1. a 60° Angle Cannot Be Trisected by Ruler and Compass
ثبت نشده
چکیده
§10.1. Number Fields and Field Extensions One of the classic non-constructability problems is that of trisecting an angle. Ruler and compass constructions exist for certain angles, such as 60, and constructions exist that approximately trisect any angle, but there are none that exactly trisect any angle. This is not an open question, waiting for some brilliant mathematician to solve it. It is a logical impossibility. One can be as sure that no such construction will be produced as one can be sure that no-one will ever find a solution to the equation 0.x = 1. We’ll demonstrate the impossibility of trisecting an angle by translating it into an algebraic problem, using the concept of a number field.
منابع مشابه
Geometric Cryptography Identi cation by Angle Trisection Draft
We propose the eld of geometric cryptography where messages and ciphertexts may be represented by geometric quantities such as angles or intervals and where computation is performed by ruler and compass constructions We describe a elegant little zero knowledge identi cation scheme based on the impossibility of trisecting an angle using ruler and compass operations While geometric cryptography m...
متن کاملConic Construction of a Triangle from the Feet of Its Angle Bisectors
We study an extension of the problem of construction of a triangle from the feet of its internal angle bisectors. Given a triangle ABC, we give a conic construction of points P which are the incenter or excenters of their own anticevian triangles with respect to ABC. If the given triangle contains a right angle, a very simple ruler-and-compass construction is possible. We also examine the case ...
متن کاملTriangles, Ruler and Compass
A triangle can be specified by giving three independent geometric relationships defined between its elements. Usually, these relationships are distances between two vertices, angles between two sides, and heights. For each triangle specified by a set of three of such relationships, we present a procedure that constructs the triangle using ruler and compass alone. ∗While on leave in Computer Sci...
متن کاملMagnetic Calibration of Three-Axis Strapdown Magnetometers for Applications in Mems Attitude-Heading Reference Systems
In a strapdown magnetic compass, heading angle is estimated using the Earth's magnetic field measured by Three-Axis Magnetometers (TAM). However, due to several inevitable errors in the magnetic system, such as sensitivity errors, non-orthogonal and misalignment errors, hard iron and soft iron errors, measurement noises and local magnetic fields, there are large error between the magnetometers'...
متن کامل1 4 N ov 2 00 1 Constructing Fresnel reflection coefficients by ruler and compass
A simple and intuitive geometical method to analyze Fresnel formulas is presented. It applies to transparent media and is valid for perpendicular and parallel polarizations. The approach gives a graphical characterization particularly simple of the critical and Brewster angles. It also provides an interpretation of the relation between the reflection coefficients for both basic polarizations as...
متن کامل