Congruent Triangles Research Articles

Construction of a Holliday Junction in Small Circular DNA Molecules for Stable Motifs and Two‐Dimensional Lattices

Design rules for DNA nanotechnology have been mostly learnt from using linear single-stranded (ss) DNA as the source material. For example, the core structure of a typical DAO (double crossover, antiparallel, odd half-turns) tile for assembling 2D lattices is constructed from only two linear ss-oligonucleotide scaffold strands, similar to two ropes making a square knot. Herein, a new type of coupled DAO (cDAO) tile and 2D lattices of small circular ss-oligonucleotides as scaffold strands and linear ss-oligonucleotides as staple strands are reported. A cDAO tile of cDAO-c64nt (c64nt: circular 64 nucleotides), shaped as a solid parallelogram, is constructed with a Holliday junction (HJ) at the center and two HJs at both poles of a c64nt; similarly, cDAO-c84nt, shaped as a crossed quadrilateral composed of two congruent triangles, is formed with a HJ at the center and four three-way junctions at the corners of a c84nt. Perfect 2D lattices were assembled from cDAO tiles: infinite nanostructures of nanoribbons, nanotubes, and nanorings, and finite nanostructures. The structural relationship between the visible lattices imaged by AFM and the corresponding invisible secondary and tertiary molecular structures of HJs, inclination angle of hydrogen bonds against the double-helix axis, and the chirality of the tile can be interpreted very well. This work could shed new light on DNA nanotechnology with unique circular tiles.

SAS and SSA Conditions for Congruent Triangles

SAS and SSA Conditions for Congruent Triangles

A new class of monohedral pentagonal spherical tilings with GeoGebra

By a monohedral spherical tiling we mean a decomposition of the sphere by geodesic congruent polygons. Here, making use of GeoGebra, a well known free interactive mathematics software, we show how to generate new classes of monohedral non-convex triangular and new non-convex pentagonal spherical tilings, changing the side gluing rules of the regular spherical tetrahedral tiling, by means of a local action of particular subgroups of spherical isometries. In both cases each face has \pi as area measure. In relation to the new class of pentagonal tilings, we describe some of their properties and show the existence, in a special case, of an associated dihedral triangular spherical tiling, that is, a tiling composed by two sets of congruent triangles. These classes of spherical tilings have emerged as a result of an interative construction process, only possible by the use of newly produced GeoGebra tools and the dynamic interaction capabilities of this software.

Distinguishing the Plane from the Punctured Plane Without Homotopy

This is a note about topology, meant to illustrate a specific technique. If you do not know what topology is about, think of it this way: if you look at two congruent triangles in the plane, you co...

Congruent triangles in arrangements of lines

We study the maximum number of congruent triangles in finite arrangements of l lines in the Euclidean plane. Denote this number by f (l) . We show that f (5) = 5 and that the construction realizing this maximum is unique, f (6) = 8 , and f (7) = 14 . We also discuss for which integers c there exist arrangements on l lines with exactly c congruent triangles. In parallel, we treat the case when the triangles are faces of the plane graph associated to the arrangement (i.e. the interior of the triangle has empty intersection with every line in the arrangement). Lastly, we formulate four conjectures.

Why Do All Triangles Form a Triangle?

We provide a geometric explanation, based on symmetry, for why the moduli space of all triangles up to similarity is itself a triangle. Symmetries occur because the lengths of the sides define triples in ℝ3 so are acted on by the symmetric group 핊3, which is isomorphic to the symmetry group 픻3 of an equilateral triangle. The moduli space for triangles is a fundamental domain for the action of 픻3 on an equilateral triangle in ℝ3 determined by all triangles with unit perimeter and is chosen from a subdivision into six congruent triangles. Isosceles and equilateral triangles occupy special locations determined by their symmetries. The sides of a right triangle lie on one of three double cones in ℝ3, and those of unit perimeter lie on a segment of a hyperbola in the moduli space.

Students\u2019 understanding of the structure of deductive proof

While proof is central to mathematics, difficulties in the teaching and learning of proof are well-recognised internationally. Within the research literature, a number of theoretical frameworks relating to the teaching of different aspects of proof and proving are evident. In our work, we are focusing on secondary school students learning the structure of deductive proofs and, in this paper, we propose a theoretical framework based on this aspect of proof education. In our framework, we capture students’ understanding of the structure of deductive proofs in terms of three levels of increasing sophistication: Pre-structural, Partial-structural, and Holistic-structural, with the Partial-structural level further divided into two sub-levels: Elemental and Relational. In this paper, we apply the framework to data from our classroom research in which secondary school students (aged 14) tackled a series of lessons that provided an introduction to proof problems involving congruent triangles. Using data from the transcribed lessons, we focus in particular on students who displayed the tendency to accept a proof that contained logical circularity. From the perspective of our framework, we illustrate what we argue are two independent aspects of Relational understanding of the Partial-structural level, those of universal instantiation and hypothetical syllogism, and contend that accepting logical circularity can be an indicator of lack of understanding of syllogism. These findings can inform how teaching approaches might be improved so that students develop a more secure understanding of deductive proofs and proving in geometry.

