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[3] viXra:2407.0150 [pdf] submitted on 2024-07-25 05:57:54
Authors: G. Freeman
Comments: 18 Pages.
We explore the practical application of isoperimetric inequality (L² ≥ 4πA) to classical methods of circle measurement. Exampling Archimedes' n-gon approach, we compare it to a real-world kinematic scenario of a unit diameter circle rolling on a flat plane surface. Using annular geometry, we demonstrate that π can be derived algebraically by solving for the linear distance its centre travels per full revolution. Unlike exhaustive methods involving non-circular figures, our annular approach begins with isoperimetric equality by deriving a right triangle (with π its lone unknown side) & applying the Pythagorean theorem to it. This algebraic approach to π reveals unexpected yet significant connections between it and the golden ratio. We further explore more assumptions underlying 3.14159... discovering its embedment in an unbounded plane to be catastrophic & remedy with a bounded one. Finally, we close with a fresh new perspective on the notoriously unsolved Riemann Hypothesis problem. Our result suggests both a need for physical experimentation, as well as a need to re-evaluate the general reliability of non-circular methods in rigorously bounding and/or converging on the circle constant π.
Category: Geometry
[2] viXra:2407.0083 [pdf] submitted on 2024-07-12 03:43:41
Authors: James A. Smith
Comments: 9 Pages.
Using GA's capacities for rotating and reflecting vectors, we solve the classic 2-D version of the Snellious-Pothenot surveying problem. The method used here provides two solutions, which can be averaged to better estimate the location of the unknown point P. A link to a GeoGebra worksheet of the solutions is provided so that the reader may test the validity of the method.
Category: Geometry
[1] viXra:2407.0030 [pdf] submitted on 2024-07-04 07:48:03
Authors: Andrey V. Voron
Comments: 4 Pages.
A non-formulaic method has been found for calculating Euler parallelepipeds of the second family based on the values of Pythagorean triples of Euler parallelepipeds of the first family, the largest common divisors. To do this, three triangles with integer values of the sides are allocated in the figure. Next, Pythagorean triples are determined from the obtained triangles by selecting the values of their greatest common divisors. These triples are entered in the table. By using a cross-arrangement in the table of two values (out of three) of Pythagorean triples (using the described algorithm of mathematical operations), the values of the three sides of the "derivative" Euler parallelepiped are calculated.
Category: Geometry
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