians sorted the list of brainstormed ideas into conceptually similar groups and rated them on importance. Sorted data were analyzed to create maps showing each idea as a point, with lines around groups of points to create clusters of wellness. Mean importance scores for each cluster were calculated and compared using pattern match. ResultsPediatricians in this study identified eight clusters of wellness: 1) Experiencing belonging and support at work, 2) Alignment in my purpose, my work, and my legacy, 3) Feelings of confidence and fulfillment at work, 4) Skills and mindset for emotional well-being, 5) Harmony in personal, professional, and community life, 6) Time and resources to support holistic sense of self, 7) Work boundaries and flexibility, and 8) Organizational culture of inclusion and trust. There were no significant differences in mean cluster rating score; the highest rated cluster was Harmony in personal, professional and community life (3.62). ConclusionPediatricians identified eight domains of wellness, spanning professional and personal life, work, and individual factors.

Visual analysis of acupuncture point selection patterns and related mechanisms in acupuncture for hypertension.

Hypertension has become one of the most pathogenic diseases in the world. This paper summarizes and analyzes the acupuncture point combinations and treatment principles of acupuncture for hypertension in a systematic way by means of big data mining. The literature for this paper was obtained from CNKI, Wanfang, VIP, SinoMed and PubMed, Embase, Cochrane Library, Web of Science, and Ovid databases. Thedata were collected to obtain combinations of acupoints with strong associations through association rule analysis, complex networks for screening to obtain core acupoint nuclei, and cluster analysis to derive treatment principles. A total of 127 acupuncture prescriptions involving 66 acupoints were included in this study. Tai-chong (LR3), Qu-chi (LI11), Zu-san-li (ST36), Feng-chi (GB20), and He-gu (LI4) were the most commonly used acupoints. The large intestine meridian was the preferred meridian, and most of the extremity acupoints, especially the lower extremities, were selected clinically. The association rule reveals that Qu-chi (LI11) and Zu-san-li (ST36) are the dominant combination acupoints. 3 core association points obtained after complex network analysis, the 1st association, Bai-hui (DU20), Tai-xi (KI3), Gan-shu (BL18), Shen-shu (BL23); The 2nd association, Qu-chi (LI11), He-gu (LI4), San-yin-jiao (SP6), Zu-san-li (ST36), Feng-chi (GB20), Tai-chong (LR3); The 3rd association, Qi-hai (RN6), Guan-yuan (RN4), Zhong-wan (RN12), Zhao-hai (KI6), Tai-yang (EX-HN5), Lie-que (LU7), Yang-ling-quan (GB34), Xing-jian (LR2), Yin-ling-quan (SP9). Cluster analysis yielded the treatment principles of nourishing Yin and submerging Yang, pacifying the liver and submerging Yang, tonifying Qi and Blood, and calming the mind and restoring the pulse, improving clinical outcomes. By means of big data mining, we can provide reference for acupuncture point grouping and selection for clinical acupuncture treatment of hypertension.

Development of an umbilical cord blood transplantation–specific nonrelapse mortality risk assessment score

AbstractHigher rate of nonrelapse mortality (NRM) remains yet to be resolved in umbilical cord blood transplantation (UCBT). Considering that UCBT has some unique features compared with allogeneic hematopoietic cell transplantation from other graft sources, a UCBT-specific NRM risk assessment system is required. Thus, in this study, we sought to develop a UCBT-specific NRM Risk Assessment (CoBRA) score. Using a nationwide registry database, we retrospectively analyzed 4437 recipients who had received their first single-unit UCBT. Using the backward elimination method, we constructed the CoBRA score in a training cohort (n = 2687), which consisted of recipients age ≥55 years (score 2), hematopoietic cell transplantation–specific comorbidity index ≥3 (score 2), male recipient, graft-versus-host disease prophylaxis other than tacrolimus in combination with methotrexate, performance status (PS) 2 to 4, HLA allele mismatch ≥ 2, refined Disease Risk Index high risk, myeloablative conditioning, and CD34+ cell doses < 0.82 × 105/kg (score 1 in each). The recipients were categorized into 3 groups: low (0-4 points), intermediate (5-7 points), and high (8-11 points) groups according to the CoBRA score. In the validation cohort (n = 1750), the cumulative incidence of NRM at 2 years was 14.9%, 25.5%, and 47.1% (P < .001), and 2-year overall survival (OS) was 74.2%, 52.7%, and 26.3% (P < .001) in the low, intermediate, and high groups, respectively. In summary, the CoBRA score could predict the NRM risk as well as OS after UCBT. Further external validation will be needed to confirm the significance of the CoBRA score.

