Longest Edge Research Articles

A proof of convergence and an error bound for the method of bisection in 𝑅ⁿ

Let S = ⟨ X 0 , . . , X m ⟩ S = \langle {X_0},..,{X_m}\rangle be an m-simplex in R n {{\mathbf {R}}^n} . We define "bisection" of S as follows. We find the longest edge ⟨ X i , X j ⟩ \langle {X_i},{X_j}\rangle of S, calculate its midpoint M = ( X i + X j ) / 2 M = ({X_i} + {X_j})/2 , and define two new m-simplexes S 1 {S_1} and S 2 {S_2} by replacing X i {X_i} by M or X j {X_j} by M. Suppose we bisect S 1 {S_1} and S 2 {S_2} , and continue the process for p iterations. It is shown that the diameters of the resulting Simplexes are no greater then ( 3 / 2 ) ⌊ p / m ⌋ {(\sqrt 3 /2)^{\left \lfloor {p/m} \right \rfloor }} times the diameter of the original simplex, where ⌊ p / m ⌋ \left \lfloor {p/m} \right \rfloor is the largest integer less than or equal to p / m p/m .

A lower bound on the angles of triangles constructed by bisecting the longest side

Let Δ A 1 A 2 A 3 \Delta {A^1}{A^2}{A^3} be a triangle with vertices at A 1 , A 2 {A^1},{A^2} and A 3 {A^3} . The process of "bisecting Δ A 1 A 2 A 3 \Delta {A^1}{A^2}{A^3} " is defined as follows. We first locate the longest edge, A i A i + 1 {A^i}{A^{i + 1}} of Δ A 1 A 2 A 3 \Delta {A^1}{A^2}{A^3} where A i + 3 = A i {A^{i + 3}} = {A^i} , set D = ( A i + A i + 1 ) / 2 D = ({A^i} + {A^{i + 1}})/2 , and then define two new triangles, Δ A i D A i + 2 \Delta {A^i}D{A^{i + 2}} and Δ D A i + 1 A i + 2 \Delta D{A^{i + 1}}{A^{i + 2}} . Let Δ 00 {\Delta _{00}} be a given triangle, with smallest interior angle α > 0 \alpha > 0 . Bisect Δ 00 {\Delta _{00}} into two new triangles, Δ 1 i , i = 1 , 2 {\Delta _{1i}},i = 1,2 . Next, bisect each triangle Δ 1 i {\Delta _{1i}} , to form four new triangles Δ 2 i , i = 1 , 2 , 3 , 4 {\Delta _{2i}},i = 1,2,3,4 , and so on, to form an infinite sequence T of triangles. It is shown that if Δ ∈ T \Delta \in T , and θ \theta is any interior angle of Δ \Delta , then θ ⩾ α / 2 \theta \geqslant \alpha /2 .

En Flintplads i Qatar

En Flintplads i Qatar

Flint Harpoon-Barbs

Though Sir John Evans described a series of flint implements as “single-barbed arrowheads,” the shape of the majority precluded any such explanation of their use, and it was not until Dr. A. E. Relph put forward the suggestion in “The Antiquary” for September, 1907, that these implements were in reality harpoon-barbs, that students of prehistoric archæology were satisfied that their nomenclature was correct. They are all triangular, the longest edge being thin and unworked, and this, it is suggested, was fixed in the side of a bone or wooden shaft, probably tipped by a triangular or tanged and barbed arrowhead. In most of the specimens the remaining two edges are carefully chipped, the longer being usually straight, and the shorter concave. This curved portion would be fixed pointing to the handle of the harpoon, the long sloping edge enabling the weapon to pierce the object aimed at, while the curved barb would prevent it slipping out.