ARCHIMEDES IN THE CLASSROOM

Eureka! Eureka! What better way to excite and engage students than to bring Archimedes into the classroom? Archimedes is widely regarded as the greatest mathematician of antiquity. Very little is known about Archimedes’ personal life. Archimedes was born in Syracuse, Sicily around 287 B.C., and he was the son of an astronomer. He probably studied in Alexandria, Egypt under followers of Euclid. Today we know Archimedes as a brilliant mathematician and scientist, he spent much of his career inventing war machines. Despite orders not to harm Archimedes, his life was cut short in 212 B.C. when the Romans invaded Syracuse and killed Archimedes at age 75, more information on the history of Archimedes can be found in the article, Archimedes, by Chris Rorres of New York University. This paper will take a closer look at some of Archimedes’ most brilliant discoveries, and how they can be brought into the classroom. Unlike Plato and other great minds before him, Archimedes solved problems with anything and everything. That type of innovation and outside of the box thinking is exactly the type of skills students will need to succeed in today’s world

The Trisection of an Arbitrary Angle: A Classical Geometric Solution

This paper presents an elegant classical geometric solution to the ancient Greek's problem of angle trisection.  Its primary objective is to provide a provable construction for resolving the trisection of an arbitrary angle, based on the restrictions governing the problem.  The angle trisection problem is believed to be unsolvable for compass-straightedge construction.  As stated by Pierre Laurent Wantzel (1837), the solution of the angle trisection problem corresponds to an implicit solution of the cubic equation x cubed minus 3x minus 1 equals 0, which is algebraically irreducible, and so is the geometric solution of the angle trisection problem.  The goal of the presented solution is to show the possibility to solve the trisection of an arbitrary angle using the traditional Greek's tools of geometry (a classical compass and straightedge) by changing  the problem from the algebraic impossibility classification to a solvable plane geometrical problem.  Fundamentally, this novel work is based on the fact that algebraic irrationality is not a geometrical impossibility.  The exposed methods of proof have been reduced to the Euclidean postulates of classical geometry

Mathematical Surprises

This is open access book provides plenty of pleasant mathematical surprises. There are many fascinating results that do not appear in textbooks although they are accessible with a good knowledge of secondary-school mathematics. This book presents a selection of these topics including the mathematical formalization of origami, construction with straightedge and compass (and other instruments), the five- and six-color theorems, a taste of Ramsey theory and little-known theorems proved by induction. Among the most surprising theorems are the Mohr-Mascheroni theorem that a compass alone can perform all the classical constructions with straightedge and compass, and Steiner's theorem that a straightedge alone is sufficient provided that a single circle is given. The highlight of the book is a detailed presentation of Gauss's purely algebraic proof that a regular heptadecagon (a regular polygon with seventeen sides) can be constructed with straightedge and compass. Although the mathematics used in the book is elementary (Euclidean and analytic geometry, algebra, trigonometry), students in secondary schools and colleges, teachers, and other interested readers will relish the opportunity to confront the challenge of understanding these surprising theorems

Mathematical Surprises

This is open access book provides plenty of pleasant mathematical surprises. There are many fascinating results that do not appear in textbooks although they are accessible with a good knowledge of secondary-school mathematics. This book presents a selection of these topics including the mathematical formalization of origami, construction with straightedge and compass (and other instruments), the five- and six-color theorems, a taste of Ramsey theory and little-known theorems proved by induction. Among the most surprising theorems are the Mohr-Mascheroni theorem that a compass alone can perform all the classical constructions with straightedge and compass, and Steiner's theorem that a straightedge alone is sufficient provided that a single circle is given. The highlight of the book is a detailed presentation of Gauss's purely algebraic proof that a regular heptadecagon (a regular polygon with seventeen sides) can be constructed with straightedge and compass. Although the mathematics used in the book is elementary (Euclidean and analytic geometry, algebra, trigonometry), students in secondary schools and colleges, teachers, and other interested readers will relish the opportunity to confront the challenge of understanding these surprising theorems

The last chapter of the Disquisitiones of Gauss

This exposition reviews what exactly Gauss asserted and what did he prove in the last chapter of {\sl Disquisitiones Arithmeticae} about dividing the circle into a given number of equal parts. In other words, what did Gauss claim and actually prove concerning the roots of unity and the construction of a regular polygon with a given number of sides. Some history of Gauss's solution is briefly recalled, and in particular many relevant classical references are provided which we believe deserve to be better known.Comment: 13 page

A historical survey of methods of solving cubic equations

It has been said that the labor-saving devices ot this modern age have been made possible by the untiring efforts of lazy men. While working with cubic equations, solving them according to the standard methods appearing in modern text-books on the theory of equations, it became apparent, that in many cases, the finding of solutions was a long and tedious process involving numerical calculations into which numerous errors could creep. Confessing to laziness, and having been told at an impressionable age that any fool can do it the hard way but it takes a genius to find the easy way , it became of interest to find a simpler method of solution. It eventually became clear that it is necessary to find out what has been done in the past to accomplish this. The information is found to be interesting and varied, but scattered among many sources. These sources are brought together here, not only in the hope that this history will be helpful in learning about the development of cubic equations, but also to challenge the reader to find solutions of his own, which will not only reduce the labor and errors involved, but also will explain the true mathematical meaning of a cubic equation. This history brings together the most important aspects in the development of solutions to the cubio equation and presents a selected list of literature and notes