Fractional quantum ferroelectricity

For an ordinary ferroelectric, the magnitude of the spontaneous electric polarization is at least one order of magnitude smaller than that resulting from the ionic displacement of the lattice vectors, and the direction of the spontaneous electric polarization is determined by the point group of the ferroelectric. Here, we introduce a new class of ferroelectricity termed Fractional Quantum Ferroelectricity. Unlike ordinary ferroelectrics, the polarization of Fractional Quantum Ferroelectricity arises from substantial atomic displacements that are comparable to lattice constants. Applying group theory analysis, we identify 28 potential point groups that can realize Fractional Quantum Ferroelectricity, including both polar and non-polar groups. The direction of polarization in Fractional Quantum Ferroelectricity is found to always contradict with the symmetry of the “polar” phase, which violates Neumann’s principle, challenging conventional symmetry-based knowledge. Through the Fractional Quantum Ferroelectricity theory and density functional calculations, we not only explain the puzzling experimentally observed in-plane polarization of monolayer α-In2Se3, but also predict polarization in a cubic compound of AgBr. Our findings unveil a new realm of ferroelectric behavior, expanding the understanding and application of these materials beyond the limits of traditional ferroelectrics.

Geomorphology and Morphometry of the Uttar Mand River Basin, Satara District, Maharashtra, India

In the present investigation various morphometric parameters of the Uttar Mand river basin are outlined. The Survey of India Toposheet numbers 47 G/15, 47 K/3 on the scale of 1: 50,000 were used for the present study. Stream ordering method as suggested by the Strahler has been employed. According to morphometric analysis of the Uttar Mand river basin, it is a fifth-order basin that is transitioning from an early mature to an older stage of the fluvial geomorphic cycle. The drainage pattern of the basin is dendritic in nature. A particular order's mean channel segment length is more than the next lower order but less than the next higher order. The relationship between stream order and mean stream length provides a group of points that lie roughly along a curve line, indicating that the area is not subject to severe structural influence.

Quantized ferroelectricity in multivalent ion conductors with non-polar point groups.

Ferroelectricity with switchable polarizations is generally associated with small ion-displacements and occurs only in 10 specific polar point groups, which is not a necessary requirement for ion conduction where the ions can also be electrically displaced but by much longer distances. Herein, through first-principles calculations, we predict the formation of unconventional ferroelectricity based on previous experimental reports on topotactic reaction with an aliovalent cation between trigonal layers of ion conductors. In such systems, the multivalent cations are surrounded by vacant sites that can simultaneously migrate by a much larger distance compared with conventional displacive ferroelectricity, giving rise to a quantized change in polarization even if the crystal lattices do not belong to the 10 polar groups. The deviation from classical principles can be attributed to the long ion displacements in ferroelectric ion conductors during switching that can lead to the transformation between multiple equivalent symmetrical stable states, which cannot be realized by the relatively small ion displacements in current ferroelectrics. The evenly distributed vacant sites due to Coulomb repulsion do not break the insulativity of the systems, while their inhomogeneous distribution under an electric field or in ferroelectric domain walls will give rise to high electrical conductance, which may be utilized for constructing nanoscale artificial ionic synapses that enable neuromorphic computing.

A General Framework for Comparing Embedding Visualizations Across Class-Label Hierarchies.

Projecting high-dimensional vectors into two dimensions for visualization, known as embedding visualization, facilitates perceptual reasoning and interpretation. Comparing multiple embedding visualizations drives decision-making in many domains, but traditional comparison methods are limited by a reliance on direct point correspondences. This requirement precludes comparisons without point correspondences, such as two different datasets of annotated images, and fails to capture meaningful higher-level relationships among point groups. To address these shortcomings, we propose a general framework for comparing embedding visualizations based on shared class labels rather than individual points. Our approach partitions points into regions corresponding to three key class concepts-confusion, neighborhood, and relative size-to characterize intra- and inter-class relationships. Informed by a preliminary user study, we implemented our framework using perceptual neighborhood graphs to defne these regions and introduced metrics to quantify each concept. We demonstrate the generality of our framework with usage scenarios from machine learning and single-cell biology, highlighting our metrics' ability to draw insightful comparisons across label hierarchies. To assess the effectiveness of our approach, we conducted an evaluation study with fve machine learning researchers and six single-cell biologists using an interactive and scalable prototype built with Python, JavaScript, and Rust. Our metrics enable more structured comparisons through visual guidance and increased participants' confdence in their fndings.

