Radiographic assessment of the sagittal inclination of the first metatarsal (1M) is essential for evaluating foot disorders. However, 1M varus deviation may influence these measurements. This study aimed to determine whether hallux valgus (HV) varus deformity influences the sagittal radiographic inclination of the 1M by comparing weightbearing radiography (WBR) and weightbearing computed tomography (WBCT) images. Eighty-four feet were analyzed, including HV cases with intermetatarsal angle (IMA) >15° and control feet without HV. The sagittal inclination of the 1M, its base height, and ground lines form a rectangular scalene triangle, in which perspective changes can modify the lengths and angles of its sides. The first metatarsal declination angle (FMDA) and length of the first metatarsal (L1M) were used to assess the influence of 1M varus on sagittal alignment differences between WBR and WBCT. FMDA values showed no significant differences within or between groups, with a mean difference of 0.39° (P = .98). In contrast, L1M measurements differed significantly between imaging modalities and between HV and control groups, with a mean difference of 2.48 mm (P < .05). Agreement analysis demonstrated strong concordance between WBR and WBCT measurements, indicating comparable values with minimal systematic bias. In patients with IMA >15°, forefoot geometric changes were reflected with modest L1M differences, although their clinical impact should be interpreted cautiously. Importantly, these changes did not affect FMDA, which remained stable across imaging modalities. The strong agreement between WBR and WBCT supports FMDA as a reliable parameter for assessing 1M sagittal alignment and planning realignment procedures, irrespective of HV severity. Level IV, cross-sectional study.

Abstract To address the issues of susceptibility to local buckling instability and unstable deformation modes in star-shaped honeycomb (SSH) structures under quasi-static compression, this study proposes a novel star-triangle honeycomb (STH) structure with a negative Poisson’s ratio (NPR). The core innovation of this work lies in the geometric coupling of a triangular stabilizing frame (right-angled triangle) with the SSH unit, which actively suppresses the inherent asymmetric torsional deformation mode of the SSH that leads to performance degradation. Through a combination of experimental and finite element methods, the in-plane compressive mechanical properties, deformation mechanisms, and energy absorption capacity of the STH structure were systematically investigated. Under identical geometric dimensions, the STH structure demonstrates significantly superior deformation stability, initial stiffness, and plateau stress compared to the SSH structure. Benefiting from this strategy of ‘actively locking’ the unstable deformation mode, the STH structure achieves a synchronized enhancement in its NPR effect (reaching −0.64, an improvement of approximately 82.9%) and a near doubling of its energy absorption efficiency (with specific energy absorption increased by 99%). Parametric analysis further reveals the influence of the star-shaped angle ( θ ₁), triangular angle ( θ ₂), and their respective wall thicknesses ( t ₁, t ₂) on the mechanical response. By employing an active geometric design strategy, this study simultaneously enhances multiple key properties of the auxetic structure, providing a novel approach for designing metamaterials with both high energy absorption efficiency and stable deformation modes.

Summary We provide a method for explicitly finding the line that simultaneously halves the areas of two triangles; such a line is guaranteed to exist by the Pancake Theorem and is unique if the triangles are not overlapping. We also address the question of whether this halving line is always constructible with straight-edge and compass. This was answered negatively by Berele and Catoiu in a 2024 paper, where they provided an example of two non-congruent right-angle triangles whose halving line is non-constructible. Our approach is different from that of Berele and Catoiu, treating the problem of constructibility of the common tangent lines of two hyperbolas. In particular, we obtain an example of two congruent right-angle triangles with non-constructible halving line. We also point out some relatively unexpected cases where the halving line is constructible. Some remarks on the uniqueness and non-uniqueness of the halving line for a pair of overlapping triangles are also made.

