Geometric Constructions

    This module is designed to familiarize teachers with the close connection that exists between compass and straightedge constructions and the real number system. Most geometry teachers are aware that certain geometric constructions such as trisecting certain angles or squaring a circle (i.e. constructing a square whose area is equal to that of a given circle) are known to be impossible, but they may not know why.
    This module explains the relationship between geometric constructions and a set of real numbers called the constructible real numbers, and uses this relationship to decide the possibility or impossibility of certain geometric constructions. The methods used in this module avoid the intricate machinery that is sometimes used in discussing these problems in college level courses in abstract algebra.
    This module is divided into the following units:

    Although these units deal with topics directly taught high school mathematics, their content is not intended for use directly in the high school classroom. However, teachers will find that the content of these units directly informs and influences their teaching of geometry in the high school classroom.

    What do we mean by a geometric construction?

    A geometric construction is a finite succession of compass and straightedge constructions that begins with a given geometric figure G (e.g. a line segment AB, a triangle ABC or an angle ABC), and ends with a desired geometric figure F (e.g. the circle F centered at the midpoint of the given line segment AB with AB as its diameter, the circle F passing through the vertices of the given triangle ABC, or the trisector of the given angle ABC).
    The mathematical heart of this module is that the finite succession of straightedge and compass constructions used to construct a required geometric figure F from a given figure corresponds exactly to a finite expanding succession of subfields of the field of all real numbers. In each successive step of the geometric construction, a given subfield F of the field R of real numbers is extended to a subfield F[c] which contains F as well as all numbers of the form p + q Sqrt(c) where p, q and c are numbers in F.

    Some familiarity with the concept of fields and subfields of the real number system should be considered prerequisite background for this module. Teachers who have taken a course in abstract algebra or other college level courses that discussed number fields should have the necessary background for this module. The content and homework of the module will give such teachers an opportunity review, apply and extend their previous knowledge of number fields. 

Credit: 2 graduate semester hrs.

Common Core Standards for Mathematical Practice that are emphasized include:

    See Step-by-Step Instructions for more information about enrollment options and instructions for completing this module.    The author of this module is Tony Peressini, Professor Emeritus of Mathematics at the University of Illinois. This module was completed in March 2003.

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.

Unit 2: Geometric constructions and the real number system.

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.

Required Materials

     In addition to the general requirements for participating in Math Teacher Link, this module also has the following requirement:

 

Step by Step Instructions

Enrollment and Credit Options:
    Mathematics teachers who have completed an undergraduate major in mathematics or mathematics education or the equivalent can enroll in this module for two semester hours of University of Illinois Graduate Credit.  Practicing mathematics teachers in Illinois have the option of enrolling or six Continuing Education Units, which would be applicable to teacher recertification renewal in Illinois. Within 5 working days of completion of enrollment, these teachers will be issued a log-in and password that will provide them access to the Module Working Environment where they submit required assignments and retrieve the graded results.
    Mathematics teachers may also use the online materials as an MTL Guest (free). As a guest you have access to the instructional materials but no access to instructional support, or to the Module Working Environment.

    Teachers enrolled for graduate credit must complete and submit the seven assignments for the module described in Steps #1 through #7 below and an approved Final Project (See Step #8 below for more details about this requirement.) 

Estimated Time Requirements for completion of the module:  We estimate that the work for this module will take participating teachers an average of about 90 hours to complete. Technically, you have a 16-week enrollment period to complete the module after you enroll in that unit. However, we recommend that you have as a goal to complete each of the two units of the module in eight weeks or less. That should be possible even when you are teaching if you set aside about six hours each week to work on the module.

 After you have registered for Module 13 follow these steps:
    This module is divided into the following units:

•    Unit 1: Discovering geometry through geometric constructions. Steps 1-4
•    Unit 2: Geometric constructions and the real number system. Steps 5-8

Step 1: Geometric Construction Basics

Assignment 1.1: Geometric Construction Basics (This assignment is designed to establish the technical meaning of straightedge and compass construction in Euclidean geometry.)

Instructions: Assignment 1.1 must be submitted by e-mail to both of the module instructors at the following e-mail addresses:
    “Anderson, Thomas E” teanders@illinois.edu
    “Peressini, Anthony L” anthonyperessini@gmail.com
with subject line: Module 13 Assignment 1.1

Note: All other module assignments are submitted electronically through the Module Hand-In System.  

 What do we mean by geometric constructions?

    The first three postulates in Book I of Euclid’s Elements describe which constructions were allowed in his development of geometry:

•    Postulate 1: A straight line can be drawn from any point to any other point.
•    Postulate 2: A finite straight line can be produced continuously in a straight line.
•    Postulate 3: A circle can be described with any center and radius.

    For Euclid and the other mathematicians of his time, the construction of the lines in Postulates 1 and 2 were carried out with an unmarked straightedge, while the circle in Postulate 3 was described with a compass. They restricted the use of these tools to the following specific tasks:

  1. Given two points P and Q in the plane, a straightedge can be used to construct the line passing through P and Q.
  2. Given a point P and a line segment RS of length |RS| = r, a compass can be used to construct a circle centered at P of radius r.

    A geometric figure F is constructible from a given geometric figure G, if all of the components of F can be obtained from those in G in a finite sequence of constructions of the sort described in 1) and 2) above. We say that the figure F results from the figure G by a geometric construction and the finite sequence of constructions is called a construction algorithm for G given F.
    This meaning for the term geometric construction is still current today! Although we now have computer software such as Geometer’s Sketchpad that can replace the compass and straightedge for actually carrying out geometric constructions, such software is designed to produce constructions that are exactly the same as those that can be achieved with a straightedge and compass. Such computer programs also have capabilities to:

•    Measure angles, lengths, distances, and areas in a figure.
•    Enhance the display of a geometric figure with shading, color, labels, and explanatory text.
•    Perform geometric transformations such as rotations, translations, dilations and reflections on geometric figures.
•    Change a geometric construction dynamically, that is, change the positions and distances between components of the given figure while still maintaining the prescribed relationships between constructed components of the figure.

    Nevertheless, the geometric constructions that can be produced by these high-tech wonders are, by design, essentially the same in mathematical content as those achievable with a simple compass and straightedge.

Why are geometric constructions important for geometry students and teachers?

    Trying to learn geometry without using geometric construction is like trying to learn chemistry or biology without using laboratories. Basic knowledge and skills on geometric constructions help students to discover and explore geometric relationships and interpret geometric concepts and theorems. They can also help the teachers to transform the static and confusing array of definitions and theorems typically found in geometry textbooks into an active and exploratory investigation of geometric relationships. Computer-based geometric construction programs such as Geometers Sketchpad enable both students and teachers to explore geometric relationships dynamically and to create very complex and yet very precise geometric constructions and diagrams.
    As any mathematics teacher knows, and as most students soon come to realize, geometric figures are nearly always helpful for the analysis and solution of real world problems and for learning new mathematical ideas. Although the geometric constructions as we have defined them are not, strictly speaking, necessary for such graphical descriptions of problems or ideas, their precision can often reveal aspects of the problem or idea that may not be evident in an informal paper or blackboard sketch. Teachers usually find that extra time spent in producing excellent graphical representations and figures for class handouts and exams is rewarded by better student understanding and interpretation of problems and ideas.

Just Do It 1.1.1

1.1.1 Just Do It!

            The following diagram displays a sequence of geometric constructinss that begins with a right triangle with sides of length 1. A second right triangle is then constructed with the hypotenuse of the first triangle as one side and the second side of length 1. This process is continued through six steps to produce the following diagram:

            

Question :What are the lengths of the segments marked with the question marks?

Give this question a little thought with pencil and paper because you will find that it has an interesting solution. After you have thought it through you can check your work below.

JDI 1.1.1 Answer

The lengths of the successive segments marked with ? in clockwise order are

If you continued that process indefinitely, the length of the segment with the nth-question mark would be Isn’t that pretty? As we will show later, this also tells us something interesting about the numbers that can be lengths of line segments in a geometric construction.

Assignment 1.1

Assignment Completion and Submission Directions:

    To complete this assignment, you will need the following items: Several sheets of plain white paper, a pencil-point compass to draw circles centered at any point, any convenient straightedge to draw lines, a protractor to measure angles, a ruler to measure lengths, and a fine point black pen for labeling constructions and writing text.
    Using these construction and measurement tools, carry out each of the following construction problems on separate sheets of paper. Be sure to label the diagrams clearly and precisely. Write careful descriptions of the construction algorithm you use as well as the requested conjectures on the same page as the constructions:

Problem 1. Given a line segment AB:
 i) construct the midpoint of AB;
 ii) construct an equilateral triangle with side AB;
iii) For a given point P not on AB, construct a line segment PQ parallel to AB such that |AB|=|PQ|. On the same sheet of paper, write two geometric conjectures based on the resulting diagram.  