Uniaxial-strain tuning method in study of iron-based superconductors

In the study of quantum materials, introducing pressure and strain that can change lattice parameters and symmetry is an effective experimental method for manipulating the electronic properties of the system. In measurements under hydrostatic pressure or in-plane epitaxial strain, the changes in lattice parameters will lead to significant changes in the electronic structure, thereby triggering off novel quantum phenomena and phase transitions. By comparison, the in-plane uniaxial strain, which has been widely employed in recent years, not only changes lattice parameters, but also directly destroys and controls the symmetry of the system, thereby affecting the electronic ordering state and even collective excitation of the system. This article provides a comprehensive overview of the basic concepts of uniaxial strain, the development of experimental methods, and some research progress in using these methods to regulate superconductivity and electronic nematicity in iron-based superconductors. This review contains six sections. Section 1 focuses on a genetral introduction for the uniaxial strain techque and the arrangement of this paper. Section 2 is devoted to the basic concepts and formulas related to elastic moduli and the decomposition of uniaxial strain into irreducible symmetric channels under &lt;i&gt;D&lt;/i&gt;&lt;sub&gt;4&lt;i&gt;h&lt;/i&gt;&lt;/sub&gt; point group. Section 3 gives iron-based superconductors (FeSCs) and discusses the uniaxial-pressure detwinning method and related research progress. Section 4 introduces the establishment of the elastoresistance as a probe of the nematic susceptibility and discusses the key researches in this direction. Section 5 describes the research progress of the effects of uniaxial strain on superconductivity and nematicity. In sections 4 and 5, key experimental techniques, such as elastoresistance, are discussed in detail. Section 6 extends the discussion to several types of quantum materials suitable for uniaxial-strain tuning method beyond the FeSCs. Finally, we provide a brief summary and outlook on the uniaxial strain tuning technique. Overall, this review article provides valuable resources for the beginners in the field of FeSC and those who are interested in using uniaxial strain to modulate the electronic properties of quantum materials. By summarizing recent advancements and experimental techniques, this review hopes to inspire further research and innovation in studying electronic materials under uniaxial strain.

Planes geometry verification program written in Python

In recent years, there has been significant progress in sensor and scanning technologies in the computer vision and 3D visualization field. This has led to an increased interest in processing and analysing point clouds, which are groups of 3D points collected by laser scanners, stereo cameras, and drones. The data obtained from these devices has a wide range of applications. This article focuses on using point cloud data to verify planar structural elements of buildings. The article aims to present a methodology for analysing planar point cloud objects using 3D models in IFC format as reference data. The research includes processing data in IFC format, exporting geometric information, segmenting plane objects from a point cloud obtained by laser scanning, and analysing the data obtained. We provide the necessary graphic materials and code to enhance the content of the article. We also evaluate the results and create an outline for future work.

Seeing Your Stories: Visualization for Narrative Medicine.

Importance: Narrative medicine (NM), in which patient stories play a crucial role in their diagnosis and treatment, can potentially support a more holistic approach to patient care than traditional scientific ones. However, there are some challenges in the implementation of narrative medicine, for example, differences in understanding illnesses between physicians and patients and physicians' increased workloads and overloaded schedules. This paper first presents a review to explore previous visualization research for narrative medicine to bridge the gap between visualization researchers and narrative medicine experts and explore further visualization opportunities. Highlights: The review is conducted from 2 perspectives: (a) the contexts and domains in which visualization has been explored for narrative medicine and (b) the forms and solutions applied in these studies. Four applied domains are defined, including understanding patients from narrative records, medical communication, medical conversation training in education, and psychotherapy and emotional wellness enhancement. Conclusions: A future work framework illustrates some opportunities for future research, including groups of specific directions and future points for the 4 domains and 3 technological exploration opportunities (combination of narrative and medical data visualization, task-audience-based visual storytelling, and user-centered interactive visualization). Specifically, 3 directions of future work in medical communication (asynchronous online physician-patient communication, synchronous face-to-face medical conversation, and medical knowledge dissemination) were concluded.

