Compass & Straightedge Construction and the Impossible Constructions

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Creating a regular hexagon with a ruler and compass
This image shows the step by step construction of a hexagon inscribed in the circle using a compass and a unmarked straightedge.

Contents

Basic Description

Let's assume we only have a compass and a unmarked straightedge. What can we construct and how can we construct them? That was the problems that Euclid pondered not only because those were probably the only instruments that he had at his time but also he wanted to construct his theorems with as few assumptions, or axioms, as possible. In the main image of this page, we want to divide the circle into six equal arcs and then connect consecutive points to form the hexagon. It seems to be a fairly simple construction but it should prompt you to ask a question: is every polygon constructible, that is able to be constructed using only compass and straightedge? To extend the question, what are constructible and what are not? This issue is addressed in this page.

A More Mathematical Explanation

Note: understanding of this explanation requires: *A little Geometry and Some Abstract Algebra

What is Compass & Straightedge Constructions

Introduction

We start by familiarizing [...]

What is Compass & Straightedge Constructions

Introduction

We start by familiarizing ourselves with Euclid's three Postulates in his books Elements.

Let it be granted
  1. that a straight line may be drawn from any one point to any other point;
  2. that a line segment may be extended into a straight line;
  3. that given any straight line segment, a circle may be described having the segment as radius and one endpoint as center.
It should be carefully noted that Euclid started with two given points and produced a line segment, from where he could extend into a straight line if he pleased. Then ONLY from the original two points, he could use one point as center and ONLY spread the legs of compass the distance of the line segment to produce a circle. He could not say "I wanted to spread the legs of the compass \pi centimeters apart (or any specified denominations)". Therefore, he could not (so couldn't we) however, choose any two points as he pleased. In all, Compass and Straightedge Constructions only allow us to start with points (and hence lengths) we have been given, or constructed from given points, and create the ones we don't. Thus, we define Compass & Straightedge Construction as the construction of points, lengths, angles, and circles using only idealized straightedge and compass. A straightedge is infinite in length, has no markings on it and only one edge. A compass has two legs, one end of which is fixed on the plane of construction and the other end is of given distance away and maintains the distance throughout the construction. It collapses when lifted from the page, so may not be directly used to transfer distances. However, it turns out that this restriction makes no difference due to the Compass Equivalence Theorem which was stated as Proposition II of Book I of Euclid's Elements. It stated that from a given point, it was possible to construct a line segment equal to a given line segment using collapsible compass. Euclid's proof for the Compass Equivalence Theorem will be presented after the section of Basic Construction.

Some Basic Constructions

The constructions below are some basic ones from where many more constructions are possible and they are by no means exhaustive. The resulting products are in red. The proofs for these constructions are relatively simple and only require the knowledge of congruent triangles. Euclid derived the theories on congruency and congruent triangles directly from his Postulates. Try proving the theorems yourself!

Line Segment Bisection

Given points A and B and the straight line passing through it. Construct a line that bisects line segment AB.

  1. Draw a circle centered at point A with radius equaling AB.
  2. Next, draw a circle centered at point B with the same radius.
  3. Where the two circles intersect, call those points C and D.
  4. Draw a line through points C and D.
CD intersects AB at the midpoint. It should be noted that CD \perp AB as well. Line CD is the perpendicular bisector of line segment AB.


Angle Bisection

Given angle \angle AOB, construct a line that bisects the angle.

  1. Construct a circle centered at point O with radius OA. This circle should intersect at points A and B.
  2. Keeping the same radius, draw a circle at points A and B. Where they intersect each other, call the point C.
  3. Draw a line through points C and O. This line bisects \angle AOB.

Perpendicular through a point

Given a point M on a line AB, construct a line that is perpendicular to the given line through M.

  1. Draw a circle centered at M.
  2. Where the circle intersects the original line, construct a perpendicular bisector.

Parallel

Given two points, A and B and the straight line passing through them, construct a line that is parallel to the given line through another given point C.

  1. Draw a circle at A, crossing C. Where the circle A intersects AB, call the point D.
  2. Centered at points C and D, draw circles crossing point A. Where these two circles intersect each other, call it E.
  3. Connect points C and E with a line.
CE is parallel to AB

Tangent Line to a Circle

Euclid's Proof of Compass Equivalence Theorem

This part refers back to the previous section about the issue of compass being collapsible. Euclid's proof is presented in its originality. Additional comments are contained in the parenthesis.

From a given point to draw a line segment equal to a given line segment.

Let A be the given point, and BC the given straight line : it is required to draw from the point A a straight line equal to BC.