Sample Solution for Problem 1(i)

Problem 2. Given an acute angle ABC with vertex B:
i) Construct the ray BD from B that bisects the angle ABC;
ii) Construct an isosceles triangle with ABC as one base angle;
iii) Construct a parallelogram ABCD with angle ABC as one interior angle. Then measure the approximate lengths of the side and diagonals of the parallelogram ABCD. On the same sheet of paper, write two geometric conjectures based on the diagram.
 
Problem 3. Given a triangle ABC:
 i) Construct its medians;
 ii) Construct its angle bisectors;
iii) Construct its altitudes.
Write conjectures at least two conjectures about the resulting diagram.
 iv) Construct the circles inscribed in triangle ABC and circumscribed about triangle ABC. Observe, conjecture and test conjectures.

Problem 4. Given a quadrilateral ABCD, construct the quadrilateral whose vertices are the successive midpoints of the four sides. Write two geometric conjectures and use the measurement tools to test them.

When you finish this assignment, scan the pages for Problems 2, 3, 4 and attach them to an e-mail message with subject line Assignment 1.1 to the instructors at the following e-mail addresses:
    “Anderson, Thomas E” teanders@illinois.edu
    “Peressini, Anthony L” anthonyperessini@gmail.com

Congratulations, you have just completed Assignment 1.1!

Step 2: Geometer's Sketchpad Tutorial

Comments on Step #2: If you already able to use all of the menu capabilities of Geometer's Sketchpad, Version 3.0 or higher,proceed directly to Assignment 1.2. Be sure to follow the directions given at the beginning of the assignment for completing and submitting Assignment 1.2

Note: Facility with Geometers Sketchpad is a prerequisite for this entire module. If you do not have this prerequisite skill, you can acquire it either completing Module 4: Using the Geometer's Sketchpad for graduate  credit, or as an MTL Guest (free!) for that module and completing it on your own. If you have used Geometer's Sketchpad in the past but currently do not use it frequently, we strongly recommend that you refresh your background by completing Module 4 as an MTL Guest.

For this module The Geometer's Sketchpad application (version 3 or higher) needs to be installed on the computer you will use.
Go to the Geometer's Sketchpad home page of Key Curriculum Press, the makers of the software.  This page has information, resources and support for The Geometer's Sketchpad software. Follow thei information and pricing link to order the current version of Geometer's Sketchpad software for either your Mac or Windows machine ($69.95 - single user). Install the program on your machine according to the instructions that come with the program.

Once Geometer's Sketchpad is on your computer and you are familiar it either from Module 4: Using the Geometer's Sketchpad or elsewhere please complete the following assignments:

Just Do It 1.2.1

In 1471, Regiomontanus posed the following problem in a letter to a professor at the University of Erhart "At what point on the ground does a perpendicularly suspended rod appear largest? (See Trigonometrric Delights by Eli Maor, Princeton University Press, 1998 ISBN 0-691-o5754-0). Equivalent forms of the problem appear in most calculus books, often in the context of viewing a picture on the wall. Here is a diagram with a more precise description of the problem:

 

 

Click here for an animation of the problem that should help you to formulate a conjecture.

Question Can you prove the conjecture by making use of results from high school geometry?

Assignment 1.2

Assignment Completion and Submission Directions: Prepare a single Microsoft Word document with GSP figures inserted that presents the problems in Assignment 1.2 in a manner that is

  • technically correct and clearly described, and
  • attractively displayed and organized.

Download the template Word document by clicking on the link "assg_1.2_templete.doc" at the bottom of this page. Use a separate page for each problem.  Your grade on this assignment will be based on the extent to which the file you submit meets both of these criteria.

When you complete Assignment 1.2, submit your file through the Module Working Environment  (also referred to as Moodle). Select Module 13 and enter your log-in and password. Then follow the directions there for submitting your assignment.

Problem 1. Given a line segment AB, use only the compass and straightedge constructions tools (i.e. Point, Line, Circle) in Geometer's Sketchpad to:

i) construct the midpoint of AB;
ii) construct an equilateral triangle with side AB;
iii) For a given point P not on AB, construct a line segment PQ parallel to AB such that |AB|=|PQ|

Problem 2. Given an angle ABC with vertex B, use only the compass and straightedge constructions in Geometer's Sketchpad to:

i) construct the ray BD from B that bisects the angle ABC.
ii) for an acute ABC, construct an isosceles triangle with ABC as one base angle.
iii) construct a parallelogram ABCD with ABC as one interior angle. Then use the Measure menu to measure the angles, lengths of the sides and diagonals of ABCD. State and verify at lest two geometric conjectures based on your measurements.

Problem 3. Given a triangle ABC, use only the compass and straightedge construction tools (i.e. Point, Line, Circle) in Geometer's Sketchpad to:

i) Construct its medians, its angle bisectors and its altitudes. State at least two geometric conjectures based on these constructions. Then verify these conjectures by using the Measure menu.
ii) Construct the circles inscribed in triangle ABC and circumscribed about triangle ABC. State at least two geometric conjectures based on these constructions. Then verify these conjectures by using the Measure menu.

Problem 4. Given a quadrilateral ABCD, construct the quadrilateral PQRS whose vertices P, Q, R, and S are the successive midpoints of the four sides. Observe, conjecture and test conjectures

Congratulations! You have finished Assignment 1.2. Submit the Assignment 1.2 file to us at Math Teacher Link according to the instructions at the beginning of the assignment and then begin Part 3 of this unit.

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Step 3: Constructing Regular Polygons

Comments on Step #3:

     Regular polygons are a standard topic in the geometry component of the middle school and high school curriculum. Although geometry books often discuss the construction of equilateral triangles, squares, hexagons and octagons, the general question of constructing regular n-gons for integers n > 2 is not considered. The purpose of this section is to discuss most of what is known about this fascinating subject. (By the way, although we will show in Unit 2 that a regular septagon (7-gon) cannot be constructed with compass and straightedge alone, a middle school student discovered the remarkably close approximation displayed on the Unit 2 home page!)

What do we mean by constructing a regular polygon with n sides?

     For each postive integer n > 2, a regular polygon with n sides (or simply a regular n-gon) is a polygon in a plane whose vertices lie on a circle C in a plane and whose sides are congruent line segments. If A is the center of the circle C, then A is the center of the regular n-gon. The radius of the regular n-gon is the radius of the circle C. We also use the term radius to refer to any line segment joining the center of a regular n-gon to one of its vertices.

     Our objective in this part of Unit 1 is to discuss the following problem:

Construction of the Regular n-Gon Problem:

     Given a line segment AB and a positive integer n > 2, construct a regular polygon RP[n] with n sides such that A is the center of the polygon and |AB| is the common length of all the line segments joining the center A to any vertex B of RP[n].

     In 1.3.1 Just Do It! we showed that, given a line segment AB, it is possible to construct a regular polygon RP[3], (i.e. an equilateral triangle) with radii of length |AB|.

     In Problem 1 of Assignment 1.3, you are asked to contruct RP[4] (a square) with a given radius. In Problem 2 of Assignment 1.3, you are asked to show that for each integer n > 2, RP[n] can be constructed with radii lengths equal to that of a given segment AB if and only if RP[2n] can be so constructed. Therefore, since an equilateral triangle RP[3] and a square RP[4] can be constructed with radii of a given length, so can RP[6], RP[8], RP[12], RP[16], and in general RP[n] for any postive n that is a multiple of 2 or 3 by a power of 2. The first value of n for which the construction of the corresponding regular polygon RP[n] is not included in the peceding list is n = 5. Problem 2 of Assignment 1.3 shows that the problem of constructing a regular pentagon RP[5] is equivalent to constructing a regular decagon RP[10]. This raises the problem:

Regular Decagon Construction Problem:

     Given a line segment AB, can we construct a regular decagon RP[10] or a regular pentagon RP[5] with radius |AB| using compass and straightedge only?

     The answer is yes, but, as we shall see, the construction is not nearly as simple as that of the cases discussed above. However, Greek geometers of Euclid’s time were well aware of a construction because Proposition 10 of Book IV of Euclid’s Elements provides a construction for an angle of 36 degrees, which is the central angle subtended by any side of a decagon. If we can construct the isosceles triangle BAP with vertex angle of measure of 36 degrees, then we can duplicate that tringle ten times to produce the decagon.

     The following diagram describes the required construction:

     But how do we know that the angle we have constructed has a measure of 36 degrees? You can verify this informally with Geometer's Sketchpad by completing the following

Which regular polygons are constructible with straightedge and compass?

