Equal Triangles Research Articles

On truth and polarity in negation processing: language-specific effects in non-linguistic contexts.

This study examines how negation is processed in a nonverbal context (e.g., when assessing ▲ ≠ ▲) by speakers of a truth-based system like Mandarin and a polarity-based system like English. In a truth-based system, negation may take longer to process because it is typically attached to the negation as a whole (it is not true that triangle does not equal triangle), whereas in polarity-based systems, negation is processed relatively faster because it is attached to just the equation symbol (triangle does not equal triangle), which is processed relatively faster. Our hypothesis was that negation processing routines previously observed for verbal contexts, namely that speakers of Mandarin get slowed down more when processing negative stimuli than positive stimuli compared to speakers of English, also extend to contexts when language use is not obligatory. To test this, we asked participants to agree/disagree with equations comprising simple shapes and positive '=' or negative '≠' equation symbols. English speakers showed a response-time advantage over Mandarin speakers in negation conditions. In a separate experiment, we also tested the contribution of equation symbols '≠'/'=' to the cognitive demands by asking participants to judge shape sameness in symbol-free trials, such as ▲ ■. This comparison allowed us to test whether crosslinguistic differences arise not because of shape congruence judgement but arguably due to negation attachment. The effect of the '≠' symbol on shape congruence was language-specific, speeding up English speakers but slowing down Mandarin speakers when the two shapes differed. These findings suggest language-specific processing of negation in negative equations, interpreted as novel support for linguistic relativity.

On the notion of equal figures in Euclid

Euclid uses an undefined notion of "equal figures", to which he applies the common notions about equals added to equals or subtracted from equals. When (in previous work) we formalized Euclid Book~I for computer proof-checking, we had to add fifteen axioms about undefined relations "equal triangles" and "equal quadrilaterals" to replace Euclid's use of the common notions. In this paper, we offer definitions of "equal triangles" and "equal quadrilaterals", that Euclid could have given, and prove that they have the required properties. This removes the need for adding new axioms. The proof uses the theory of proportions. Hence we also discuss the "early theory of proportions", which has a long history.

Resection and reconstruction for radial polydactyly Type IV-D in 206 cases: a retrospective clinical analysis

BackgroundRadial Polydactyly Type IV-D deformity is difficult to treat because of the most complex bone and soft tissue anomalies. Resection and reconstruction for one of the two thumbs was an option for treatment.ObjectiveThe study was to present our method of resection and reconstruction with a new incision for radial polydactyly Type IV-D and evaluate the clinical efficacy comprehensively using Rotterdam assessment system in a large sample.Methods206 cases of type IV-D thumb duplication underwent resection and reconstruction surgical treatment between 2010 and 2019. Two equal triangle flap incisions were designed around the radial thumb. The radial thumb was resected and the ulnar thumb was reconstructed in aspects of bone, tendons, ligaments and abductor pollicis brevis.The clinical results were evaluated using Rotterdam assessment system.ResultsThe mean follow–up period was 2.2 years (SD 1.5). The mean age of the patients was 9 months (SD 1.8) at the time of operation. The mean ranges of active IP and MP joint flexion and extension were 110° and 26°. The mean angulations for IP and MP joint instabilities were 3° and 11°, relatively. Angulation for palmar abduction was 58°. The mean appearance domain score was 8.9. The average parental satisfaction score was 2.5 and the average patient-reported pain score was 2.1. The mean functional domain score for all patients was 6.6. The average appearance domain score was 8.9. The mean patient-reported domain score was 4.5. The mean Rotterdam outcome score was 20.0, equivalent to 67% of the full score. The postoperative score of patients over two years old was significantly lower than that of patients under two years old.ConclusionResection and reconstruction method with two equal triangle flap incisions was a recommended treatment for radial polydactyly Type IV-D.Level of evidenceIV

Triangle Tiling Billiards and the Exceptional Family of their Escaping Trajectories: Circumcenters and Rauzy Gasket