  1. From the point A to B draw the straight line AB and on it describe the equilateral triangle DAB(this is done the same way as bisecting a line segment), and produce the straight lines DA, DB to E and F.
  2. From the center B, at the distance BC, describe the circle CGH, meeting DF at G.
  3. From the center D, at the distance DG, describe the circle GKL, meeting DE at L.
  4. AL shall be equal to BC.


Because the point B is the center of the circle CGH, BC is equal to BG. And because the point D is the center of the circle GKL, DL is equal to DG and DA, DB parts of them are equal therefore the remainder AL is equal to the remainder BG.


But it has been shewn that BC is equal to BG ; therefore AL and BC are each of them equal to BG. But things which are equal to the same thing are equal to one another. Therefore AL is equal to BC.


Wherefore from the given point A a straight line AL has been drawn equal to the given straight line BC.∎

It should be noted that from AL, we could "duplicate" AL in all directions by construct a circle centered at A with radius AL.

Algebraicization of Compass & Straightedge Constructions

A simple derivation

From the few basic constructions, you would have probably realized that the different possibilities seems infinite. However, by intuition, we know that the possibility could not be infinite. Hence, mathematician are curious to find out what are constructible and what aren't and for this purpose, the language of pure geometry seems to have "limited vocabulary". Back in ancient times, mathematicians had limited algebraic knowledge and were more familiar with geometry. But in modern times, the reverse is true. Hence, today's mathematicians go back to their familiar realm of Algebra and try to find the link between geometry and algebra.


Algebraicization is the translation of any problem statements into algebraic problems. In the case of Compass & Straightedge construction, we algebraicize each step of a straightedge and compass construction, and consequently obtaining general results about the nature of constructibility. Hilda P. Hudson put it aptly in his lecture Ruler & Compasses,

"each step of a ruler and compass construction is equivalent to a certain analytical process; it is found that the power to use a ruler corresponds exactly to the power to solve linear equations, and the power to use compasses to the power to solve quadratics. For this reason, problems that can be solved with ruler only are called linear problems, and those that can be solved with ruler and compasses are called quadratic problems. Since each step of a ruler and compass construction is equivalent to the solution of an equation of the first or second degree, we consider that these algebraic processes can lead to , when combined in every possible way, and that enables us to answer the question before us and say that those problems and those problems alone can be solved by ruler only, which can be made to depend on a linear equation, whose root can be calculated by carrying out rational operations only; and that those problems and those problems alone can be solved by ruler and compasses, which can be made to depend on an algebraic equation, whose degree must be a power of 2, and whose roots can be calculated by carrying out rational operations together with the extraction of square roots only."

Hudson lectured on this in the early 20th century and certain phrases of his could potentially cause confusion. The take-away from this paragraph is that in order to algebraicize straightedge and compass construction, we begin by designating a given point as the origin and the coordinates of another given point (we are given two points at least) as (1,0) or (0,1). Thus we have established the Cartesian Coordinates. Then, every time we construct a straight line or a circle, we think of it instead as constructing numbers representing the coordinates of all the points on them. But that is easy since we all know the expression for a line and a circle as y = ax + b and (x-m)^2 + (y-n)^2 = r^2. However, the only times we can pinpoint a point (and find its coordinates as a result) is when a line intersects with a line, or a circle, or a circle intersects with another circle in which case we can pinpoint 2 points. We then conclude that only those coordinates of the points of intersections are constructible. In this way, a geometric process is translated into an algebraic process.


Firstly, we define "1" on a straight line as stated previously. Then, once you have chosen that length to be "1", you have to stick to this specification throughout your construction. Next, it is very obvious that we could construct all the integers, that is \cdots -3,-2,-1,0,1,2,3,\cdots (or x = \{x|- \infty < x < \infty,x \in \mathbb{Z}\}). How so? Well, once we have the "1", all we have to do is to use the Compass Equivalence Theorem finite number of times to duplicate the length "1" that we previously defined. Now, that means that we could have any two random integers, a and b, and for the sake of this discussion and clarity, we are talking about positive integers here. Next, I will show that from a and b, we could construct a \pm b, a \times b and \frac {a}{b}.

To construct a \pm b, we will use a as the center and use b as radius. The two points of intersection with the line will be a+b and a-b.


To construct a \times b, we have 0, 1, a and b on the straight line.

  1. Draw any straight line through 0, call it l_1.
  2. Construct circle centered at 0 with radius b, intersecting l_1 at B.
  3. Connect B and 1, call it l_2.
  4. Construct l_3 that is parallel to l_2, intersecting l_1 at A.
  5. Construct circle centered at 0 with radius 0A, intersecting l_0 at a point.
The distance between 0 and that point is ab.