     More precisely, for which values of n is the Construction of the Regular n-Gon Problem stated at the beginning of this part solvable? The results that we have discussed so far show this problem is solvable for n = 3, 4 and 5 as well as any multiple of these values by a positive integer power of 2. The values of n between 3 and 20 not included in this list are: 7, 9, 11, 13, 17 and 19. Are the regular n-gons for these n values constructible or not?

     One of the greatest mathematicians of all time, Carl Friedrich Gauss (1777 – 1855) found a ruler and compass construction for the regular 17 – gon (called a heptadecagon) and went on to show more generally that if p is a Fermat prime number (ie. a prime number p of the form

for some non-negative integer m, then the regular p-gon is constructible with compass and straightedge only. In 1837, Laurent Wantzel showed that if a regular p–gon is constructible for a prime p, then p must be a Fermat prime. His result was combined with the result of Gauss and some other lemmas to provide the following result:

The Gauss-Wantzel Theorem: A regular n-gon can be constructed with compass and straightedge if and only if:

1) n is a Fermat prime.
2) n is a power of 2.
3) n is the product of a power of 2 and distinct Fermat primes.

     Thus, a regular 7-gon is not constructible because it is prime but not a Fermat prime, while a regular 9-gon is not constructible because 9 is the square of the Fermat prime 3. A 15-gon is constructible because 15 = (5)(3) and 5 and 3 are distinct Fermat primes. A regular 11-gon, 13-gon and 19-gon are not contructible because 11, 13 and 19 are primes but not Fermat primes.

The Gauss-Wantzel Theorem is too deep and difficult to prove here.

Assignment 1.3

Assignment Completion and Submission Directions: Prepare a single Microsoft Word document with GSP figures inserted that presents the problems in Assignment 1.3 in a manner that is

  • technically correct and clearly described, and
  • attractively displayed and organized.

Download the template Word document by clicking on the link "assg_1.3_templete.doc" at the bottom of this page. Use a separate page for each problem.  Your grade on this assignment will be based on the extent to which the file you submit meets both of these criteria.

When you complete Assignment 1.3, submit your file through the Module Working Environment (also referred to as Moodle). Select Module 13 and enter your log-in and password. Then follow the directions there for submitting your assignment.

Problem 1: Given a line segment AB, use only the compass and straightedge tools in Geometer’s Sketchpad to construct the square RP[4] centered at A with AB as its radius, and write a precise description of your construction algorithm.

Problem 2: Suppose that n is an integer greater than 1 and that AB is a given line segment. Explain why the following statement is true: A regular polygon RP[2n] with 2n sides centered at A with the given line segment AB as one radius can be constructed if and only if a regular polygon RP[n] with n sides centered at A with the given line segment AB as one radius can be constructed.
(You need to explain how, given RP[2n] for some n > 1, you can construct RP[n], and how, given RP[n] for some n > 1, you can construct RP[2n])

Problem 3: Given a line segment AB, use the Basic Toolbar and Construct menus only in Geometer’s Sketchpad to construct a regular octagon with radius AB. Describe your construction procedure briefly.

Problem 4: Find simple formulas for the length of each side, and for the perimeter and the area of a regular polygon with n sides and with radius R.

Problem 5: For each integer n between n = 20 and n=30, decide if a regular n-gon RP[n] with n sides can be constructed with a given line segment AB as its radius.

     The following two problems describe an approach to the “Duplication of a Cube Problem developed by the Greek geometer Hippocrates of Chios, a Greek shipping merchant (not the Greek physician, Hippocrates of Cos who authored the famous Hippocratic Oath). Hippocrates of Chios lived in the fifth century B.C. and supported himself by teaching mathematic after his ship was captured by pirates around 450 B.C. He made several discoveries in the field of geometry including the one described in Problem 6 below.

     Problem 6 sets up Problem 7 by discussing the meaning of the meaning of the first and second mean proportionals of four positive numbers A, B, C, D that are in proportions: A is to B as B is to C as C is to D. These mean proportionals are a generalization of the geometric mean B = (A C)1/2 of two positive numbers A and C.  (See 8.3.1 in MFHST)

 Problem 6: (Mean Proportionals) Given four positive numbers A, B, C, D such that A is to B as B is to C as C is to D, then we call B the first mean proportional between A and D and C is the second mean proportional between A and D. Given A = 2, B = 4, C = 8, and D = 16, construct the first and second mean proportionals between A and D.

Problem 7: (Hippocrates’ Reduction of the Squaring of the Cube Problem to constructing mean proportionals.) If A is the volume V of a given cube and D = 2V is the volume of the cube that duplicated cube, then explain why the side length S of a side of the duplicated cube is S = p s, where p the first mean proportional between V and 2V and s is the side length of the original cube.

Congratulations! You have finished Assignment 1.3. Submit the Assignment 1.3 file to us at Math Teacher Link according to the instructions at the beginning of the assignment and then begin Part 4 of this unit.

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Step 4: Some Famous Impossible Geometric Constructions

A Little History

     The period between about 475 B.C. and 325 B.C. saw enormous growth in theoretical mathematics led by many great mathematicians in the Greek sphere of influence. These advances were primarily in five subject areas:

1. The theory of numbers.
2. Metric geometry (focusing primarily on the development of formulas for the computation of areas and volumes of a wide variety of geometric figures).
3. The development of non-metric geometry (focusing primarily on geometric constructions with compass and straightedge).
4. The theory of music (an investigation led by the Pythagoreans).
5. The theory of reasoning, mathematical proof and axiomatics.

     The publication of Euclid’s Elements certainly served as a culmination of these investigations.

Related Web Resources:

     For more information about two of the most famous mathematicians of this period, see Euclid and Pythagoras. Also, Professor David E. Joyce of Clark University has developed an excellent onl-line resource on Euclid's Elements that samples results from all thirteen books of the Elements. (The diagrams in the Elements are enabled with Java so that they are dynamic.)

In their investigations of that compass and straightedge constructions, these Greek mathematicians were able to achieve many remarkable feats. However, they and succeeding generations of mathematicians were frustrated in their attempts to solve the following problems:

  • Squaring a circle: Given any circle, find a compass and straightedge construction for a square of equal area to the circle
  • Trisecting an angle: Given any angle, find a compass and straightedge construction for an angle of one-third the measure of the given angle.
  • Duplicating a cube: Given any cube, find a compass and straightedge construction for a cube with twice the volume of the given cube.

     Mathematicians had come to believe that these constructions were actually impossible; they were not able to prove that until the nineteenth century.

How do you prove that a geometric construction actually works or that a proposed geometric construction is impossible?

     Proving that a certain geometric construction is impossible seems an especially daunting task. Of course, it is not sufficient to simply say “Many capable people have tried for many years so it must be impossible”.  After all, many capable engineers from Galileo onward tried to build a flying machine and failed until the Wright brothers succeeded late in the nineteenth century at Kitty Hawk.

    In mathematics, proving that a certain compass and straightedge construction is impossible is quite different from what impossible usually means in engineering or science. Mathematical results require deductive reasoning in proof science and engineering normally use inductive reasoning.  This means that even if many famous mathematicians fail to find a certain geometric construction over many centuries, which allow one to conclude that no such construction exists.

    As we will soon see, the key to the impossibility proofs for the three compass and straightedge constructions listed above lies in a correspondence that exists between geometric constructions and certain algebraic structures called fields and field extensions.    

    We will develop this correspondence later in this part of Unit 1 and in Unit 2.

    In our discussion of the construction of a regular decagon in Part 3 of this unit, we first described a geometric construction that claimed to construct, for a given line segment AB, an isosceles triangle ABP with a vertex angle at A of 36 degrees, and with equal sides of length |AB|. Once this triangle is constructed, it is an easy matter to proceed to construct a regular decagon and a regular pentagon of radius AB.

     When you carried out the construction of the isosceles triangle ABP in Geometer’s Sketchpad and measured the angle at A in 1.3.2 Just Do It!, you found that the angle at A did indeed measure 36 degrees. As we pointed out in the Answer to that problem, such a measurement may not prove that the given construction algorithm is correct or mathematically valid.

     However, we did provide a rigorous algebraic-geometric proof of the validity of the construction. We did this by introducing a rectangular coordinate system with the given line segment AB as the segment joining (0, 0) and (1, 0), and then following the construction algorithm algebraically by identifying the coordiates of each successive point constructed by the algorithm. The end result of this analysis was that the cosine of the vertex angle A of the triangle PBA is given by

     We then showed, again by an algebraic-geometric argument that the value of the cosine of 36 degrees has the same exact value, and so the measure of angle A is exactly 36 degrees as required.

