Symmetry Line Research Articles

Identifying Jittering Jet-shaped Ejecta in the Cygnus Loop Supernova Remnant

Analyzing images of the Cygnus Loop, a core-collapse supernova (CCSN) remnant, in different emission bands, we identify a point-symmetrical morphology composed of three symmetry axes that we attribute to shaping by three pairs of jets. The main jet axis has an elongated S shape, appearing as a faint narrow zone in visible and UV. We term it the S-shaped hose, and the structure of three symmetry lines the point-symmetric wind rose. The two other lines connect a protrusion (an ear or a bulge) with a hole on the opposite side of the center (a nozzle or a cavity), structures that we identify in the X-ray, UV, visible, IR, and/or radio images. There is a well-known blowout at the southern end of the S-shaped hose, and we identify a possible opposite blowout at the northern end of the S-shaped hose. The point-symmetrical morphology of the Cygnus Loop is according to the expectation of the jittering jets explosion mechanism (JJEM) of CCSNe, where several to few tens of pairs of jets with stochastically varying directions explode the star. The three pairs of jets that shaped the wind-rose structure of the Cygnus Loop are the last energetic pairs of this series of jets. Our study further supports the JJEM as the main explosion mechanism of CCSNe.

Regular reflection to Mach reflection (RR–MR) transition in short wedges

Regular reflection (RR) to Mach reflection (MR) transitions ( ${\rm RR}\leftrightarrow {\rm MR}$ ) on long wedges in steady supersonic flows have been well studied and documented. However, in a short wedge where the wedge length is small, the transition prediction becomes really challenging owing to the interaction of the expansion fan emanating from the trailing edge of the wedge with the incident shock and the triple/reflection point. The extent of this interaction depends on the distance between the wedge trailing edge and the symmetry line (Ht). This distance is a geometric combination of the distance of the wedge leading edge from the symmetry line $(H)$ , the wedge angle ( $\theta$ ) and the wedge length $(w)$ . In the present study, we used the method of characteristics to model the complete wave interactions which accurately predicted shock curvatures and the reflection configurations for all ranges of the incoming flow Mach number. In the case of short wedges, the transition criterion strongly depends on the wedge length, which can be so adjusted even to eliminate the ${\rm RR}\rightarrow {\rm MR}$ transitions till the wedge angle reaches the no-reflection domain. Transition lines for both the detachment criterion and von Neumann criterion are also drawn to investigate the dual solution domain, and the reflection configurations were verified experimentally for the first time on short wedges. By using proper input configuration parameter ( $w/H$ ), various types of shifts in the dual solution domain for short wedges are studied and categorised into three types, namely Type I, Type II and Type III.

Analysis of the Variation in Salaries of Employees of Government Ministries in Iraq by Using of The Lorenz Curve and The Gini Coefficient*H.

The disparity in the distribution of income is one of the main reasons for the failure to achieve social justice among members of society, which creates an unhealthy environment for workers in government sectors and institutions and also works to reduce the quality of work and lower the level of productivity. Conversely, reducing disparity leads to increased productivity and creates a suitable environment. It contributes to increasing productivity rates and improving the quality of institutional performance. The research discussed an important topic related to the extent of the discrepancy between the salaries of employees in Iraqi government institutions using the Lorenz curve and the Gini coefficient through a presentation of the research problem, which is the extent of the discrepancy in the salaries of state employees according to the approved salary scale, and what impact this has on social justice in the distribution of income, assuming that the adoption of A salary scale set by the authorities is based on reducing financial disparities between state employees with equal qualifications, such as educational attainment and work tasks, and reduces disparities of discrimination between workers in various government sectors and achieves social justice in income distribution. The research concluded that the actual distribution curve was very close to the symmetry line curve, in addition to the presence of a very small gap between the two lines, which indicates fairness in the distribution of monthly salaries among workers in government ministries who are in the fourth degree/first stage of the approved salary scale. The value of the Gini coefficient complementary to the work of the Lorenz curve was about (0.4), which means that the value of the Gini coefficient is close to zero. This indicates that the distribution of salaries of employees of government ministries is close to equitable distribution.