Ruminant herbivores interact dynamically with their food resources, especially in deserts, where plant availability fluctuates sharply across seasons. We evaluated how seasonal food availability and the nutritional traits of preferred plants shape the diet and macronutrient niche of a desert mule deer (Odocoileus hemionus) population in the buffer zone of the Médanos de Samalayuca protected area, northern Mexico. From 2021 to 2022 we quantified seasonal food plant availability and characterized mule deer diet using microhistological fecal analysis and the nutrient content by right-angled mixture triangles. Mule deer diets were consistently low in diversity and dominated by grass, but preferred species shifted seasonally among shrubs, succulents, and grasses. Deer strongly selected some plant species that were scarce in the landscape, particularly during the cold-dry season. Preferred plants generally had high carbohydrate and variable protein contents, with the highest protein proportions in the temperate-dry season. Mixture triangles showed a narrow, carbohydrate-biased macronutrient niche, with the broadest range of nutrient mixtures in the temperate-dry season. Overall, our results support an opportunistic foraging strategy in which mule deer consume what is seasonally available while selectively using key plant species to maintain a relatively constant nutritional balance under limited and variable food resources.

Esophageal hiatus closure during hiatal hernia repair is essential. Improper closure can lead to recurrence and high patient morbidity. Our aim is to introduce an easy and reproducible method of calculating the surface area (hiatal surface area [HSA]) of the esophageal hiatus. Standardization of this value will enable surgeons to have an evidence-based approach for hiatal hernia closure during laparoscopic repair. We developed a measurement of HSA of the esophageal hiatus corresponding to a right-angle triangle: Area = (1/2) base × height. The height was defined as the left crus of the diaphragm. The base was defined as perpendicular to the crus and tangential to the medial edge of the esophagus. The mean esophageal hiatus surface area was calculated from deceased patients without a hiatal hernia undergoing full autopsies and compared to patients undergoing laparoscopic repair. A total of 237 (37 cadaveric) hiatuses were measured. The median HSA defect in the cadaveric group was 1.97 cm2 with an interquartile range (IQR) of 1.13 to 3.0 cm2 was significantly larger compared to 5.0 cm2 with an IQR of 3.5 to 7.5 cm2 in the operative group (P < .001). Multivariate linear regression demonstrated an overall significant positive correlation between esophageal hiatus surface area defect and the variables of age and weight but not with gender. This study demonstrated a new reproducible method of measurement for esophageal hiatus. The significant difference between the two groups suggests that our formula can be utilized to develop a standardized value for the surface area of the esophageal hiatus.

The Qibla direction is a fundamental aspect of Muslim prayer, requiring precise determination. This study evaluates the accuracy of the Qibla direction at the As-Sakinah Mosque in Surabaya by comparing three instruments: a Theodolite, a Mizwala, and a Right Triangle (Segitiga Siku-siku). Employing field experimental methods, the measurement data were analyzed using a comparative descriptive approach. The findings reveal significant variations in accuracy among the instruments: the Theodolite and Mizwala detected deviations of 5° 56' 48.71" and 5° 35' 4.21" respectively, whereas the Right Triangle method showed a substantial deviation of 25°. It is concluded that the mosque's current orientation deviates by approximately 5 degrees to the North, highlighting that digital optical instruments offer superior accuracy compared to simple manual methods in urban environments.

This article confronts the persistent challenge of determining the exact perimeter of an ellipse. It proposes a high-accuracy geometric approximation centred on a uniquely defined Measuring Right-Angled Triangle (MRAT). Constructed with specific spatial and angular properties, the MRAT is positioned at a distance of 2b/π from the centre of a reference circle and terminates at its circumference at a 45° angle. The ellipse's center is co-located with the circle's center. The resulting values were rigorously compared against classical Ramanujan approximations, and the PRI test and high-precision graphical analysis were used to confirm significant accuracy. This high-accuracy geometric approximation method offers a computationally efficient alternative to traditional algebraic methods, enhancing both theoretical understanding and applied precision.

Abstract In 1985, Linderholm posed the following problem: does every convex body K in the plane admit a right-angled triangle T that contains K and whose area is at most twice that of K ? Assuming that K has constant width, we prove that such a triangle T exists with area at most 2.07016 times the area of K .