     This algebraic-geometric approach to proving that the proposed construction of an angle of 36 degrees is valid is also the key to proving that certain geometric constructions are impossible. Here is the general idea behind the algebraic-geometric approach to proving that a given geometric construcion is either poosible or impossible:

  • Step 1: For the given geometric figure G, identify enough of the given geometric objects in G to completely determine G with real numbers.
  • Step 2: For the geometric figure F that is to to be constructed, find real numbers for components of that figure that determine that fgure completely, and express these numbers in terms of real numbers that completely described given geometric figure G in Step 1.
  • Step 3: Determine whether or not the real numbers in Step 2 can result from those in Step 1 by folloiwng an algebraic construction algorithm.

     The determination required in Step 3 is facilitated by the rich tools of the real number system including the algebraic structures as of fields and subfields of the real number system R and the algebraic theory for polynomial equations. We will review and expand these algebraic tools in Unit 2.

    In Steps 1 and 2, we might identify given points by their coordinates, given line segments by their length or by the coordinates of their endpoints, and given circles by the coordinates of their center and their radius. For example, if the objective is to construct a regular n-gon with a given line segment AB as a radius and the point A as its center, then we might describe the given segment by the real numbers that are the coordinates of its two endpoints.

    The resulting polygon F might then be described by the real numbers that are the coordinates of the endpoint C of an adjacent radii AC, or by the length of the side BC, in terms of the real number coordinates of the given objects.

    If there is a geometric construction of the required sort, then it is possible to obtain the real numbers that determine the final figure F by successively “updating” these real numbers through the steps of the construction algorithm.

    That is precisely what we did in our proof that the proposed construction of the regular decagon was correct was valid. On the other hand, if we can show that no sequence of updates of the real numbers determining the given figure G can result in the real numbers that determine the final figure F of the construction, then we would conclude that there is no such geometric construction.

Application: The Squaring the Circle Problem.

     For example, in the Squaring The Circle Problem, the relative positions of the given circle C and the desired square S are unimportant, only the equality of their areas is significant. Consequently, the given circle can be determined by a single real number r, its radius, and the desired square S by another real number s, its side length.

Squaring the Circle Animation.

     The circle C and the square S have the same area if and only if

     Thus, if we assume that the circle has radius 1, the Squaring the Circle Problem reduces to the following problem:

Application: Duplicating the Cube Problem found in Just Do It 1.4.1

Application: The Angle Trisection Problem.

     Let’s begin with three important observations about the angle trisection problem:

  • 1) Given a line segment AB and a line segment of length 1, there is a geometric construction for an angle ABC if and only if there is a geometric construction for a line segment of length equal to the cosine of angle ABC.

This can be verified by considering the following diagram:

     Observation 1) has the following important consequece:

  • 2) Suppose that m is the measure of an angle and that an angle of measure 3m can be constucted from a given line segment of length 1. Then an angle of measure 3m can be trisected given a line segment of length 1 if and only if a line segment of length cos(m) can be constructed given a line segment of length 1.

     Explanation of 2): Certainly, if a line segment of length cos(m) can be constructed given a line segment of length 1, then an angle of measure m can be constructed by 1) and that angle would trisect the angle of measure 3m. Conversely, if the angle of measure 3m can be trisected given a line segment of length 1, then an angle of measure m can be constructed (by composing the trisection construction and the construction of the angle of measure 3m) and so by 1) a line segment of length cos(m) can be constructed given a line segment of length 1.

     Angles of measure 180 degree and 90 degrees can be trisected because angles of mesure 60 degrees and 30 degrees are constucted by constructing an equilateral triangle and subdividing into two congruent right triangles with acute angles of mesure 30 degrees and 60 degrees. However, we will show that:

  • 3) Although an angle of 60 degrees is constructible, it cannot be trisected given a line segment of length 1.

     According to 2), we can prove 3) by showing that given a line segment of length 1, it is not possible to construct a line segment of length cosine of 20 degrees. To prove that, we proceed as follows:

     a) Find a polynomial equation with rational number coefficients that has the cosine of 20 degree as one of its roots.

     b) Prove that if x = r is a root of a polynomial equation of the sort decribed in a), then a segment of length r cannot be constructed given a segment of length 1.

     You are asked to do Part a) in Problem 1 of Assignment 1.4. Part b) is a consequence of a general result about constructible roots of cubic polynomials that will be discussed in Step 7

Just Do It 1.4.1

Problem: Find a reformualtion of the problem similar to that of the Squaring the Circle Problem for:

Duplicating a Cube Problem: Given any cube, find a geometric construction for a cube with twice the volume of the given cube.

as a geometric construction problem for a given line segment of length 1.

JDI 1.4.1 Answer

Answer:

Because the given cube and the duplicated cube do not have specified relative positions, each is completely determined by specifying, for example, the length of any of its edges. If we assume that the edges of the given cube U have length 1 and that the edges of the duplicatd cube K have length h, then, because the volume of a cube is the cube of the length of any of its edges, the Duplicating a Cube Problem is equivalent to the following geometric construction problem for line segments:

Assignment 1.4

Assignment Completion and Submission Directions: Prepare a single Microsoft Word document with GSP figures inserted that presents the problems in Assignment 1.4 in a manner that is

  • technically correct and clearly described, and
  • attractively displayed and organized.

Download the template Word document by clicking on the link "assg_1.4_templete.doc" at the bottom of this page. Use a separate page for each problem.  Your grade on this assignment will be based on the extent to which the file you submit meets both of these criteria.

When you complete Assignment 1.4, submit your file through the Module Working Environment  (also referred to as Moodle). Select Module 13 and enter your log-in and password. Then follow the directions there for submitting your assignment.


Problem 1:

     Express cos(3 m) as a combination of positive integer powers of cos(m). (Hint: Use this expression to obtain a cubic polynomial p(x) with rational number coefficients such that x = cos(m) is a root. Graph the resulting polynomial for m = 20 degrees on a window that displays all three of the roots.

Problem 2: Trisecting Angles with a Marked Ruler.

     Archimedes discovered the following procedure for trisecting an arbitrary angle using a compass and a marked ruler, Suppose that the given angle PQR has measure 3m and that there are two marks on the ruler at points A and B at a distance r apart, Draw a semicircle C of radius r centered at the vertex Q of the given angle. Extend the side PQ of the angle PQR to a line L. Let D be the point of intersection of the circle C and the side QR of the angle PQR. Place the marked ruler so that it passes through the point D and so that one mark B is on C while the other A is on the line L as in the figure below in the following diagram:

     Prove that the angle n at A is m, one-third the measure of the given angle PQR.

     (Hint: Look for isosceles triangles in the figure and then for combinations of angles whose measures add up to that of a straight angle.)

Problem 3: Nested Stars.

    The next figure shows a regular pentagon ABCDE for which the chords joining each of the five vertices A, B, C, D, E to its two neighboring vertices creates a second pentagon A’B’C’D’E’ interior to the pentagon ABCDE. (See Basic Star Diagram (a).
    If we iterate this construction inside the pentagon A’B’C’D’E’, we obtain a chain of pentagons of decreasing radii.
 (See Shrinking Star Diagram (b) in figure b) below. The interesting mathematical feature of this chain of shrinking pentagons is that their radii approach 0 but remain positive.

                      Basic Star Diagram (a)          Shrinking Star Diagram (b)
    

Use the figures a) and b) above to solve the following problem parts:
    a) For the Basic Star Diagram in a), explain why any ‘outer pentagon triangle” such as ACD is similar to any “inner pentagon triangle” such as BB’A’.    
    b) If the edges of the outer pentagon ABCDE are 1 unit long, what is the length of the edges of the inner pentagon A’B’C’D’E’.
    c) Use a similar triangle analysis to prove that the pentagon ABCDE is similar to the pentagon C’D’E’A’B’.
    d) Explain why BC = BE’ and A’A = A’D’ and that A’A=CA-BC.
    e) Explain why the radii of Shrinking Stars in b) are all positive but approach 0.

    The Pythagoreans believed that all real numbers were “commensurable”; that is, given two line segments of lengths M and N, one can always find a rational number p/q such that M/N = p/q. When they discovered that there were “incommensurable” segments, it caused a huge crisis for the Pythagorean society that led suicides and banishments among its members.

    In modern mathematical terminology, the Pythagoreans discovered that there are irrational numbers. How, when and what prompted this discovery is not clear. The following problem demonstrates one way in which the to show that there are irrational numbers. It makes use of:

    The Fundamental Theorem of Arithmetic: Every positive integer n can be factoered as the product of prime factors and this factorization is unique except for the order of factors in the factorization.

Problem 4: Use the Fundamental Theorem of Arithmetic to solve the following problem parts:
    a) Prove that √2 is not a rational number.
    b) More generally, prove that if n is any positive integer, then n is either a perfect square or √n is not a rational number.

