We define hyperbolic Heron triangles (hyperbolic with “rational” side-lengths and area) parametrize them in two ways via rational points of certain elliptic curves. show that there are infinitely many one angle α area A for any (admissible) choice A; particular, the congruent number problem has always solutions setting. also explore question a median bisector (median splitting triangle half).

Cognitive control is by now a large umbrella term referring collectively to multiple processes that plan and coordinate actions to meet task goals. A common feature of paradigms that engage cognitive control is the task requirement to select relevant information despite a habitual tendency (or bias) to select goal-irrelevant information. At least since the 1970s, researchers have employed propo...

A Heron quadrilateral is a cyclic quadri­ lateral whose area and side lengths are rational. In this work, we establish a correspondence between Heron quadri­ laterals and a family of elliptic curves of the form y2 = 3 2 x + αx2 − n x. This correspondence generalizes the no­ tions of Goins and Maddox who established a similar connec­ tion between Heron triangles and elliptic curves. We further s...

A hinged dissection of a set of polygons S is a collection of polygonal pieces hinged together at vertices that can be rotated into any member of S. We present a hinged dissection of all edge-to-edge gluings of n congruent copies of a polygon P that join corresponding edges of P . This construction uses kn pieces, where k is the number of vertices of P . When P is a regular polygon, we show how...

For a given triangle T and a real number ρ we define Ceva’s triangle Cρ(T ) to be the triangle formed by three cevians each joining a vertex of T to the point which divides the opposite side in the ratio ρ ∶ (1 − ρ). We identify the smallest interval MT ⊂ R such that the family Cρ(T ), ρ ∈ MT , contains all Ceva’s triangles up to similarity. We prove that the composition of operators Cρ, ρ ∈ R,...

For a given triangle T and a real number ρ we define Ceva’s triangle Cρ(T ) to be the triangle formed by three cevians each joining a vertex of T to the point which divides the opposite side in the ratio ρ ∶ (1 − ρ). We identify the smallest interval MT ⊂ R such that the family Cρ(T ), ρ ∈ MT , contains all Ceva’s triangles up to similarity. We prove that the composition of operators Cρ, ρ ∈ R,...

a r t i c l e i n f o a b s t r a c t 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 penta...

Mohammed Ben Alhocain, in an Arab manuscript of the 10th century, stated that the principal object of the theory of rational right triangles is to find a square that when increased or diminished by a certain number, m becomes a square [Dickson LE (1971) History of the Theory of Numbers (Chelsea, New York), Vol 2, Chap 16]. In modern language, this object is to find a rational point of infinite ...