Prospective randomised clinical study. To compare the effectiveness of the Right Triangle (RT) method with the traditional UBE(unilateral biportal endoscopic) method in UBE surgery for lumbar disc herniation or stenosis, focusing on surgical efficiency and radiation exposure minimization. UBE technique is a minimally invasive approach for spinal surgeries. This study introduces the RT method as an alternative during UBE procedures, aiming to enhance surgical efficiency and minimize radiation exposure. We conducted a prospective analysis of 91 consecutive patients who underwent UBE surgery using either the traditional method (43 cases) or the Right Triangle (RT) method (48 cases) between Dec 2022 and Dec 2023. All data were collected prospectively with a minimum follow-up of 12 months. Both groups demonstrated comparable improvements in VAS and ODI scores. However, the RT method exhibited significant advantages in terms of reduced operative time (62.7±5.7min for traditional UBE vs. 56.8±6.9min for RT) and fewer fluoroscopy shots (3.9±0.74 for traditional UBE vs. 2.6±0.61 for RT). The RT method within UBE practice offers a promising alternative that effectively shortens operative time and decreases reliance on intraoperative x-ray fluoroscopy. As operator proficiency increases, operative time is expected to decrease further, particularly in cases with left-sided symptoms. Level 3.

Abstract An embedded twisted paper cylinder of aspect ratio λ is a smooth isometric embedding of a flat λ × 1 cylinder into ℝ 3 such that the images of the boundary components are linked. We prove that for such an object to exist we must have λ &gt; 2 and that this bound is sharp. We also show that any sequence of examples having aspect ratio converging to 2 must converge to a (non-smooth) 4-fold wrapping of a right-angled isosceles triangle.

The kisrhombille tiling is the dual tessellation of one of the semi-regular tilings, composed of right-angled triangles arranged in 12 distinct orientations. A coordinate system has been employed to formally describe the grid of tiles. In this paper, two tiles are defined as neighbors if they share an edge in their boundary. The concept of digital distance is introduced as the minimum number of steps required to traverse between two tiles, and the corresponding distance formula is derived by constructing minimal paths.

We propose an estimation method of the relative position of a light emitting diode (LED) from a rolling-shutter (RS) camera with a cross filter (CF). The CF is set in front of the camera lens so that incident light can be scattered into two straight lines extending about <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pm \pi /4$</tex-math></inline-formula> from the horizontal row of the CMOS RS image sensor (IS). With the geometry of a right-angled isosceles triangle, our method makes it possible to estimate the coordinates of the LED in the IS plane when the scattered light is captured at two points in a scanline. With the CF, our method continues to update the estimated coordinate of the LED as long as the row captures scattered light at two points. Naturally, our method makes it possible row by row when multiple LEDs are used or when they are moving. In contrast, without the CF, estimation is not possible until the scanline reaches the row which captures the whole LED, and the estimated coordinates are not updated once it is done for that frame. Experiments verify that our estimation method works with the estimation error less than five pixels, and extends the region of a frame in which estimated coordinates are output, even in a moving environment, while the conventional method (without CF) does the estimated coordinates only once per frame.

Visual representation of data can ease their understanding and interpretation, particularly when more variables are considered together. The ternary plot, based on a barycentric coordinate system and commonly used to represent compositional variables, may be difficult to interpret due to its structural complexity—stemming in part from the 60° axis projection and the need for indirect value estimation. This article presents its new alternative, the right-angled triangle plot, and compares the two plot types in the representation of a compositional variable of three components. According to a theoretical comparison, the right-angled plot has several technical advantages (e.g., larger plotting area, direct axis reading), resulting from its construction being based on the Cartesian coordinate system. To verify this hypothesis from an empirical point of view, a survey was conducted involving 441 researchers to assess the effectiveness in correctly interpreting the data presented on both plots. The bias was lower, and the precision was higher on the right-angled plot. Indeed, this study’s results, particularly the higher accuracy (more than 95% when a minimal tolerance was allowed) in determining the values of individual variables (X, Y, and Z), as well as the correctly identification of all three variables simultaneously in the right-angled plot compared with the ternary plot, suggest that the former may hold potential for improving the visualization of compositional data with three components. However, introducing this new type of plot would require familiarizing potential users with it, since the majority of respondents (63.2%) still considered the ternary plot easier to use, likely due to its long use and the novelty of the new plot type.