    Note: Most Advanced Algebra and Precalculus textbooks include a proof of the irrationality of √2 and some follow this with the result in part b) as an exercise. 

    The Fundamental Theorem of Arithmetic is a deep result about system N of natural numbers that is often stated and used in Advanced Algebra and Precalculus textbooks. However, its proof is beyond the scope of Precalculus Mathematics. Proofs of the Fundamental Theorem of Arithmetic are usually reserved for post-calculus collegiate mathematics courses in Number Theory or Abstract Algebra or Real Analysis.

    We now return to the result mentioned the Historical Note following Problem 4

Problem 5: Use Part e) of Problem 4 to explain why the side S of a regular pentagon is not commensurate with the length D of its diagonal; that is, the real number S/D is not not a rational number.

Extra Credit Problem: Squaring Lunes.

Squaring A Lune:

     A lune is a moon-shaped plane figure determined by a given isosceles triangle as follows: Given the isosceles triangle ABC with base BC, the corresponding lune is the plane figure bounded by two circles, one with the base BC as its diameter, and the other with the equal sides AB and AC as radii. If the isosceles triangle determining the lune is a right triangle, we call the lune a right lune.

     The diagram below displays two lunes (shaded in yellow) determined by two different isosceles triangles. The one on the right is a right lune.

     Hippocrates of Chios (ca. 460 - 380 B.C.) , who worked on the problems of squaring the circle and duplicating a cube was able to obtain some interesting related results. In particular, he was able to square a right lune. [Note: This is not the physician Hippocrates who authored the Hippocratic Oath, but rather a geometer who wrote the first systematic account of the theorems of geometry with proofs. His book is lost. See Section 3.4 in A Contextual History of Mathematics by Ronald Calinger (Prentice-Hall, Upper Saddle River, NJ 1999 ISBN 0-02-318285-7)].

    Hippocrates was able to obtain some interesting geometric  results on lunes related to the The Circle-Square Problem . In particular, he was able to square a right lune. He also showed that if any lune can be squared, then any circle can also be squared. 

    Finally, we will now show how to construct a square that has the same area as that of a given right lune. The figure displayed below shows a right lune with base isosceles right triangle OAC.

     You can construct the square with the same area as this lune by completing the following steps:

1) Explain why the following equation holds:

2) Use a) to explain why the area of the quarter-circle centered at O of radius |OA| is equal to the area of the semi-circle centered at D of radius |AD|.

3) Use b) to explain why the area of the lune is equal to the area of the isosceles right triangle AOC.

4) Construct a square with the same area as the area of the isosceles right triangle AOC and display it in the diagram of the lune based on the isosceles right triangle AOC.

5) Find a formula for the area A of the right lune determined by an isosceles right triangle with equal sides of length x.

Congratulations! You have finished Unit 1 and Assignment 1.4. Submit Assignment 1.4 file to us at Math Teacher Link according to the instructions at the beginning of the assignment and then proceed to Unit 2 of this module.

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Step 5: Constructible Real Numbers

What do we mean by a constructible real number?

In Step 4, we showed that geometric constructions can be reformulated in terms of real numbers. For example, the construction of a regular n-gon from a given line segment is possible if and only if a line segment of length equal to the cosine of (360degrees)/n can be constructed from a line segment of length 1. This correspondence between constructions and real numbers leads us to the following definition.

Definitions: A real number c is constructible if and only if a line segment of length |c| can be obtained from a given line segment of length 1 by a Euclidean construction. The set of all constructible real numbers is denoted by the symbol E.

Some imporatnt observation about constructible real numbres:

Fact 1: A real number is constructible if an only if its absoloute value is constructible.

Fact 2: If a and b are constructible nambers, then so are their sum a + b, their difference a - b, their product ab and their quotient a/b if b is not zero.

Because of Fact 1, Fact 2 can be verified by showing that for given line segments of length a, b and 1, it is possible to construct line segments of length a + b, b - a if b > a, ab , and a/b if b > 0. You are asked to do that (with a little help!) in Just Do It 2.1.1

Fields of real numbers

The set R of all real numbers is a field for the usual operations of addition and multiplication. More precisely,

  • the operations of addition and multiplication are commutative and associative;
  • multiplication is distributive over addition; that is, for all a, b and c in R, a (b + c) = a b + a c;
  • the number 0 is the additive identity, the number 1 is the multiplicative identity; that is, a + 0 = a and a 1 = a for all a in R;
  • each a in R has an additive inverse, - a, and each non-zero a in R has a multiplicative inverse 1/a so that
    a + (-a) = 0 and a (1/a) = 1;

Any subset F of R that has these properties (with R replaced by F ) is a subfield of R.

To show that a subset F of R is a subfield of R, it is only necessary to show that:
a. the sum or product of any two numbers in F are in F;
b. for each a in F, the number - a is in F, and, if a is non-zero, the number 1/a is in F.
This is true because the remaining properties of a field are inherited by any subset of R.

Examples of subfields of the field of real numbers are:i) the set Q of all rational numbers;
ii) the set E of all constructible numbers (this follows fom Facts 1 and 2 above.

Fact 3: If F is any subfield of the field of real numbers R and if b is any positive real number in F with a square root that is not in F, then the following set is also a subfield of R.

 

This field is called the square root extension field F[b] of F by b or simply a square root extensiom field of F.

                                       Click here for an Explanation of Fact 3.

Fact 4: If a is a constructible, non-negative real number, then so is its square root.

Fact 4 can be concluded from the following diagram:

Important Consequence of Facts 1 through 4:

Theorem: Given a line segment of length 1, and any rational number d, it is possible to construct a line segment whose length is |d| or whose length is the square root of |d|. Therefore the set E of constructible numbers includes all rational numbers and all square roots of positive rational numbers

The following discussion describes the key connection between constuctible real numbers and the construction of geometric figures in a plane.

Constructability of points in a coordinate plane.

Given a line segment of unit length in a plane, a rectangular coordinate system can be introduced in the plane through Euclidean construction in such a way that the endpoints of the given segment have coordinates (0, 0) and (1, 0). Then a point P(a, b) in this plane is constructible with compass and straightedge from the given line segment if and only if the coordinates a and b are constructible numbers.

Fact 5: Given a line segment of length 1, a point P(a, b) in a coordinate plane is constructible if and only if both of the coordinates a and b of P are constructible numbers.

You are asked to explain why Fact 5 is true in Just Do It 2.1.2

Just Do It 2.1.1

Problem: Show that for given line segments of length a, b and 1, it is possible to construct line segments of length a + b, b - a if b > a, ab , and a/b if b > 0.

Hint: In the diagram below, if you assign the line segments of length a, b and 1 to three of the segments in the triangle, a segment of length ab will also be in the figure. The same idea works for a/b.

JDI 2.1.1.Answer

In the figures below, the given line segment are displayed in the upper left-hand corner and the constructions of the sum and difference are shown in the upper right hand corner. On the lower half of the diagram, the given line segments are displayed on the triangle described in the Hint with "swing out" copies of the segments to clearly identify them.

Just Do It 2.2.1

Explain why the following result holds

Fact 5): Given a line segment of length 1, a point P(a, b) in a coordinate plane is constructible if and only if both of the coordinates a and b of P are constructible numbers.

Hint: A geometric construction that begins with a figure that includes a line segment of length 1 followed by another geometric construction is also a geometric construction that begins with the same given figure.

JDI 2.2.1 Answer

If the point P (a, b) can be constructed from the given line segment of length 1 with a finite sequence of compass and straightedge constructions, then by following this construction with the construction of the perpendicular segments joining P to each coordinate axis, we obtain the coordinates a and b of P as constructible numbers. Conversely, if the coordinates a and b of P are contructible numbers, then the points (a, 0) and (0, b) are constructible points. But then the point P(a, b) is the point of intersection of the lines perpencicular to the coordinate axes at the points (a, 0) and (0, b), and so P(a, b) is a constructible point.

Assignment 2.1

Assignment Completion and Submission Directions: Prepare a single Microsoft Word document with GSP figures inserted that presents the problems in Assignment 2,1 in a manner that is

  • technically correct and clearly described, and
  • attractively displayed and organized.

Download the template Word document by clicking on the link "assg_2.1_templete.doc" at the bottom of this page. Use a separate page for each problem.  Your grade on this assignment will be based on the extent to which the file you submit meets both of these criteria.

When you complete Assignment 2.1, submit your file through the Module Working Environment  (also referred to as Moodle). Select Module 13 and enter your log-in and password. Then follow the directions there for submitting your assignment.

Problem 1: Suppose that r is a real number that is a root of a quadratic equation whose coefficients are constructible real numbers. Explain why r is a constructible number.