Convex symmetric rectangular pentagon central configurations

A convex rectangular pentagon, also called house-shaped, is a pentagon with the added restriction that two non-adjacent sides have equal lengths, each of which forms a right angle with the intervening side. In this paper, we focus on the existence of central configurations of the 5-body problem, where the five bodies are in a symmetric house-shaped configuration. That is, when the five bodies are located at the vertices of a convex rectangular pentagon, and moreover, the body not belonging to the two parallel sides lies along the symmetry line.

Ab Initio Studies of Mechanical, Dynamical, and Thermodynamic Properties of Fe-Pt Alloys.

The density functional theory (DFT) framework in the generalized gradient approximation (GGA) was employed to study the mechanical, dynamical, and thermodynamic properties of the ordered bimetallic Fe-Pt alloys with stoichiometric structures Fe3Pt, FePt, and FePt3. These alloys exhibit remarkable magnetic properties, high coercivity, excellent chemical stability, high magnetization, and corrosion resistance, making them potential candidates for application in high-density magnetic storage devices, magnetic recording media, and spintronic devices. The calculations of elastic constants showed that all the considered Fe-Pt alloys satisfy the Born necessary conditions for mechanical stability. Calculations on macroscopic elastic moduli showed that Fe-Pt alloys are ductile and characterized by greater resistance to deformation and volume change under external shearing forces. Furthermore, Fe-Pt alloys exhibit significant anisotropy due to variations in elastic constants and deviation of the universal anisotropy index value from zero. The equiatomic FePt showed dynamical stability, while the others showed softening of soft modes along high symmetry lines in the Brillouin zone. Moreover, from the phonon densities of states, we observed that Fe atomic vibrations are dominant at higher frequencies in Fe-rich compositions, while Pt vibrations are prevalent in Pt-rich.

Research on point pressing process of curved parts of thick plates with asymmetric double curvature of high strength steel

Abstract High-strength steel asymmetric double-curvature thick plate curved parts are widely used in shipbuilding, nuclear power and other industries, but their accurate forming has been a challenge due to their high strength, large size, asymmetric structure and serious springback. In this paper, the simulation analysis study of spot press forming is carried out with large-size double-curvature spherical thick shells as the research object, and the effects of mould surface size, mould radius of curvature on the results of pressing and forming of asymmetric double-curvature curved thick plate parts made of high-strength steel are analyzed. The results show that the pressing route of pressing the centre of the slab first and then carrying out point pressing along the line of symmetry to the surroundings has the best forming effect. Different mould profile dimensions and radius of curvature of the profile have a great influence on the simulated forming results.

Using international large-scale assessment for learning: Analyzing U.S. students’ geometry performance in TIMSS

The data from international large-scale assessments, such as The Trends in International Mathematics and Science Study (TIMSS), is often designed and used as an assessment of learning rather than an assessment for learning. This research employed TIMSS 2011 data, focusing on the geometric performance of fourth-grade students in the United States, to demonstrate how large-scale assessments can be utilized qualitatively to identify students' learning challenges, providing valuable insights to inform geometry teaching practices. Using van Hiele’s levels of geometric thinking as a guiding framework, the 24 released geometry items were analyzed to identify students’ struggle points in geometry nationwide. Our analysis revealed that students performed well on tasks that were visual, contextually grounded, and had clearly defined manipulative expectations.. However, students needed further support to solve items requiring a deep understanding of geometrical concepts, such as finding the perimeter of a square. Additionally, students excelled at solving tasks involving the identification of relative positions of objects on a map and recognizing lines of symmetry in regular shapes. However, they struggled with measurement-related tasks. Our analysis also identified a range of item features that might cause difficulties that impacted the way students responded to these items. Instructional implications for elementary mathematics education are discussed.