Similarly, we could construct \frac {a}{b}.

  1. Draw any straight line through 0, call it l_1.
  2. Construct circle centered at 0 with radius b, intersecting l_1 at B.
  3. Connect B and 1, call it l_2.
  4. Construct circle centered at 0 with radius a, intersecting l_1 at A.
  5. Construct l_3 that is parallel to l_2, intersecting l_0 at a point.
The distance between 0 and that point is \frac {a}{b}.


I will leave the proofs to you since they are very simple using similar triangles.


Therefore, it has been proven that we could construction all the rational numbers since a and b are any arbitrary integers.


The natural question to ask right now is that what else is possible to construct? It is not hard to think of numbers that are not rational. For example, \sqrt{2} is constructible. Construct a unit square and the diagonal is of length \sqrt{2}. So is it possible to construct \sqrt{a} given any constructible number a? It turns out that we could. See below for method.

  1. Construct a+1.
  2. Construct \frac {1}{2}(a+1).
  3. Construct circle at \frac {1}{2}(a+1) with radius \frac {1}{2}(a+1).
  4. Draw perpendicular through a, intersecting the circle at point A.
aA=\sqrt {a}.Again, I will leave the proof to you as well using similar triangles.


Next, we moved to the general solution of the problem.


Assume we have two points P_1 and P_2 with coordinates (x_1, y_1) and (x_2, y_2). Take an arbitrary point X (x,y) on the line.

By similar triangle, \frac {x_1-x}{x-x_2} = \frac {y-y_1}{y_2-y}.

Rearranging the above we have

x(y_1-y_2)+y(x_2-x_1)=x_2y_1-x_1y_2.

Since x_1, x_2, y_1 and y_2 are constant we can express this as ax+by=p which is the general expression of a straight line.


Now, if we have two lines specified by four given points, P_1,....,P_4 with coordinates (x_i,y_i). The intersection of the two lines, P (x,y) will satisfy two equations

\begin{cases}
    x(y_1-y_2)+y(x_2-x_1)=x_2y_1-x_1y_2\\
    x(y_3-y_4)+y(x_4-x_3)=x_4y_3-x_3y_4\\
\end{cases}

You may say that the there might not be a solution. True the two lines do not have to intersect. But if they do, we only need the operations of addition, subtraction, multiplication and division to find the point.

Now, we move onto circle. Say we have circle centered at some point P_5 with coordinates (x_5, y_5) and radius r. We know that the explicit expression for a circle is (x-x_5)^2+(y-y_5)^2=r^2. Hence, if that circle intersects with one of the straight lines, then the points of intersection will satisfy

\begin{cases}
    x(y_1-y_2)+y(x_2-x_1)=x_2y_1-x_1y_2\\
    (x-x_5)^2+(y-y_5)^2=r^2\\
\end{cases}

To solve for the points of intersection, we only need the operations of addition, subtraction, multiplication and division along with the extraction of square roots. Therefore, from this analysis, we have turned geometric problem into algebraic problem and come to the conclusion that a number is constructible if and only if it may be obtained from the integers by repeated use of addition, subtraction, multiplication, division and the extraction of square roots.


A Rigorous Proof

What I have presented above is a simplified version of the derivation towards the theorem. To see a rigorous proof of this theorem at a college level, refer to the text below which is mainly taken from I. N. Herstein's Topics in Algebra, Second Edition. You need some knowledge in Linear Algebra and/or Abstract Algebra. Also see Constructible Numbers. You should not be discouraged should you find it hard to understand. Instead, you should be marveled by the simplicity and elegance of the algebraic proof.

We have proven that if a and b are constructible numbers, then so are a \pm b, ab, and when b \ne 0, \frac {a}{b}. Therefore, the set of constructible numbers form a subfield, W, of the field of real numbers.


In particular, since 1 \in W, W must contain F_0, the field of rational numbers. If w \in W, we can reach w from the rational field by a finite number of constructions.


Let F be any subfield of the field of the field of real numbers. Consider all the points (x,y) in the real Euclidean plane both of whose coordinates x and y are in F; we call the set of these points the plane of F. Any straight line joining two points in the plane of F has an equation of the form ax+by+c=0 where a,b,c are all in F. Moreover,any circle having as center a point in the plane of F and having as radius an element of F has equation of the form x^2+y^2+ax+by+c=0, where all of a, b, c are in F.