Consider a periodic tiling of a plane by equal triangles obtained from the equilateral tiling by a linear transformation. We study a following tiling billiard: a ball follows straight segments and bounces of the boundaries of the tiles into neighboring tiles in such a way that the coefficient of refraction is equal to –1. We show that almost all the trajectories of such a billiard are either closed or escape linearly, and for closed trajectories we prove that their periods belong to the set We also give a precise description of the exceptional family of trajectories (of zero measure): these trajectories escape nonlinearly to infinity and approach fractal-like sets. We show that this exceptional family is parametrized by the famous Rauzy gasket. This proves several conjectures stated previously on triangle tiling billiards. In this work, we also give a more precise understanding of fully flipped minimal exchange transformations on 3 and 4 intervals by proving that they belong to a special hypersurface. Our proofs are based on the study of Rauzy graphs for interval exchange transformations with flips.

Improved SURF Descriptor Based on Triangle Partition

In order to improve the robustness and real time performance of SURF based image matching algorithms, a new descriptor is proposed. We compute the new descriptor in a rectangle local region (the side set to 20s). Firstly, the local region is divided into 8 equal triangle subregion. Secondly, local region location grid is rotated to align its dominate orientation to a canonical direction. The keypoint dominate orientation and its orthogonalorientation is defined as the x and y directions of the descriptors local coordinate system.Thirdly, compute the Haar wavelet response in x and y directions within the keypoint local region. In order to reduce the boundary effect and outer noise, Haar wavelet response in the same Grid of different triangle is both assigned to each triangle in different weight, and then a gaussian weighting function is used. Compute the histogram of Haar wavelet response and absolute Haar wavelet response, so each triangle subregion constitutes a vector with 4 dimensions. Finally, a descriptor with 32 dimensions is constituted and the descriptor is normalized to achieve illumination invariance. The experimental results show that the performance of the new descriptor is even better than SURF descriptor.

Modified high-order neural network for invariant pattern recognition

A modification for high-order neural networks (HONN) is described. The proposed modified HONN takes into account prior knowledge of the binary patterns that must be learned. This significantly reduces hence computation time as well as memory requirements for network configuration and weight storage. An “approximately equal triangles” scheme for weight sharing is also proposed. These modifications enable the efficient computation of HONNs for image fields of greater that 100×100 pixels without any loss of pattern information.

Proposal for a parametric conceptual CAD model of a mono-modular inertial fusion reactor

The present work tries to solve the problem of realizing a parametric conceptual CAD model of a modular reactor for future inertial fusion power plants. The choice of a modular structure seems to be a good solution for efficiency and economic requirements. On the other hand, the realization of a parametric-variational CAD model is very useful to optimize nuclear and mechanical parameters and to permit the shift from the conceptual to the final model. First, geometric solutions for a modular reactor are analysed; the most interesting is that of a 20-face regular polyhedron (icosahedron). The subdivision of each face into six equal triangles consents to obtain a mono-modular reactor with 120 modules (called ‘ICO120’). This solution should be easy, efficient and cheap. Secondly, the work proposes a conceptual CAD model of the ICO120 reactor in which special attention is put on the parametrization. Starting from such parametric model it will be possible to develop and optimize icosahedral reactors with different features, sizes and performances.

Unique decomposition of element stiffness numeric matrices for three‐noded plate bending triangles