Operative slip systems and their critical resolved shear stresses in η-Fe2Al5 investigated by micropillar compression at room temperature

The plastic deformation behavior of single crystals of orthorhombic η-Fe2Al5 has been investigated by micropillar compression at room temperature as a function of crystal orientation and specimen size. Plastic flow is observed even at room temperature by the operation of six slip systems; (001)<010>, (001)<110>, (001)<130>, {22‾3}<110>, {311}<1‾03> and {301}<1‾03>. The CRSS values for the six identified slip systems are very high all in the range of 1.1∼1.5 GPa and do not vary much with specimen size. In the middle of the stereographic projection, the (001)<010>, (001)<110>, (001)<130> and {22‾3}[110] slip systems operate according to the relative Schmid factors with the similar CRSS values in the range of 1.08∼1.23 GPa. In orientations close to [001], the {311}<1‾03> slip system as well as the {301}<1‾03> slip system operate with a much higher CRSS values around 1.5 GPa, producing wavy slip traces due to the occurrence of frequent cross-slip among these slip planes. In orientations close to the [100]-[110]-[010] symmetry line, on the other hand, premature failure occurs without the operation of any slip systems, although, the Schmidt factor-wise, the {311}<1‾03> and {301}<1‾03> slip systems could operate. The selection of slip systems, their CRSS values and the possible dislocation dissociation modes are discussed based on the overlapped atomic volume that occurs during shear along the slip direction on the slip plane, taking into account the partial occupancies of Al atoms in the linear atomic chain along the orthorhombic c-axis direction.

Enhanced Frequency Conversion in Parity-Time Symmetry Line.

Non-Hermitian degeneracies reveal intriguing and nontrivial behaviors in open physical systems. Examples like parity-time (PT) symmetry breaking, topological encircling chirality, and enhanced sensing near an exceptional point (EP) are often associated with the abrupt nature of the phase transition around these degeneracies. Here we experimentally observe a cavity-enhanced second-harmonic frequency (SHG) conversion on a PT symmetry line, i.e., a set consisting of open-ended isofrequency or isoloss lines, both terminated at EPs on the Riemann surface in parameter space. The enhancement factor can reach as high as 300, depending on the crossing point whether in the symmetry or the broken phase of the PT line. Moreover, such enhancement of SHG enables sensitive distance sensing with a nanometer resolution. Our works may pave the way for practical applications in sensing, frequency conversion, and coherent wave control.

Toughening of intrinsically brittle materials by inserting arrays of voids

The aim of the current paper is to find optimum arrangements of voids that are able to catch and trap all possible cracks that might originate from a free surface. The idea is to investigate whether it is possible to apply this crack trapping mechanism to enhance the fracture toughness of intrinsically brittle materials. In doing so, the void volume shall be as small as possible. The investigation applies the crack trajectory interpolation (CTI) method, which is a new, computationally efficient approach to approximate crack trajectories near a single void or stiff particle. Under the assumption of linear superposition, the CTI-method is extended to predict the crack trajectories near arrangements of voids. This is done for uniaxial and biaxial loading conditions. The investigation shows that a symmetrical arrangement of voids should be avoided, since cracks near the symmetry line can bypass the array. Staggered void arrangements provide better conditions for crack trapping. By using elliptical voids with an aspect ratio of 2 instead of circular voids, it is possible to achieve a higher crack trapping efficiency and a lower void volume. A damage-based approach is used to determine the improvements in fracture toughness due to crack trapping for various void arrangements. The Python code of the CTI method is presented in the Appendix of this work.

Data-driven reconstruction of chaotic dynamical equations: The Hénon–Heiles type system

In this study, the classical two-dimensional potential VN=12mω2r2+1NrNsin(Nθ), N∈Z+, is considered. At N=1,2, the system is superintegrable and integrable, respectively, whereas for N>2 it exhibits a richer chaotic dynamics. For instance, at N=3 it coincides with the Hénon–Heiles system. The periodic, quasi-periodic and chaotic motions are systematically characterized employing time series, Poincaré sections, symmetry lines and the largest Lyapunov exponent as a function of the energy E and the parameter N. Concrete results for the lowest cases N=3,4 are presented in complete detail. This model is used as a benchmark system to estimate the accuracy of the Sparse Identification of Nonlinear Dynamical Systems (SINDy) method, a data-driven algorithm which reconstructs the underlying governing dynamical equations. We pay special attention at the transition from regular motion to chaos and how this influences the precision of the algorithm. In particular, it is shown that SINDy is a robust and stable tool possessing the ability to generate non-trivial approximate analytical expressions for periodic trajectories as well.