Стандартизація еліптичних кривих: аналіз та впровадження в криптографічні протоколи

The purpose of the article is to consider the current state of elliptic cryptography, the prerequisites for its use, as well as the requirements of modern standards related to the use of elliptic cryptography The use of elliptic curves in cryptography is considered one of the most promising areas of development of modern security algorithms. This mathematical approach is based on the complexity of solving the discrete logarithm problem in a group of points of an elliptic curve over a finite field. The use of cryptography on elliptic curves allows you to ensure the security of data exchange using effective encryption algorithms and the creation of digital signatures (DI). This study examines elliptic curves for cryptographic purposes, and provides basic operations on the point group of elliptic curves. Special attention is paid to Elliptic curve Diffie-Hellman (ECDH) and Elliptic Curve Digital Signature Algorithm (ECDSA) key exchange algorithms. The standards regulating the use of elliptic curves in cryptographic systems are also analyzed, and the advantages of this cryptographic paradigm compared to the main asymmetric algorithms are considered. Potential threats and vulnerabilities of cryptographic algorithms based on elliptic curves are investigated. Examples of popular standardized curves recommended by relevant organizations, such as NIST, used in real-world cryptographic applications are also provided. Elliptic curve cryptography (ECC) is currently one of the foundations for the development of modern public-key cryptographic algorithms. ECC has gained recognition in cryptography for providing a high level of security with shorter key lengths (compared to other cryptographic approaches), high speed, resource savings, and versatility, giving it an advantage over other methods such as RSA and others. It provides a secure network connection, generates secret keys for TLS servers and their clients, and is also used to create digital signatures that guarantee the authenticity of transactions in cryptocurrency systems.

Evidence for field induced quantum spin liquid behavior in a spin-1/2 honeycomb magnet

&lt;p&gt;One of the most important issues in modern condensed matter physics is the realization of fractionalized excitations, such as the Majorana excitations in the Kitaev quantum spin liquid. To this aim, the 3&lt;i&gt;d&lt;/i&gt;-based Kitaev material Na&lt;sub&gt;2&lt;/sub&gt;Co&lt;sub&gt;2&lt;/sub&gt;TeO&lt;sub&gt;6&lt;/sub&gt; is a promising candidate whose magnetic phase diagram of &lt;b&gt;B&lt;/b&gt; // &lt;b&gt;a*&lt;/b&gt; contains a field-induced intermediate magnetically disordered phase within 7.5 T &lt; |B| &lt; 10 T. The experimental observations, including the restoration of the crystalline point group symmetry in the angle-dependent torque and the coexisting magnon excitations and spinon-continuum in the inelastic neutron scattering spectrum, provide strong evidence that this disordered phase is a field induced quantum spin liquid with partially polarized spins. Our variational Monte Carlo simulation with the effective K-J&lt;sub&gt;1&lt;/sub&gt;-Γ-Γ'-J&lt;sub&gt;3&lt;/sub&gt; model reproduces the experimental data and further supports this conclusion.&lt;/p&gt;

Systematic analysis of electronic barrier heights and widths for concerted proton transfer in cyclic hydrogen bonded clusters: (HF)n, (HCl)n and (H2O)n where n = 3, 4, 5.

The MP2 and CCSD(T) methods are paired with correlation consistent basis sets as large as aug-cc-pVQZ to optimize the structures of the cyclic minima for (HF)n, (HCl)n and (H2O)n where n = 3-5, as well as the corresponding transition states (TSs) for concerted proton transfer (CPT). MP2 and CCSD(T) harmonic vibrational frequencies confirm the nature of each minimum and TS. Both conventional and explicitly correlated CCSD(T) computations are employed to assess the electronic dissociation energies and barrier heights for CPT near the complete basis (CBS) limit for all 9 clusters. Results for (HF)n are consistent with prior studies identifying Cnh and Dnh point group symmetry for the minima and TSs, respectively. Our computations also confirm that CPT proceeds through Cs TS structures for the C1 minima of (H2O)3 and (H2O)5, whereas the process goes through a TS with D2d symmetry for the S4 global minimum of (H2O)4. This work corroborates earlier findings that the minima for (HCl)3, (HCl)4 and (HCl)5 have C3h, S4 and C1 point group symmetry, respectively, and that the Cnh structures are not minima for n = 4 and 5. Moreover, our computations show the TSs for CPT in (HCl)3, (HCl)4 and (HCl)5 have D3h, D2d, and C2 point group symmetry, respectively. At the CCSD(T) CBS limit, (HF)4 and (HF)5 have the smallest electronic barrier heights for CPT (≈15 kcal mol-1 for both), followed by the HF trimer (≈21 kcal mol-1). The barriers are appreciably higher for the other clusters (around 27 kcal mol-1 for (H2O)4 and (HCl)3; roughly 30 kcal mol-1 for (H2O)3, (H2O)5 and (HCl)4; up to 38 kcal mol-1 for (HCl)5). At the CBS limit, MP2 significantly underestimates the CCSD(T) barrier heights (e.g., by ca. 2, 4 and 7 kcal mol-1 for the pentamers of HF, H2O and HCl, respectively), whereas CCSD overestimates these barriers by roughly the same magnitude. Scaling the barrier heights and dissociation energies by the number of fragments in the cluster reveals strong linear relationships between the two quantities and with the magnitudes of the imaginary vibrational frequency for the TSs.