Discrete Differential Geometry of Triangles and Escher-Style Trick Art

This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles) on a triangular mesh consisting of congruent isosceles triangles. As an example, we perform global analysis of an Escher-style trick art, i.e., a simpler version of “Ascending and Descending”. After defining the local structure on the trick art, we analyze its global structure and attribute its paradox to a singular point (i.e., a singular triangle) at the center. Then, the endless “Penrose stairs” is described as a closed trajectory around the isolated singular point. The approach fits well with graphical projection and gives a simple and intuitive example of the interaction between global and local structures. We could deal with higher dimensional objects as well by considering n-simplices (n > 2) instead of triangles.

Automated registration of multi-view point clouds using sphere targets

This paper addresses the problem of automated registration of multi-view point clouds generated by a 3D scanner using sphere targets. First, sphere targets are detected from each point cloud. The centroids of the detected targets in each point cloud are then used for rough registration. Congruent triangles are computed from the centroids for the correspondence among them, with which a rigid body transformation is obtained to bring the two point clouds together as closely as possible. After the initial registration, the two point clouds are further registered by refining the position and orientation of the point clouds using the underlying geometric shapes of the targets. These registration steps are integrated into one system that allows two input point clouds automatically registered with no user intervention. Real examples are used to demonstrate the performance of the point cloud registration.

Medium-Voltage Drive for Induction Machine With Multilevel Dodecagonal Voltage Space Vectors With Symmetric Triangles

Multilevel inverters with dodecagonal (12-sided polygon) voltage space vector structure have advantages, such as complete elimination of fifth and seventh harmonics, reduction in electromagnetic interference, reduction in device voltage ratings, reduction of switching frequency, extension of linear modulation range, etc., making it a viable option for high-power medium-voltage drives. This paper proposes two power circuit topologies capable of generating multilevel dodecagonal voltage space vector structure with symmetric triangles (for the first time) with minimum number of dc-link power supplies and floating capacitor H-bridges. The first power topology is composed of two hybrid cascaded five-level inverters connected to either side of an open-end winding induction machine. Each inverter consists of a three-level neutral-point-clamped inverter, which is cascaded with an isolated H-bridge making it a five-level inverter. The second topology is for a normal induction motor. Both of these circuit topologies have inherent capacitor balancing for floating H-bridges for all modulation indexes, including transient operations. The proposed topologies do not require any precharging circuitry for startup. A simple pulsewidth modulation timing calculation method for space vector modulation is also presented in this paper. Due to the symmetric arrangement of congruent triangles within the voltage space vector structure, the timing computation requires only the sampled reference values and does not require any offline computation, lookup tables, or angle computation. Experimental results for steady-state operation and transient operation are also presented to validate the proposed concept.

A Triangle Theorem

SummaryThe medians of a triangle divide it into six parts, which can be reassembled into three congruent triangles.

Prospective Teachers' Competency in Teaching how to Compare Geometric Figures: The Concept of Congruent Triangles as an Example

Prospective Teachers' Competency in Teaching how to Compare Geometric Figures: The Concept of Congruent Triangles as an Example

기하 증명 구성에 나타나는 학생들의 사고과정 탐색

This study aims to explore secondary students' thinking while doing proof in geometry. Two secondary students were interviewed and the interview data were analyzed. The results of the analysis suggest that the two students similarly showed as follows: a) tendencies to use the rules of congruent and similar triangles to solve a given problem, b) being confused about the rules of similar and congruent triangles, and c) being confused about the definitions, partition and hierarchical classification of quadrilaterals. Also, the results revealed that a relatively low achieving student has tendency to rely on intuitive information such as visual representations.

General triangular midpoint subdivision

In this paper, we introduce triangular subdivision operators which are composed of a refinement operator and several averaging operators, where the refinement operator splits each triangle uniformly into four congruent triangles and in each averaging operation, every vertex will be replaced by a convex combination of itself and its neighboring vertices. These operators form an infinite class of triangular subdivision schemes including Loop's algorithm with a restricted parameter range and the midpoint schemes for triangular meshes. We analyze the smoothness of the resulting subdivision surfaces at their regular and extraordinary points by generalizing an established technique for analyzing midpoint subdivision on quadrilateral meshes. General triangular midpoint subdivision surfaces are smooth at all regular points and they are also smooth at extraordinary points under certain conditions. We show some general triangular subdivision surfaces and compare them with Loop subdivision surfaces.