Crowdsourced Assessment of Aesthetic Outcomes of Dorsal Preservation Rhinoplasty

Abstract Goals/Purpose The inherent subjectivity in evaluating aesthetic outcomes presents a unique challenge in assessing rhinoplasty. Crowdsourcing has provided a new metric for objective analysis. The authors designed a study to compare the aesthetic outcomes of dorsal preservation rhinoplasty versus structural rhinoplasty using a reliable and valid crowdsourcing platform. We aim to objectively quantify the relative aesthetic advantages of performing a dorsal preservation technique. Additionally, we aim to demonstrate the efficacy of using crowdsourcing as an efficient and reliable method for evaluating any plastic surgery aesthetic outcome. Methods/Technique This retrospective observational study was approved by the IRB at UCLA. A total of 64 patients who had previously undergone rhinoplasty performed by the senior author were included. All surgeries were performed at the UCLA Ronald Reagan Medical Center. Patients with prior nasal surgery or trauma were excluded. All subjects were photographed using standard rhinoplasty views during the pre-and postoperative visits. Frontal and right profile photographs were then cropped in a standardized fashion (Photos, Version 8.0, Apple Inc.) and used for evaluation. All evaluations and ratings were collected using the HIPPA compliant and encrypted software via a secure online platform (LoveMyDelta, Inc., Philadelphia, PA). All crowd workers were recruited and vetted through this platform, with internal fidelity checks performed. Each crowd worker was shown a profile and frontal view of a patient and asked to score the overall appearance of the nose, the nasal dorsum profile, the symmetry of the dorsal aesthetic lines, and the dorsal contour using a visual analog scale. Scoring was performed using a provided validated visual assessment guide. They were asked whether they believed the subject had undergone rhinoplasty and asked to provide a level of confidence in their guess. Crowdworkers’ responses were aggregated to allow for high-powered intra-rater analysis. Intra-rater reliability and confidence intervals were calculated. Aggregate data from all raters were used to generate an absolute value of aesthetic state for each cohort. A "delta" was then obtained for each value by comparing the preoperative to the postoperative state to obtain a representative value of the improvement after undergoing surgery. Raters were asked if the patient appeared to have had surgery, and each correct and incorrect response was proportionally weighted based on the raters' confidence in their answer. The delta (difference in preoperative to postoperative aesthetic state) for each parameter for the dorsal and non-dorsal preservation cohorts was calculated using non-paired T-tests to determine significance. Results/Complications A total of 64 patients were included for evaluation. The structural rhinoplasty cohort consisted of 34 patients. The dorsal preservation cohort included 30 patients. Both dorsal preservation and non-dorsal preservation rhinoplasty cohorts were associated with improved overall crowdsourced aesthetic outcomes and improved outcomes across all subparameters. Overall, delta (d) improved in both dorsal preservation (0.300, 95% CI +/- 0.047) and structural (0.377, 95% CI +/- 0.055) cohorts. When raters were asked to predict whether a patient had surgery, the correlation coefficient between the confidence in the prediction and that prediction being correct was 0.74 (95% CI +/- 0.47 in the structural group, suggesting that a crowd worker was likely to be correct in identifying whether a patient had structural rhinoplasty. However, the correlation coefficient was -0.0554 (95% CI +/- 0.058) in the dorsal preservation group. This suggests the raters were unable to consistently identify which patients had dorsal preservation rhinoplasty with any confidence. Conclusion In the ongoing challenge to achieve better aesthetic outcomes in plastic surgery, crowdsourcing is a valuable tool for helping to measure these outcomes. We compared the aesthetic outcomes of dorsal preservation rhinoplasty versus structural rhinoplasty using a reliable and valid crowdsourcing platform. We found that significant improvements in overall aesthetic outcomes were achieved with both techniques, while a more natural “unoperated” outcome was achieved when performing a dorsal preservation technique. With this type of study and assessment tool, we helped to demonstrate the efficacy of using crowdsourcing as an efficient and reliable method for evaluating any plastic surgery aesthetic outcome.