Problem 2: Explain why

is a constructible number but that it is not in any square root extension field Q[c] of the field Q of rational numbers where c is rational number whose square root is not rational.

Problem 3: Suppose that F is a subfield of the field of real numbers and that c is a real number whose square root is not in F. Show that any number d in the square root extension field F[c] is a root of a linear or quadratic polynomial P(x) with coefficients in F.Problem 4: Is the set of all real numbers of the form

where a, b, c are rational numbers a subfield of the field of real numbers? Explain your answer.

The following problem, which is a natural extension of Problem 3, is not required for completion of Assignment 2.1. However, a correct solution of this problem earns the solver "bonus points" that can raise the grade on this or other assignments.

Extra Credit Problem: Suppose that F is a subfield of the field of real numbers, that c is a number in F whose square root is not in F, and that h is a number in the square root extension field G = F[c] of F by c whose square root is not in G. Show that any number d in the square root extension field G[h] of G by h is a root of a quartic polynomial P(x) with coefficients in F.

Congratulations! You have finished Assignment 2.1. Submit your Assignment 2.1 file to us at Math Teacher Link according to the instructions at the beginning of the assignment and then proceed to Part 2 of Unit 2 of this module.

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Step 6: Euclidean Constructions and Constructible Real Numbers.

Complete and submit Assignment 2.2 according to the directions at the beginning of that assignment.

Comments on Step #6: Recall that a geometric construction is a series of operations of the following two sorts: 1) Drawing or extending a line or line segment through two points that have already been constructed; 2) Drawing a circle centered at an already constructed point and passing through another already constructed, point, or with a radius equal to the length of an already existing line segment. Each such construction step may produce new points of intersection with existing lines and circles.
   
     This section shows how these construction steps correspond to arithmetic operations in the field of constructible numbers. This correspondence provides a numerical counterpart to any Euclidean construction that enables us to decide if a proposed construction is possible or not. The Constructible Roots Theorem turns out to be the key to deciding that several famous proposed constructions are impossible.

How can we interpret a geometric construction algebraically?

    In Part 1, of this module, we began with Euclid’s definition of a geometric construction as a finite succession of compass and straightedge constructions that begins with a given geometric figure G and ends with a desired geometric figure F.   
    In Steps 1 and 2 of Part 1, we developed a number of simple examples of geometric constructions as well as others that were thought to be either very difficult or impossible.
    In Steps 3 and 4 of Part 1, we discussed the question of which regular n-gons can be constructed with compass and straightedge alone from a given figure G consisting of a single line segment AB of length 1 to obtain a regular n-gon F with side lengths or |AB|.
    In Step 5 of Part 1, we developed the basic algebraic connections between Euclidean constructions and the fields of real numbers. These connections are made by Facts 1 through 5 in this step. These facts are:
    Fact 1: A real number is constructible if and only if its absolute value is constructible.
    Fact 2 and 3: The set E of constructible real numbers is a subfield of the field of real numbers that includes the square root of any positive number in E.
    Fact 4: If F is any subfield of the field R of real numbers that  includes the field Q of rational numbers as a subfield and if p is a positive number in F then the set of all real numbers of the form:  is a subfield of the field of real numbers that includes the field F as a subfield. F[p] is called a square-root extension field of the field F by p.
    Fact 5: Given a line segment of length 1, a point P(a, b) in a coordinate plane is constructible if and only if both of its coordinates a and b are constructible numbers.

    By Fact 5, we can introduce a rectangular coordinate system in the plane with the given line segment copied as the unit segments between (0, 0) and (1, 0) on the x-axis and the unit segments between (0, 0) and (0, 1) on the y-axis.
    With this “square grid” coordinate system, any Euclidean construction can be carried out in a coordinate plane.
    By our definition of geometric construction in Part 1, each step of the construction consists of drawing a new line or circle based on the preceding parts of the construction. thereby identifying points of intersection of the new line or circle with previously constructed objects. The following result is the algebraic counterpart of one such step of a geometric construction:

Fact 6: Let F be a subfield of the field of real numbers, Suppose that L[1] and L[2] are lines in the coordinate plane each passing through two points with coordinates in F and that C[1] and C[2] are two circle each with the coordinates of its center and its radius in F. Then the coordinates of any point of intersection of any pair of these lines and circles has coordinates that are either in F or in the field F[a] where a is a positive number in F whose square root is not in F.

Explanation of Fact 6

    If a geometric construction begins with a figure that includes a line segment of unit length, then, at each step of a geometric construction, all of the numbers that represent objects that have already been constructed lie in some subfield F of the field of real numbers. According to Fact 6, the next step of the construction will result in a new object in the figure that is determined by numbers that are either in the field F or in a field F[a] which is a quadratic extension field of F by a positive number a in F whose square root is not in F.

    As the geometric construction proceeds, the coordinates each new point in the construction has coordinates that are either lie in the field containing the coordinates of all of the "old" points, or they lie in a quadratic extension of that field by an element of that field whose square root is not in that field.

    This analysis leads us to the following alternate description of constructible real numbers:

Constructible Numbers (Algebraic Description): A real number d is constructible if and only if there is a finite sequence {F[a(0)], F[a(1), F[a(2)]], ......, F[a(k)]} of subfields of the field of real numbers such that:
1.    F[a(0)] is the field Q of rational numbers.
2.    For each n < k, F[a(n)] is either equal to F[a(n+1)] or F[a(n+1)] is a quadratic extension field of F[a(n)] by an element a(n+1) of F[a(n)] whose square root is not in F[a(n)].
3.    d is an element of the last field F[a(k)] in the sequence. .
 

Constructible numbers as roots of cubic polynomials

    The key to proving that there do not exist geometric constructions to trisect any angle, square a circle or duplicate a cube (as well as other Euclidean construction problems) is the following result concerning the roots of cubic polynomials.

Theorem: (Constructible Root Theorem) If a cubic polynomial with rational coefficients has a constructible root, then it has a rational root.

 

Proof of the Constructible Root Theorem  
 

Constructible numbers are algebraic numbers

    A real number is called algebraic if it is the root of a polynomial P(x) with rational number coefficients. Real numbers that are not algebraic are called transcendental numbers.

    Two of the most famous transcendental numbers are Pi and the base e of the natural logarithm.

Historical remarks about transcendental numbers:

    The transcendence of Pi was first proved by Ferdinand Lindeman (1852 - 1939). In 1873, Charles Hermite (1822 - 1901) proved that e is transcendental. Both proofs were intricate and difficult. In 1893, Adolf Hurwitz (1859 - 1919) found a proof of the transcendence of e that used only the tools of calculus.

    The first number proved to be transcendental was the number whose decimal representation is .10100100000010.......in which there are n! 0's between the nth and (n+1)st 1. This was proved in 1844 by Joseph Liouville (1809 - 1882). In 1934, Theodor Schneider and Aleksander Gelfond independently proved that if x is an algebraic number different from 0 or 1 and if y is an irrational algebraic number, then x to the power y is a transcendental number. However, it is not known if e raised to the e power, or Pi raised to the Pi power, or Pi raised to the e power are transcendental.

    The algebraic real numbers can be shown to be a subfield A of the field of real numbers, and the field E of constructible numbers is a subfield of the field A of algebraic real numbers. The idea behind the proof that all constructible numbers are algebraic is suggested by the algebraic description of constructible numbers given above and by Problems 1, 2 and of Assignment 2.1.

 

THE OLD IS BELOW THIS - Go down to the assignments at the end of this page.  Ignore from this point on down to the assignments, it is here while we proofread the new compared to the old.

Suppose that a geometric construction begins with a geometric figure that includes a line segment of unit length. Then, by Fact 5, we can introduce a rectangular coordinate system in the plane with the given line segment copied as the unit segments on the coordinate axes..Then the geometric construction can be carried out in the coordinate plane. By our definition of geometric construction in Part 1, each step of the construction consists of drawing a new line or circle (based on the preceeding parts of the construction) and thereby identifying points of intersection of the new line or circle with previously constructed objects. The following result is the algebraic counterpart of one such step of a geometric construction:

Fact 6: Let F be a subfield of the field of real numbers, Suppose that L[1] and L[2] are lines in the coordinate plane each passing through two points with coordinates in F and that C[1] and C[2] are two circle each with the coordinates of its center and its radius in F. Then the coordinates of any point of intersection of any pair of these lines and circles has coordinates that are either in F or in the field F[a] where a is a positive number in F whose square root is not in F.

Explanation of Fact 6

If a geometric construction begins with a figure that includes a line segment of unit length, then, at each step of a geometric construction, all of the numbers that represent objects that have already been constructed lie in some subfield F of the field of real numbers. According to Fact 6, the next step of the construction will result in a new object in the figure that is determined by numbers that are either in the field F or in a field F[a] which is a quadratic extension field of F by a positive number a in F whose square root is not in F.