The conventional subject of Newtonian calculus is purely mathematical, and the concepts, approximations, and theorems employed within it have rarely been connected or correlated with the principles of physics and topology. In this research article, the entire domain of calculus has been revisited through a tripartite lens—encompassing mathematics, physics, and topology. This research article is based on the following: 1. Topological interpretation of Calculus 2. Dimensional perspective of differential calculus: how the variable changes for a mathematical function like xn when the value of n changes from 1 to 2 to 3 to 4…like this. The variable does not remain to be x only. 3. The physical significance of the ‘differential co-efficient’(DC) of a function which substantiates that the mathematical expression of DC should not retain the function itself since it is ‘co-efficient’. 4. The dimensional invalidity of the trigonometric theories at zero- or 90-degree values of angle θ of a right angled triangle. 5. Differential co-efficient of trigonometric functions derived based on a novel concept of expansion - contraction of a cuboid (being composed of perpendicular, base and hypotenuse of a right-angled triangle) through very simple algebraical steps and without using limit theorem, chain rule or quotient rule of the Newtonian Calculus. 6. The problem of ‘circularity of definition’ of the higher order differential co-efficient of the mathematical functions of Newtonian Calculus and offering a new mathematical expression of higher order differential co-efficient. 7. The concept of integration has been established as a process of hybridizatio n of a mathematical variable in space through rotations or translations. A genuine attempt has been made to unify these perspectives into a singular framework, resulting in the development of a new field: the reassessed topological version of calculus, which is now presented to the global scientific community. The limitations and flaws inherent in Newtonian calculus have been critically examined, logically addressed, and redefined within the broader scope of this newly proposed topological version of calculus.

A parametric study of longitudinal fracture in continuously inhomogeneous viscoelastic beam structures subjected to combined time-dependent loadings is described in this paper. The beams are under angle of twist and vertical displacement which change continuously with time. Beams with different cross-sections (T-shape, I-shape, trapezium, and rightangled triangle) are considered in order to assess the influence of the beam cross-section on the longitudinal fracture. The viscoelastic behaviour under bending and torsion is treated by using time-dependent modulus of elasticity and shear modulus, respectively. A viscoelastic mechanical model with two springs and a dashpot under time-dependent strains is used in order to derive the time-dependent moduli. These moduli are applied when obtaining solution of the strain energy release rate. The compliances of the beam configuration under time-dependent angle of twist and vertical displacement are analyzed in order to derive the strain energy release rate for verification. The parametric study is carried-out by using the solution of the strain energy release rate.

This article explores the geometric knowledge embedded in Vedic rituals, focusing on the construction of Agnikundas (fire altars) as detailed in the Vedic texts Sulba Sutras, Kundamandip Siddhi and also highlights the central place of geometry in Vedic rituals including the use of right-angled triangles, square roots, and precise measurements to construct ritual spaces. Through field observations and interviews with Sanskrit scholars and ritual practitioners, the research uncovers the informal transmission of this knowledge and its absence from modern Sanskrit curricula. A survey of 100 participants revealed limited awareness of these geometric principles. The knowledge of constructing Agnikundas is transmitted to the generations through participatory approach with the seniors means they learned through hands-on learning, and culturally responsive approach to learning geometry. The study indicates the need of curriculum reform on Sanskrit intuitions to include Vedic geometry used on rituals to bridge traditional knowledge and contemporary education. Integrating this ancient practice could enhance students' understanding of geometry while preserving cultural heritage.