Engineering Two-Dimensional Nodal Semimetals in Functionalized Biphenylene by Fluorine Adatoms.

We propose a band engineering scheme on the biphenylene network, a newly synthesized carbon allotrope. We illustrate that the electronic structure of the biphenylene network can be significantly altered by controlling conditions affecting the symmetry and destructive interference of wave functions through periodic fluorination. First, we investigate the mechanism for the appearance of a type-II Dirac fermion in a pristine biphenylene network. We show that the essential ingredients are mirror symmetries and stabilization of the compact localized eigenstates via destructive interference. While the former is used for the band-crossing point along high symmetry lines, the latter induces highly inclined Dirac dispersions. Subsequently, we demonstrate the transformation of the biphenylene network's type-II Dirac semimetal phase into various Dirac phases such as type-I Dirac, gapped type-II Dirac, and nodal line semimetals through the deliberate disruption of mirror symmetry or modulation of destructive interference by varying the concentration of fluorine atoms.

AHiLS—An Algorithm for Establishing Hierarchy among Detected Weak Local Reflection Symmetries in Raster Images

A new algorithm is presented for detecting the local weak reflection symmetries in raster images. It uses contours extracted from the segmented image. A convex hull is constructed on the contours, and so-called anchor points are placed on it. The bundles of symmetry line candidates are placed in these points. Each line splits the plane into two open half-planes and arranges the contours into three sets: the first contains the contours pierced by the considered line, while the second and the third include the contours located in one or the other half-plane. The contours are then checked for the reflection symmetry. This means looking for self-symmetries in the first set, and symmetric pairs with one contour in the second set and one contour in the third set. The line which is evaluated as the best symmetry line is selected. After that, the symmetric contours are removed from sets two and three. The remaining contours are then checked again for symmetry. A multi-branch tree representing the hierarchy of the detected local symmetries is the result of the algorithm.

Effect of turbulent coflows on the dynamics of turbulent twin jets

The impact of turbulent coflows on the dynamics of turbulent twin round jets is investigated experimentally. Parallel twin jets, at three jet spacing values and two Reynolds number/jet-to-coflow velocity ratios, were released into turbulent coflows with two distinct levels of turbulence intensity. Velocity measurements were made using acoustic Doppler velocimetry. An increase in the coflow turbulence intensity leads to an earlier merging and combining of the jets and also accelerates the rate of decay with downstream distance of the mean centerline excess velocity of the jets. The mean velocity on the symmetry line, for different values of jet spacing, ratios of jet exit velocity to coflow mean velocity, and coflow turbulence intensity, is self-similar when scaled by the maximum mean velocity on the symmetry line and the corresponding streamwise distance. Moreover, as the turbulence level of the coflow intensifies, the turbulence intensity along the symmetry line of the jets increases. The longitudinal integral length scale on the symmetry line of the twin jets decreases as the coflow turbulence intensity increases. The energy spectra of the coflowing twin jets show that the turbulence in the coflow transfers the energy contained by the larger scales to the smaller scales at a greater rate than that which occurs for jets in a quiescent background. However, as the jet spacing increases, less energy is transferred to the smaller scales.

Non-magnetic four-port electronic circulators based on 90∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90^\\circ $$\\end{document} non-reciprocal phase-shifters

