<abstract><p>Among the triangle congruence axioms, side-side-angle (SsA) axiom states that two triangles are congruent if and only pairs of corresponding sides angles opposite longer equal. We construct sequences in which items satisfy a modified condition. require shorter be The locus intersection points other is derived to hyperbola, generalized form defined by complete quadrilate...

An r-gentiling is a dissection of a shape into r ≥ 2 parts which are all similar to the original shape. An r-reptiling is an r-gentiling of which all parts are mutually congruent. This article shows that no acute tetrahedron is an r-gentile or r-reptile for any r < 9, by showing that no acute spherical diangle can be dissected into less than nine acute spherical triangles.

Current neutrino oscillation data indicate that the 3 × 3 lepton flavor mixing matrix V is likely to be symmetric about its Ve3-Vμ2-Vτ1 axis. This offdiagonal symmetry corresponds to three pairs of congruent unitarity triangles in the complex plane. Terrestrial matter effects can substantially modify the genuine CP -violating parameter and off-diagonal asymmetries of V in realistic long-baselin...

We consider the set of triangles in the plane with rational sides and a given area A. We show there are infinitely many such triangles for each possible area A. We also show that infinitely many such triangles may be constructed from a given one, all sharing a side of the original triangle, unless the original is equilateral. There are three families of triangles (including the isosceles ones) ...

Marching triangles is a method for producing a polygon mesh surface approximation composed of triangular facets which are approximately equilateral. This paper improves the Marching Triangles algorithm where the inputs are multiple range images of scenes. C discontinuities (fold edges) are detected and used to constrain the final triangulation, thus increasing the accuracy of the mesh at sharp ...

Uniform polyhedra consist of regular faces and congruent vertices. Allowing for nonconvex faces and vertex figures, there are 75 such polyhedra, as well as 2 infinite families of prisms and antiprisms. A recently discovered uniform way of computing their vertex coordinates is the basis for our program to display all of these solids, among which are many beautiful and stunning shapes. We will al...

By a dihedral folding tiling (f-tiling, for short) of the sphere S whose prototiles are spherical right triangles, T1 and T2, we mean a polygonal subdivision τ of S such that each cell (tile) of τ is congruent to T1 or T2 and the vertices of τ satisfy the angle-folding relation, i.e., each vertex of τ is of even valency and the sums of alternating angles around each vertex are π. In fact, the c...

An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T , overlapping only at their boundaries. The triangle T is the “tile”. The tile may or may not be similar to ABC. This paper is the first of four papers, which together seek a complete characterization of the triples (ABC,N, T ) such that ABC can be N-tiled by T . In this paper, we consider...

For a point O on the segment AB in the plane, the area surrounded by the three semicircles with diameters AO, BO and AB erected on the same side is called an arbelos. It has lots of unexpected but interesting properties (for an extensive reference see [1]). The radical axis of the inner semicircles divides the arbelos into two curvilinear triangles with congruent incircles called the twin circl...

It is well-known that the Kresling pattern of congruent triangles can be arranged either circularly on a cylinder revolution or in helical way. In both cases resulting cylindrical structures are multi-stable. We generalize these arrangements with respect to cones revolution, where our approach allows construct structures, which snap between conical realizations whose apex angles serve as design...