Results 31 to 40 of about 154,492 (75)
Complex & Intelligent SystemsThe impact of airborne pollutants present in the environment, entering the body through breathing, can cause significant risks of respiratory and heart-related health problems for individuals.
Fengyu Guo +6 more
semanticscholar +1 more source
The development of Euclidean axiomatics: The systems of principles and the foundations of mathematics in editions of the Elements in the Early Modern Age. [PDF]
Arch Hist Exact Sci, 2016 De Risi V.
europepmc +1 more source
Heronian Triangles. I. Systematic relation among the isosceles Heronian triangles.
Heronian Triangles. I. Systematic relation among the isosceles Heronian triangles.application/pdf 紀要論文 A Heronian triangle (HeT) is defined here to be a triangle whose edges and area Δ are integers. A HeT (a, a, b) with coprime a and b is called a primitive isosceles Heronian triangle (piHeT). All the piHeT's are shown to be classified into two groups, I and II, depending on the value of 2a-b, and a pair of natural numbers, (n, k ...
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We study the classical Heronian triangle with side lengths 13–14–15 and its role as a building block for exact planar tessellations with 12-fold rotational symmetry. Starting from explicit Cartesian coordinates, we construct a fundamental “kite” formed by the 13–14–15 triangle and its excentral reflection of equal area, and show how this tile generates
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The Heronian Triangle 13–14–15 as an Arithmetic, Crystallographic, and CM–Elliptic Node
We show how the Heronian triangle 13–14–15 sits at a common nodebetween four worlds: (i) the family of consecutive triangles(a−1,a,a+1) with integer area and inradius, governed by Pell equationsin the real quadratic field Q(√3); (ii) an exact periodic “corona crystal”in the Euclidean plane, whose wallpaper group is p6m and whose reflectionsubgroup is ...
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This preprint revisits the Heronian rotational mandala built from the classical 13–14–15 triangle. On the one hand, the paper develops an exact geometric construction: a lattice tessellation based on the 13–14–15 triangle and its reflected / excentral variants, together with precise rotational patterns (of order 12) that can be stated and proved ...
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In a previous preprint the author constructed a hexagonal “corona crystal” built from the Heronian triangle 13–14–15 and showed that it sits at a natural node between Pell equations in the real quadratic field Q(√3), a biquadratic field Q(√2,√3), the cyclotomic field Q(ζ12), a wallpaper group of type p6m, and a CM elliptic curve with j-invariant 0 ...
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On lattice points which become vertices of Heronian triangles (Algebras, logics, languages and related areas)openaire
On a formula of parametric version for Heronian triangles (Logic, Algebraic system, Language and Related Areas in Computer Science)openaire