This paper presents a family of four-port electronic circulators adhering to a new topology symmetry that enables linear, low-loss transistor-based circuit implementations. The underlying principle of operation employs a property of the 90∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90^\\circ $$\\end{document} non-reciprocal phase shifter (NRPS) derived in this article. Under quadrature excitation, the NRPS transfers or reflects exciting signals depending on their respective phase lead. The fundamental topology consists of two back-to-back quadrature hybrid couplers with a 90∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90^\\circ $$\\end{document} NRPS connected in parallel over the line of symmetry, interrupting the circuit’s reciprocity to achieve circular propagation by bypassing or reflecting at the NRPS but not through. We break down the circuit into three fundamental four-port sub-circuits. The transfer function of the cascaded sub-circuits enables an analysis with specific hybrid couplers. It also allows a synthesis of other four-port passive sub-circuits that, with an NRPS, achieve a four-port circulator transfer function by solving a matrix equation. Some of the mathematical solutions have circuit realizations, which are adjusted quadrature hybrid structures that differ from each other by the characteristic impedance of their arms. Two familiar solutions, including the standard quadrature hybrid and a modified design with equal Z0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z_0$$\\end{document}, λ/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda /4$$\\end{document} arms, are simulated utilizing lossless lumped element arms and a 4-Path, 65-nm NMOS 90∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90^\\circ $$\\end{document} NRPS. The simulation results verify the theoretical analysis and enable a comparison between the performance of the two circuit solutions around 1 GHz. The four-port circulator with equal arms is implemented on a PCB and measured, yielding better than 1.5 dB insertion loss between the circulator ports, over 17 dB port-to-port reverse isolation, and better than 20 dBr port matching around 1 GHz.

Exploring Geometry and Art through Tessellations

In this lesson, elementary school students explore geometric shapes and tessellations using a Cricut Maker 3. During Part 1 of the activity, students review geometric concepts of regular versus irregular polygons and lines of symmetry. This includes using shapes cut by the Cricut machine to determine which regular polygons form a tessellation when put together. Then students answer reflection questions. During Part 2, there is a discussion about how the artist MC Escher used different types of symmetry (e.g., translations, rotations, and reflections) to modify irregular shapes to create tessellations. In Part 3, students are given materials to prototype their own tessellation using regular and irregular shapes and at least one type of symmetry transformation.

Rigidity Symmetry Line for Thermodynamic Fluid Equations-of-State

Rigidity Symmetry Line for Thermodynamic Fluid Equations-of-State

Theoretical Investigations of the Structural, Dynamical, Electronic, Magnetic, and Thermoelectric Properties of CoMRhSi (M = Cr, Mn) Quaternary Heusler Alloys

The structural, dynamical, electrical, magnetic, and thermoelectric properties of CoMRhSi (M = Cr, Mn) quaternary Heusler alloys (QHAs) were investigated using density functional theory (DFT). The Y-type-II crystal structure was found to be the most stable configuration for these QHAs. Both CoCrRhSi and CoMnRhSi alloys possess a half-metallic behavior with a 100% spin-polarization as the majority spin channel is metallic. On the other hand, the minority spin channel is semiconducting with narrow indirect band gaps of 0.54 eV and 0.57 eV, respectively, along the Γ−X high symmetry line. In addition, both CoCrRhSi and CoMnRhSi alloys possess a ferromagnetic structure with total magnetic moments of 4 μB, and 5 μB, respectively, which are prominent for spintronics applications. The thermoelectric properties of the subject QHAs were calculated by using Boltzmann transport theory within the constant relaxation time approximation. The lattice thermal conductivities were also evaluated by Slack’s equation. The predicted values of the figure-of-merit (ZT) for CoCrRhSi and CoMnRhSi were found to be 0.84 and 2.04 at 800 K, respectively, making them ideal candidates for thermoelectric applications.

Symmetry and the design of self-stressed structures

The symmetry adapted counting rule for mechanisms and states of self-stress in symmetric frameworks is presented in an accessible and intuitive manner with the aim of empowering engineers who design such structures. By simply counting nodes and bars, it is possible to detect states of self-stress and mechanisms beyond the standard Maxwell-Calladine count. This methodology is first introduced without the need to understand the underlying group theory before being applied to a range of example frameworks. Design problems focusing on gridshells are discussed – it is noted that placing bars on lines of mirror symmetry tend to increase the number of states of self-stress in a framework, which can be desirable. This paper reformulates common symmetric frameworks and introduces simple rules regarding how to obtain a greater number of states of self-stress. By allowing for the design of states of self-stress, the forces in the structure can be designed with greater control.