# Detailed Description

**Module Objectives**: All of the standard content of high school geometry is part of *Euclidean geometry*, a branch of mathematics that derives its name from the fact that the first systematic treatment of this content was presented in *Euclid’s Elements* (circa 300 B.C.). Euclid did not discover all of the results of Euclidean geometry but he was the first to organize them as an axiomatic system based on what he termed *definitions*, *postulates* and *common notions*.

From this axiomatic basis, Euclid derived essentially all of the known geometry of his day as well as substantial parts of the theory of numbers and the theory of measurement. The first three of Euclid’s five postulates dealt with geometric constructions with compass and straightedge, and essentially defined what he meant by a *geometric construction*.

Geometric constructions were a central part of his development of plane and solid geometry. Given that Euclidean geometry is still the basis of modern high school geometry, it is not surprising that compass and straightedge constructions are still an important part of the geometry curriculum.

In recent years, computer programs such as **Geometer’s Sketchpad** and graphing calculators such as **TI- Nspire** allow teachers and students to carry out the geometric constructions of Euclidean geometry in an accurate, attractive and dynamic graphical environment. As a result, these technological tools have not only enhanced the relevance of compass and straightedge constructions in teaching geometry but also have led to applications of geometry to other fields.

The Greek geometers of Euclid’s time solved many difficult geometric construction problems. They also identified other geometric construction problems that they were not able to solve and that they even suspected to be impossible with straightedge and compass alone. Among the most famous geometric construction problems that were suspected to be impossible by the Greeks were the trisection of an

*arbitrary*angle (“The Angle Trisection Problem”), the construction of the side of a cube with twice the volume of a given cube (“The Doubling of the Cube” Problem), and the construction of a square of equal area to a given circle (The Squaring of A Circle” Problem).

Most teachers of geometry have heard that these and other constructions are indeed impossible because their impossibility is often mentioned in high school and college geometry books. However, even in college geometry books, these impossibility results are usually not proved because "the proofs fall outside the scope of this book". Such remarks give the impression that these results are beyond the comprehension of undergraduate mathematics majors. However, the real culprit is that undergraduate mathematics courses in geometry and abstract algebra are often too compartmentalized. The mathematics needed to discuss the possibility or impossibility of geometric constructions is actually not difficult but it does involve properties of the real number system and properties of polynomials, topics that are not usually discussed in college geometry courses, even those designed for prospective high school teachers.

This module develops this simple and elegant connection between the geometric constructions of high school geometry and the algebraic properties of the real number system. You will not only learn why the famous "impossible" geometric constructions are indeed impossible, but also more about geometry, the real number system and polynomials, all of which are useful for teaching geometry at the high school and college level.

**Structure of the Module**: This module is divided into the following two units:

Unit 1: Discovering geometry through geometric constructions.

- Part 1: Geometric construction basics.
- Part 2: Geometer's Sketchpad tutorial.
- Part 3: Constructing regular polygons.
- Part 4: Some famous impossible geometric constructions.

Unit 2: Geometric constructions and the real number system.

- Part 1: Constructible real numbers.
- Part 2: Euclidean constructions and constructible real numbers.
- Part 3: The impossibility of certain geometric constructions.

**Module Completion Requirements**: There is an assignment for each of the seven parts of the two units of this module. Each of these assignments is to be submitted to the module instructors for evaluation and feedback. Teachers enrolled in this module for University of Illinois Continuing Education Units are required to satisfactorily complete these seven assignments.

Teachers enrolled for University of Illinois graduate credit, or undergraduate secondary mathematics education majors enrolled for University of Illinois undergraduate credit, must also complete an approved Final Project.