This study examines the accuracy of the qibla direction of mosques in the Petir subdistrict. From observations made using Google Earth, many mosques do not face the Kaaba. When investigating the mosques that were indicated to be misaligned, it was found that the mosque management boards had not yet made any efforts to remeasure the qibla direction. This serves as the basis for the researcher to investigate the accuracy of the qibla direction of mosques in Petir subdistrict. This study employs a field research method. Primary data sources include interviews with ten mosque management committee (DKM) chairpersons. Data collection techniques included observation, interviews, and direct measurement of the qibla direction of the mosques. For the accuracy test of the qibla direction, the researcher used the right-angle triangle method based on the shadow of the sun. After the data was obtained, it was analyzed using qualitative descriptive techniques. The results of the study revealed that out of the ten mosques studied, only one mosque used the rubu’ mujayyab method to determine the qibla direction, while the other nine used a compass. The accuracy of the qibla direction in mosques in Petir Subdistrict shows considerable variation; one mosque is categorized as accurate, eight mosques are less accurate, and one mosque is inaccurate. This deviation occurs because during the initial measurement for mosque construction, accurate data and methods were not used. Therefore, prayers performed in mosques with inaccurate qibla directions are invalid according to the opinion of Imam Shafi'i. For this reason, it is necessary to adjust the qibla direction in accordance with the calibration results.

Education is an essential process aimed at developing the individual’s potential holistically, encompassing intellectual, emotional, and social aspects. The School of Nature, as an innovative approach to education, uses nature as the primary learning medium, offering relevant contextual experiences for students. SMP Sekolah Alam Bireuen (SABIR) implements the "Learning Together with Nature" (BBA) method, which invites students to learn through direct interaction with their surrounding environment. This approach aims to bridge the gap between theory and practice, while enhancing the understanding of mathematical concepts, including the Pythagorean Theorem. The Pythagorean Theorem, often considered abstract, can be taught more applicably through nature-based learning. This study uses the Systematic Review (SR) method to analyze literature related to the integration of the Pythagorean Theorem in BBA. The analysis results show that nature-based contextual approaches are effective in improving students' mathematical understanding, while also providing an enjoyable and meaningful learning experience. Students not only learn theory but also apply mathematical concepts in the field, such as measuring the sides of right-angled triangles, thus enhancing their understanding and problem-solving skills. This approach also instills values such as cooperation, responsibility, and environmental love, which support the holistic development of student competencies. This research is expected to serve as a reference for the development of nature-based learning methods in the context of more contextual and applicable mathematics education

This research explores the development of an interactive educational game application aimed at illustrating the Pythagorean Theorem through 2D plane figures and their associated geometric properties. The visual proof employs positional movements (translation) and rotations, strategically positioning plane figures and animating the step-by-step process from initial positions to the proof position. The selection of figures and properties depends on the proof method, incorporating angles in right-angled triangles, straight-line angles, and area substitution. The application showcases four Pythagorean Theorem proof methods: Zhou Bi Suan Jing, Henry Perigal, Garfield Trapezoid, and The Brides Chair. Developed using Unity, supported by Adobe Illustrator and Visual Studio in C#, it underwent testing on middle school students with a pretest-posttest control group method. Results analysis revealed significant average differences, indicating the application's substantial impact on students' theorem comprehension. Further examination of completion durations identified the Henry Perigal proof as the fastest, implying the ease of understanding area-based proofs over those involving analytic equations. This aligns with the perceptual simplicity of the Henry Perigal method compared to others. The study contributes an innovative educational approach, demonstrating the potential for effective theorem instruction through interactive applications.

This study aims to address situations where the base or height is unclear by creating an innovative method to calculate the area of a right-angled triangle when only the hypotenuse and one leg are known. A new computational technique is proposed to determine the missing dimension necessary for an accurate area calculation by integrating the Pythagorean theorem with the conventional area formula. The resulting formula, , offers a simplified approach to a common geometric problem. This method serves as a valuable tool for math teachers, enhances conceptual understanding of geometric relationships, and establishes a solid foundation for further research aimed at improving geometry instruction in high schools.