As the geometric construction proceeds, the coordinates each new point in the construction has coordinates that are either lie in the field containing the coordinates of all of the "old" points, or they lie in a quadratic extension of that field by an element of that field whose square root is not in that field.

This analysis leads us to the following altrnate description of constructible real numbers:

Constructible Numbers (Algebraic Description): A real number d is constructible if and only if there is a finite sequence {F[a(0)], F[a(1), F[a(2)]], ......, F[a(k)]} of subfields of the field of real numbers such that:

  1. F[a(0)] is the field Q of rational numbers.
  2. For each n < k, F[a(n)] is either equal to F[a(n+1)] or F[a(n+1)] is a quadratic extension field of F[a(n)] by an element a(n+1) of F[a(n)] whose square root is not in F[a(n)].
  3. d is an element of the last field F[a(k)] in the sequence. .

 

Constructible numbers as roots of cubic polynomials

The key to proving that there do not exist geometric constructions to trisect any angle, square a circle or duplicate a cube (as well as other Euclidean construction problems) is the following result concerning the roots of cubic polynomials.

Theorem: (Constructible Root Theorem) If a cubic polynomial with rational coefficients has a constructible root, then it has a rational root.

Proof of the Constructible Root Theorem

Constructible numbers are algebraic numbers

A real number is called algebraic if it is the root of a polynomial P(x) with rational number coefficients. Real numbers that are not algebraic are called trancendental numbers Two of the most famous transcendental numbers are Pi and the base e of the natural logarithm.

Historical remarks about traanscendental numbers: The trancendence of Pi was first proved by Ferdinand Lindeman (1852 - 1939). In 1873, Charles Hermite (1822 - 1901) proved that e is trancendental. Both proofs were intricate and difficult. In 1893, Adolf Hurwitz (1859 - 1919) found a proof of the transcendence of e that used only the tools of calculus. The first number proved to be transcendental was the number whose decimal representation is .10100100000010.......where there are n! 0's between the nth and (n+1)st 1. This was proved in 1844 by Joseph Liouville (1809 - 1882). In 1934, Theodor Schneider and Aleksander Gelfond independently proved that if x is an algebraic number different from 0 or 1 and if y is an irrational algebraic number, then x to the power y is a transcendental number. However, it is not known if e raised to the e power, or Pi raised to the Pi power, or Pi raised to the e power are transcendental.

The algebraic real numbers can be shown to be a subfield A of the field of real numbers, and the field E of constructible numbers is a subfield of the field A of algebraic real numbers.The idea befind the proof that all constructible numbers are algebraic is suggested by the algebraic description of constructible numbers given above and by Problems 1, 3 and the Extra Credit Problem of Assignment 2.1.

Assignment 2.2

Assignment Completion and Submission Directions: Prepare a single Microsoft Word document with GSP figures inserted that presents the problems in Assignment 2.2 in a manner that is

  • technically correct and clearly described, and
  • attractively displayed and organized.

Download the template Word document by clicking on the link "assg_2.2_templete.doc" at the bottom of this page. Use a separate page for each problem.  Your grade on this assignment will be based on the extent to which the file you submit meets both of these criteria.

When you complete Assignment 2.2, submit your file through the Module Working Environment  (also referred to as Moodle). Select Module 13 and enter your log-in and password. Then follow the directions there for submitting your assignment.


Problem 1: Show that

is a constructible number by finding a sequence of square root extension fields extending from the field Q of rational numbers to a square root extension field containing the number c.

Problem 2: Find the point of intersection of the line L through (1,0) and (0, -1) and the circle C with center (0, 0) and radius 3. Identify a square root extension field Q[c] of the field Q of rational numbers that contains the coordinates of this point of intersection.

Problem 3: Determine the status of each of the following real numbers among the following options:

  • Option 1: A number in Q.
  • Option 2: A number in Q[2] but not in Q.
  • Option 3: A number not in Q[2].

 

 

Congratulations! You have finished Assignment 2.2. Submit the Assignment 2.2 file to us at Math Teacher Link according to the instructions at the beginning of the assignment and then begin proceed to Part 3 of Unit 2 of this module.

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Explanation of Fact 6

Fact 6: Let F be a subfield of the field of real numbers, Suppose that L[1] and L[2] are lines in the coordinate plane each passing through two points with coordinates in F and that C[1] and C[2] are two circle each with the coordinates of its center and its radius in F. Then the coordinates of any point of intersection of any two of these lines and circles has coordinates that are either in F or in the field F[a] where a is a positive number in F whose square root is not in F..

Explanation:

    At each step of a Euclidean construction in a coordinate plane, we begin with a given set of geometric objects that are determined by points all of whose coordinates lie in some subfield F of the field E of constructible numbers. At that step, we can do only one of the following new constructions:

  • Given two existing points P and Q in the construction, construct the line passing through P and Q..
  • Given a point P and a line segment RS of length |RS| = r, construct a circle centered at P of radius r.

This step may introduce new points into our construction as the intersections of two lines or two circles or a line and a circle.

    As we will now show, the coordinates of the new points will either be in the subfield F or they will be in a square root extension field F[a] of F.

    1) Suppose that a line L passes through two poi nts P(a, b) and Q(c, d) where a, b, c, d are numbers in a subfield F of the real field R. Then (b - d)x + (c - a)y = bc - ad is an equation for L with coefficients in F. Thus, by algebraic simplification within the field F, any line L passing through two points whose coordinates are in a subfield F of the real field R, has an equation of the form px + qy + r = 0 where p, q and r are in F

    2) Suppose that a circle C has center at a point P(a, b) and passes through a point Q(c, d) where a, b, c, d are numbers in a subfield F of the real field R. Then

is an equation for L with coefficients in F. Thus, by algebraic simplification within the field F, any circle centered at a point P and passing through a point Q both of wose coordinates are in F has an equation of the form:

where p, q and r are in F.

    3) If px + qy + r = 0 and ax + by + c = 0 are the equations of two intersecting lines with coefficients in a subfield F of the real number field R, then the coordinates of the point of intersection of these lines is also in F because these two equations can be solved simultaneously within F by using the algebraic properties of a field.

    4) If

are the equations with coefficients in a subfield F of the real number field R of a line and circle that intersect, then by solving the first equation for y in terms of x and substituting this expression into the equation for the circle, we obtain a quadratic equation in x with coefficients in the field F. If we use the quadratic formula to solve for the x-coordinates of their point of intersection, this x- coordinate of the point of intersection will be in F if the discriminant of the quadratic has a square root in the field F. Otherwise, the x-coordinate will be in a square root extension field F[a] of F where a is a positive number in F whose square root is not in F. It turns out that the y-coordinate of the point of intersection will also either be in F or in the same square root extension field F[a] of F.

   

    These five cases summarize the results of any possible next step in a geometric construction based on points whose coordinates are in a number field F and prove: The coordinates of any new point will either be in F or a square root extension field F[a] where a is in F but whose square root is not in F.

Proof of the Constructible Root Theorem

The Constructible Root Theorem: Suppose that a cubic polynomial has rational coefficients. If p(x) has no rational roots, then it does not have any constructible roots.
Proof: We will assume to the contrary that p(x) has a constructible root r but that it has no rational roots. If we write p(x) in factored form and then multiply out the factors, we get: ,
,
Therefore, the coefficients a, b, c of p(x) are given in terms of the roots by:   .
    Because p(x) is assumed to have a constructible root r but no rational roots, there is a shortest expanding sequence:  Q, F[a(1)], , ……, F[a(n)] of square root extension fields such that r is in F[a(n)] but not in Q. Since r is not in Q, k must be a positive integer and:
Also, because , another (smaller) root of p(x) = 0 is  . The third root of the p(x) = 0 is given by   which is in the field F[a(n)], contrary to the assumption that F[a(1)], , ……, F[a(n)] is the shortest sequence of expanding square-root extension fields that contains r.
    This contradicts the assumption that p(x) has no rational roots but does have a constructible root. Therefore, the assumption that the cubic polynomial has a constructible root but no rational root is incorrect. We conclude that if the polynomial p(x) has a constructible root, it must also have a rational root. Equivalently, if p(x) has no rational roots, then its roots are not constructible numbers.

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Theorem: (Constructible Root Theorem) If a cubic polynomial with rational coefficients has a constructible root, then it has a rational root.