Tiling Convex Polygons with Congruent Equilateral Triangles

We study the sets $\mathcal{T}_{v}=\{m \in\{1,2,\ldots\}: \mbox{there is a convex polygon in }\mathbb{R}^{2}\mbox{ that has }v\mbox{ vertices and can be tiled with $m$ congruent equilateral triangles}\}$ , v=3,4,5,6. $\mathcal{T}_{3}$ , $\mathcal{T}_{4}$ , and $\mathcal{T}_{6}$ can be quoted completely. The complement $\{1,2,\ldots\} \setminus\mathcal{T}_{5}$ of $\mathcal{T}_{5}$ turns out to be a subset of Euler’s numeri idonei. As a consequence, $\{1,2,\ldots\} \setminus\mathcal{T}_{5}$ can be characterized with up to two exceptions, and a complete characterization is given under the assumption of the Generalized Riemann Hypothesis.

Star Covering Region Evaluation with Application to Star Tracker Design

The performance of a star tracker is closely linked to the characteristics of the guide star catalog. It is an essential part of design to estimate the number of guide stars observed by the star tracker in any given boresight direction. This paper reports on the star covering region evaluation method, which is used to find ranges of orientation where the star tracker can observe a star. With this method, a conservative number of guide stars observed by the star tracker in any given boresight direction can be quantitatively evaluated and credible ranges of orientation can be found, where the star tracker can observe enough guide stars to perform autonomous star identification; thus, verifying the design of the star tracker. The celestial sphere is divided into 131,072 approximately congruent spherical triangles, such that the star’s covering region can be approximately depicted by triangles in which the circumcenters are close enough to the star. Then, the stars observed by the tracker, in which the boresight is located anywhere inside the circumcircle of the triangle, can be found by identifying where the triangle is located in a given star’s covering region. A guide star catalog containing 2,873 stars was generated and the star covering region evaluation was performed. It was found that the star tracker with a circular field of view (FOV) radius of 4° could measure at least three guide stars when its boresight direction was located in 63,554 spherical triangles. The evaluation results for 10,000 random orientations were used to compare the counting numbers of practically observed stars. The mean error is 0.6415 and the maximum error is 5, which results from the stars not considered because they are on the edge of the FOV. The results of a simulation have validated the method.

Hitting probabilities for random convex bodies and lattices of triangles

In the first part of this paper, we obtain symmetric formulae for the probabilities that a plane convex body hits exactly 1, 2, 3, 4, 5 or 6 triangles of a lattice of congruent triangles in the plane. Furthermore, a very simple formula for the expectation of the number of hit triangles is derived. In the second part, we calculate the hitting probabilities in the cases where the convex body is a rectangle, an ellipse and a half disc. Already known results for a line segment (needle) follow as special cases of the rectangle and the ellipse.

Reasoning-and-proving in geometry in school mathematics textbooks in Japan

In Japan it is in Grades 7–9, and primarily in geometry, that school students are introduced to the significance and methodology of proof in mathematics. As textbooks play a central role in everyday lessons in Japan, this paper presents an analysis of the geometry chapters of a selected mathematics textbook currently in common use with students aged 13–14 in Grade 8. We show that the emphasis in the textbook is on ‘direct’ proofs of geometrical statements, accompanied by activities which encourage students to form conjectures. Based on our analysis, we raise critical issues related both to the strengths and weaknesses of such a textbook design on students’ understanding of reasoning-and-proving. The strengths, as evidenced by Japanese national data, are that most Grade 8 students in Japan are able to construct suitable proofs – usually based on congruent triangles. The weaknesses, as verified by other research, are that the same students may not fully appreciate the necessity or generality of mathematical proof.

Spherical tiling by 12 congruent pentagons

The tilings of the 2-dimensional sphere by congruent triangles have been extensively studied, and the edge-to-edge tilings have been completely classified. However, not much is known about the tilings by other congruent polygons. In this paper, we classify the simplest case, which is the edge-to-edge tilings of the 2-dimensional sphere by 12 congruent pentagons. We find one major class allowing two independent continuous parameters and four classes of isolated examples. The classification is done by first separately classifying the combinatorial, edge length, and angle aspects, and then combining the respective classifications together.

Tilings of Convex Polygons with Congruent Triangles

By the spectrum of a polygon A we mean the set of triples (α,β,γ) such that A can be dissected into congruent triangles of angles α,β,γ. We propose a technique for finding the spectrum of every convex polygon. Our method is based on the following classification. A tiling is called regular if there are two angles of the triangles, α and β such that at every vertex of the tiling the number of triangles having angle α equals the number of triangles having angle β. Otherwise the tiling is irregular. We list all pairs (A,T) such that A is a convex polygon and T is a triangle that tiles A regularly. The list of triangles tiling A irregularly is always finite, and can be obtained, at least in principle, by considering the system of equations satisfied by the angles, examining the conjugate tilings, and comparing the sides and the area of the triangles to those of A. Using this method we characterize the convex polygons with infinite spectrum, and determine the spectrum of the regular triangle, the square, all rectangles, and the regular N-gons with N large enough.