This work is concerned with the numeric stiffness matrices of three-noded triangular plate bending finite elements; in particular with those numeric stiffness matrices, which are freedom-deficient and comply with the conditions of the patch test Subsequent to initial transformation of the rotation connectors for such matrices it is evident that there must exist an unique decomposition to the stiffness matrix K̆6∈ℝ6×6 of the corresponding Kirchhoff six-noded constant bending moment triangle. In K̆6 all six trial functions are themselves synonymous with those which describe the patch test. The transformation matrices of decomposition, and subsequent restoration upon modification or design, are derived explicitly and are succinct in application. Decomposition of the numeric stiffness matrix leads to exceptional versatility in objective modification, e.g. design of the matrix by adaptive process. Attention is confined here to the stiffness matrices K̆9∈ℝ9×9 of nine-degree-of-freedom three-noded Kirchhoff plate bending triangles with their single-degree-of-freedom deficiency. The decomposition of the element stiffness matrix immediately reveals those six coefficients which are available for design. They control the effect of transverse shear and are the constituents of a symmetric positive-definite matrix M3∈ℝ3×3 which is designated the ‘mechanism restraint’ matrix. It is necessary only that the designed coefficients are such that the matrix M3 remains symmetric and positive definite so as to ensure retention of patch test satisfaction on restoration to the newly designed K9. The illustrative examples provide a first perception of the leap in expectation which is enabled by design of the numeric K9 when uninhibited by formal method. Thus, the feasibility is illustrated of simple adaptive design of K9 with objective to recover cubically varying w displacements over an equilateral patch of equal triangles. This recovery is readily achieved by ad hoc inverse method but raises the issue of uniqueness in design. In highlighting the characteristics of M3 it is evident that there remains a wealth of opportunity for further research before adaptive design of the element stiffness matrix, within an arbitrary prevailing w displacement field, can become a practical reality. An appendix lists the Fortran computer codings which are used in the examples to calculate the stiffness matrix K̆6 of the six-noded constant bending moment Kirchhoff triangle as well as the explicit transformation matrices for decomposition and restoration of the numeric K9 stiffness matrix. © 1998 John Wiley & Sons, Ltd.

Unique decomposition of element stiffness numeric matrices for three-noded plate bending triangles

This work is concerned with the numeric stiffness matrices of three-noded triangular plate bending finite elements; in particular with those numeric stiffness matrices, which are freedom-deficient and comply with the conditions of the patch test Subsequent to initial transformation of the rotation connectors for such matrices it is evident that there must exist an unique decomposition to the stiffness matrix K6∈ℝ6×6 of the corresponding Kirchhoff six-noded constant bending moment triangle. In K6 all six trial functions are themselves synonymous with those which describe the patch test. The transformation matrices of decomposition, and subsequent restoration upon modification or design, are derived explicitly and are succinct in application. Decomposition of the numeric stiffness matrix leads to exceptional versatility in objective modification, e.g. design of the matrix by adaptive process. Attention is confined here to the stiffness matrices K9∈ℝ9×9 of nine-degree-of-freedom three-noded Kirchhoff plate bending triangles with their single-degree-of-freedom deficiency. The decomposition of the element stiffness matrix immediately reveals those six coefficients which are available for design. They control the effect of transverse shear and are the constituents of a symmetric positive-definite matrix M3∈ℝ3×3 which is designated the ‘mechanism restraint’ matrix. It is necessary only that the designed coefficients are such that the matrix M3 remains symmetric and positive definite so as to ensure retention of patch test satisfaction on restoration to the newly designed K9. The illustrative examples provide a first perception of the leap in expectation which is enabled by design of the numeric K9 when uninhibited by formal method. Thus, the feasibility is illustrated of simple adaptive design of K9 with objective to recover cubically varying w displacements over an equilateral patch of equal triangles. This recovery is readily achieved by ad hoc inverse method but raises the issue of uniqueness in design. In highlighting the characteristics of M3 it is evident that there remains a wealth of opportunity for further research before adaptive design of the element stiffness matrix, within an arbitrary prevailing w displacement field, can become a practical reality. An appendix lists the Fortran computer codings which are used in the examples to calculate the stiffness matrix K6 of the six-noded constant bending moment Kirchhoff triangle as well as the explicit transformation matrices for decomposition and restoration of the numeric K9 stiffness matrix. © 1998 John Wiley & Sons, Ltd.

Skeletal tilings of the plane into equal triangles

Skeletal tilings of the plane into equal triangles

Biwinged excision for closure of rounded defect.

A new technique for closure of rounded defects is described. The biwinged procedure consists of excising two equal triangles on either side of the defect, one above and one below the dissecting line. Closure is in the form of a step. It can be made to parallel in part a wrinkle or contour line, such as a nasolabial fold, a temporal region above and beside the eyebrow, an area over the lower border of the mandible, or on the dorsum of the hand. A mathematical comparison of the amount of healthy tissue in the biwinged excision and in six other procedures is presented.