Given two lines in F which intersect in the real plane, then their intersection point is a point in the plane of F. On the other hand, the intersection of a line in F and a circle in F need not yield a point in the plane of F. But, using the fact that the equation of a line in F is of the form ax+by+c=0 and that of a circle in F is of the form x^2+y^2+dx+ey+f=0, where a, b, c, d, e, f are all in F, we can show that when a line and circle of F intersect in the real plane, they intersect either in a point in the plane of F or in the plane of F(\sqrt {r}) for some positive r in F. Finally, the intersection of two circles in F can be realized as that of a line in F and a circle in F, for if these two circles are x^2+y^2+a_1x+b_1y+c_1=0 and x^2+y^2+a_2x+b_2y+c_2=0, then their intersection is the intersection of either of these with the line (a_1-a_2)x+(b_1-b_2)y+(c_1-c_2)=0, so also yields a point either in the plane of F or of F(\sqrt{r}) for some positive r in F.


Thus lines and circles of F lead us to points either in F or in quadratic extensions of F. If we now are in F(\sqrt{r_1}) intersect in in points in the plane of F(\sqrt{r_1},\sqrt{r_2}) where r_2 is a positive number in F(\sqrt{r_1}). A point is constructible from F is we can find real numbers r_1,...,r_n, such that r_1^2 \in F, r_2^2 \in F(r_1), r_3^2 \in F(r_1,r_2),...,r_n^2 \in F(r_1,...,r_{n-1}), such that the point is in the plane of F(r_1,...,r_n). Conversely, if r \in F is such that \sqrt{r} is real then we can realize r as an intersection of lines and circles in F. Thus a point is constructible from F is and only if we can find a finite number of real numbers r_1,...,r_n, such that

  1. [F(r_1:F)] = 1 or 2;
  2. [F(r_1,...,r_i):F(r_1,...,r_{i-1}]=1 or 2 for i=1,2,...,n;

and such that our point lies in the plane of F(r_1,...,r_n).


We have defined a real number a to be constructible if by use of straightedge and compass we can construct a line segment of length a. But this translates, in terms of the discussion above, into: a is constructible if starting from the plane of the rational numbers, F_0, we can imbed a in a field obtained from F_0 by a finite number of quadratic extensions.


And therefore,

  1. If a is constructible then a lies in some extension of the rationals of degree a power of 2.
  2. If the real number a satisfies an irreducible polynomial over the field of rational numbers of degree k, and if k is not a power of 2, then a is not constructible.


Why is it interesting?

What is Impossible to Construct (of course, using compass and straightedge alone)?

Below is the brief introduction of a few of the impossible constructions. Remember that a number is constructible if and only if it may be obtained from the integers by repeated use of addition, subtraction, multiplication, division and the extraction of square roots.

  1. \pi is transcendental since it does not satisfy any rational polynomials. That means that \pi is not a solution of any polynomials with rational coefficients. Too see complete proof that \pi is transcendental, see Transcendental number and The 15 Most Famous Transcendental Numbers.
  2. From the above impossible construction, it follows that it is impossible to "square the circle (that is to construct a square that has the same area as a given circle)" because given a circle with radius 1 (the radius has to be constructible in the first place so that means it cannot, for example, be \frac {1}{\sqrt {\pi}}), the area of the circle will be \pi and we have to construct square with sides equal to \sqrt \pi which is not constructible.
  3. We could not double the volume of a given cube because we could not construct \sqrt[3]{2}. It should be noted that we have to start with a cube whose sides are constructible in the first place. For example, we cannot even have a cube with sides equal to \sqrt [3]{5} and thus volume 5, let alone doubling it. Prove that we cannot double the cube.
    Say we start with cube of volume 1. Then we have to construct cube of volume 2, which means we have to construct sides of \sqrt [3]{2} which is impossible to construct. So we can double the cube.
  4. We generally can not trisect any given angle because the process involves taking cube root. For example, it is impossible to trisect 60^\circ. See below for proof. For more, refer to Trisection of an Anglefor explanation in great detail. Prove that 60^\circ is impossible to trisect.
    If we could trisect 60^\circ by compass and straightedge, then the length a = \cos 20^\circ would be constructible. Since \cos 3\theta = 4\cos^3\theta - 3 \cos \theta. Substituting \theta = 20^\circ and \cos60^\circ=\frac {1}{2}, we obtain 4a^3-3a=\frac {1}{2}. Thus a is a root of a cubic polynomial over the rational field. Since this polynomial is irreducible over the rational field and its degree is 3, a is not constructible. Thus 60^\circ cannot be trisected.
  5. There are certain polygons that are impossible to construct. See Constructible polygon for more detail.

Number 2, 3 and 4 are the so-called Geometric Problems of Antiquity. Though they have been proven impossible to construct with straightedge and compass, it does not deter amateur mathematicians to come up with false proofs even today.




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