 

Proof:

Suppose that a cubic polynomial P(x) has a contructible root but no rational roots. Then there is a shortest sequence of successively larger square root extension fields starting with the rational subfield Q, F[a(1)], , ……, F[a(n)], such that there is a root of P(x) in F[a(n)] but not in F[a(n-1)]. Any r in F[a(n-1)] can be expressed as follows:

Thus, the assumption that P(x) had a costructible root but no rational root led to a contradiction. We conclude tha if a polynomial P(x) with rational coefficients has no rational roots, then it has no constructible roots.

Just Do It!

Problem: Make a list of ten (really) different trancendental numbers.

Answer

Of course, Pi and e are two easy ones. But the result of Gelfond and Schneider:

Theorem: If x is an algebraic number different from 0 or 1 and if y is an irrational algebraic number, then x to the power y is a transcendental number.

provides other different examples of transcendental numbers such as:


Step 7: The Impossibility of Certain Geometric Constructions.

Complete and submit Assignment 2.3 according to the directions at the beginning of that assignment.

Comments on Step #7: In this section, we show how the Constructible Root Theorem can be applied to show that it is not possible to trisect an angle of 60 degrees, or duplicate a cube, or square a circle with straightedge and compass alone. In Assignment 2.3 you will also conclude the impossibility of several other constructions by using the tools we have developed. This is the payoff section for the whole module!

Impossibility of trisecting a 60 degree angle with compass and straightedge alone.

We have already shown that if an angle 3m is constructible, then this angle can be trisected to obtain an angle m with compass and straightedge alone if and only if a segment of length equal to the cosine of the angle m can be constructed from a segment length 1. (See bullet item 2) in Part 4 of Unit 1.) If we can find a polynomial expression in powers of x = cos(m) for cos(3m), and if we can show that this polynomial has no constructible roots when m = 20 degrees, then we can conclude that the angle of measure 60 degrees cannot be trisected with straightedge and compass. To this end, we can use standard trigonometric identities to verify that the cosine of 20 degrees satisfies a cubic polynomial equation as follows:

If we let m = 20 degrees, we see that the cosine of 20 degrees is a root of cubic polynomial:

In Problem 2 of Assignment 2.3 for this unit, you are asked to show that P(x) has no rational roots. Consequently, by the Constructible Root Theorem, P(x) has no constructible roots. We conclude that the cosine of 20 degrees is not a constructible number. Therefore, an angle of 60 degrees cannot be trisected with compass and straightedge alone.

Impossibility of duplicating a cube with compass and straightedge alone.

This one is for you to do! Just Do It! 2.3.1

Impossibility of squaring a circle.

In Part 4 of Unit 1, we showed that, given a line segment of length 1 and a circle of radius r, the problem of finding a geometric construction for the side of a square with the same area as the given circle is equivalent to the following problem:

Of course, if a segment of length Pi can be constructed from a segment of length 1, then a segment of length equal to the square root of Pi can also be constructed. Consequently, the problem of squaring the circle is equivalent to that of determining whether or not Pi is a constructible number. However, in Part 2 of Unit 2, we pointed out that every constructible is an algebraic number; that is, it is the root of a polynomial with integer (or rational number) coefficients. However, Pi is well-known to be a transcendental (ie. not an algebraic) number. Consequently, we can conclude that a square of area equal to that of a given circle cannot be constructed with compass and straightedge alone.

Now try giving an impossibility proof of you own by completing the assignment 2.3 below and submitting it.


You have been or will be sent feedback on your seven assignments by return e-mail. After all of these assignments have been completed and graded, the University of Illinois Division of Academic Outreach will send you an official letter of completion for the module.

Those of you enrolled for graduate credit still need to complete an approved Final Project. Go to Step 8 to find out how to do that.

 

2.3.1 Just Do It!

Problem: Use the Constructible Root Theorem to prove that it is impossible to duplicate a cube using compass and straightedge alone

Hint: See the discussion of the problem in Part 4 of Unit 1.

Answer

In Part 4 of Unit 1, we showed that, given a line segment of unit length the problem of duplicating a cube is equivalent to the ffollowing problem:

This problem is, in turn, equivalent to:

The only possible rational roots of this equation are x = 1, x = 2 and their negatives by the Rational Root Test, and none of these choices is a root. Consequently, by the Constructible Root Theorem, P(x) has no constructible roots either. Therefore, it is impossible to duplicate a cube with compass and straightedge only

2.3.2 Just Do It!

Problem: Prove that it is impossible to construct a segment of length equal to the radius of a sphere with a volume of twice that of a given sphere using compass and straightedge alone.

Answer

If s is the radius of the given sphere, then we must construct a segment of length r where r is the radius of a sphere whose volume is twice that of the given sphere; that is the radii r and s are related as follows:

Because the radius of the given sphere is unspecified, we can assume that s = 1. Thus, the given problem again reduces to the same problem as duplicating the cube

As we have already seen, this polynomial does not have a constructible root, so the required construction is impossible using compass and straightedge alone.

Assignment 2.3

Assignment Completion and Submission Directions: Prepare a single Microsoft Word document with GSP figures inserted that presents the problems in Assignment 2.3 in a manner that is

  • technically correct and clearly described, and
  • attractively displayed and organized.

Download the template Word document by clicking on the link "assg_2.3_templete.doc" at the bottom of this page. Use a separate page for each problem.  Your grade on this assignment will be based on the extent to which the file you submit meets both of these criteria.

When you complete Assignment 2.3, submit your file through the Module Working Environment  (also referred to as Moodle). Select Module 13 and enter your log-in and password. Then follow the directions there for submitting your assignment.


Problem 1: Show that an angle of measure 45 degrees can be trisected using only a compass and straightedge.

Hint:15 degrees = 60 degrees - 45 degrees.Hint: Use trig identities to verify that:

  

is a number in the square-root extension field Q[2].

Problem 2: Prove that the polynomial

     has no rational roots.
Hint: Apply the Rational Root Test.

Problem 3: Is it possible to inscribe an isosceles triangle in a given circle of radius 1 in such a way that the area of the triangle is 1 and the center of the circle is inside the triangle?
Hint: See the following diagram:

    

    Recall that if z is a complex number of absolute value equal to 1, then the nth-roots of unity are given by De Moivre’s Theorem as:

     

The nth- roots of unity are distributed around the unit circle U = {z : | z |   = 1} as vertices of a regular n-gon. Only one of these, , is on the positive real axis.

    The next problem discusses the following well-known impossible compass and straightedge construction: It is not possible to construct a regular 7-gon with compass and straightedge only from a given line segment of length 1 in the plane.  Our proof will use the 7th-roots of unity as the vertices of a regular 7-gon in the complex plane.

Problem 4: Complete the following steps to conclude the impossibility of constructing a regular 7-gon with compass and straightedge only:

    Step 1: Show that the six non-real 7th-roots of unity are the solutions of the 6th-degree equation:

     z6 + z5 +z4 + z3 + z2 + z =0 .   Hint: Long divide z7 - 1  by z - 1.  

    Step 2: Use the substitution to show that 6th-degree equation in Step 1 can be written as a cubic equation p(x) = x3 + x2 - 2x - 1 = 0 .

 

Congratulations! You have finished Assignment 2.3 and Unit 2 of this module. Submit the file to us at Math Teacher Link according to the instructions at the beginning of this assignment.

If you are enrolled for Graduate Credit, you are now ready to propose a classroom project related to the module. See the Module 13 Step-by-Step Instructions for more information on the final project.

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Step 8: Propose and complete an approved Final Project

Propose and complete an approved Final Project consisting of developing a classroom unit for one of your classes based on information and techniques related to the content of this module.

Final Project Topic: You are given a great deal of latitude in the choice of topic for this unit because we want the choice to reflect your teaching situation and your inteests. We require only that the unit is based on the module. Of course, this module is written for teachers and your final project is for high school students, so the level and emphasis of your final project will likely be different than what is found in the module.

Required Final Project Proposal: After you have selected the topic for your final project, compose an e-mail message describing in a paragraph or two how you want to develop your topic, and send it to us to us in an e-mail message at:

geomConst [at] mtl [dot] math [dot] uiuc [dot] edu
We will respond with suggestions or ask for further explanation. Once your final project plan is approved, you can proceed with the development of the project.

Details about the Final Project: Your final project should be a classroom unit thaat might require 2 - 4 days to discuss in class. It must include a lesson plan, any necessary student worksheets or handouts and any documents for classroom demonstrations that you develop for the project. The project should be documented well enough so that another teacher can use it without further explanation. If practical, we would also like you to teach and evaluate the unit as part of your report, but that is not an absolute requirement. The final project will count for at least one-fourth of your final grade in the module, so it should reflect a corresponding effort and time commitment.

When you have completed your Final Project, submit all docments electronically through the MTL Module Working Environment.

You are done! The Math Teacher Link instructional staff will review and provide feedback on your assignments and Classroom Project.