Symmetry breaking through molecular engineering to achieve "on-off-on" nonlinear optical switch in a [SnCl6]2- framework.

The impact of molecular point groups on phase transition materials has been seldom discussed. Two organic cations with different symmetries were incorporated into a [SnCl6]2- framework. By breaking the symmetry of cations, a novel hybrid material with two successive phase transitions featuring a rare "on-off-on" nonlinear optical switch was obtained.

Permutation symmetry in spin-adapted many-body wave functions.

In the domain of exchange-coupled polynuclear transition-metal (PNTM) clusters, local emergent symmetries exist which can be exploited to greatly increase the sparsity of the configuration interaction (CI) eigensolutions of such systems. Sparsity of the CI secular problem is revealed by exploring the site permutation space within spin-adapted many-body bases, and highly compressed wave functions may arise by finding optimal site orderings. However, the factorial cost of searching through the permutation space remains a bottleneck for clusters with a large number of metal centers. In this work, we explore ways to reduce the factorial scaling, by combining permutation and point group symmetry arguments, and using commutation relations between cumulative partial spin and the Hamiltonian operators, . Certain site orderings lead to commuting operators, from which more sparse wave functions arise. Two graphical strategies will be discussed, one to rapidly evaluate the commutators of interest, and one in the form of a tree search algorithm to predict how many and which distinct site permutations are to be analyzed, eliminating redundancies in the permutation space. Particularly interesting is the case of the singlet spin states for which an additional reversal symmetry can be utilized to further reduce the number of distinct site permutations.

Design and synthesis of a potential candidate molecule designated as Diosgenin-Aceclofenac ester as an anticancer agent on SiHa cell line, it's spectroscopic, DFT and molecular docking studies

In the present study a new steroidal derivative i.e. Diosgenin-Aceclofenac conjugate was synthesized using Steglich esterification. The ester derivative obtained was purified through column chromatography and its structure was confirmed using various spectroscopic techniques like 1H NMR, 13C NMR, IR and mass spectrometry. Computational data was obtained using Gaussian 09 program using B3LYP function and 6-31G (d, p) basic set and the molecular structure was visualized with the help of Gauss View. The energy for the optimized compound 3 was found to be -3096.35 au which possessed a C1 point group symmetry. The pymol and discovery studio softwares were used for Molecular Docking. The results show increase in first static hyperpolarizability and activity with incorporated acceptors being responsible for structural variation. The synthesized compound was evaluated for anti-cancer activity on SiHa cell line which demonstrated an appreciable activity as indicated by the IC50 values. The computational analysis data and experimental data were in conformation with each other.

Stereographic Projection of Theoretical Orientation Relationships Between Crystals of Any Types in Phase Transformation and Precipitation

AbstractStereographic projection is a classic technology to represent the angular relationships of lattice planes and directions. In some cases, it is necessary to project arbitrary lattice planes or directions of a crystal onto an arbitrary lattice plane of another crystal, e.g., when representing orientation relationships (ORs) in phase transformation or precipitation. Commercial EBSD (electron backscatter diffraction) software cannot illustrate theoretical orientation relationships based purely on the parallelism conditions. This work presents a generalized and unified formulation for stereographic projection of any lattice planes or directions onto any lattice planes of any crystals in the 32 point groups, which is utilized to represent the theoretical orientation relationships between any crystal types. Examples are given to illustrate the correlations of common orientation relationships in both cubic and hexagonal crystals. A procedure is also established to color code and simultaneously project all low‐index Miller/Bravais planes or directions of a crystal onto an arbitrary lattice plane of the same crystal, which can be utilized to visualize the crystal symmetry and determine the standard stereographic triangle (SST). All these illustrations are realized in a computer program written in C++ using the OpenGL graphic libraries.