Download Compass and Straightedge Constructions

Geometry Supplement 1
Compass and Straightedge Constructions
Background: The Greeks were the first to treat mathematics as a formal system, using Euclid’s axiomatic system.
At that time there was no system for manipulating expressions and solving equations with symbols like we use
today in algebra. Instead they chose to ‘construct’ numbers as line segments, where the number corresponds to the
segment’s length, based on Euclid’s postulates. The first three postulates are:
Postulate 1: A straight line can be drawn connecting any two distinct points.
Postulate 2: A straight line can be extended indefinitely.
Postulate 3: A circle can be drawn with any point as its center and any distance as its radius.
They could emulate these postulates physically using an unmarked straightedge for drawing straight lines and a
compass to draw circles and arcs following these two rules:
1. Draw a line through any two points using our straightedge
2. Draw a circle or arc from any center point with any radius using our compass
Examples
Some of the basic constructions were things such as: copy a segment, copy an angle, bisect an angle, construct a
line parallel to a given line, construct a perpendicular bisector to a given segment, and many others. We will do
some of these in class. They are also in the textbook or at http://www.mathopenref.com/tocs/constructionstoc.html .
In particular, given two numbers a and b, represented as segments with lengths a and b, we can construct segments
of length: a + b, a − b, a ⋅ b, a / b, a 2 , a
1. Products: to construct the product of given lengths a and b, we would construct the following triangles with
parallel sides:
a
1
b
x
using similar triangles you can show x = ab
2. Quotients: the quotient of two given lengths a and b is given by the following triangles with parallel sides:
b
a
1
x
using similar triangles you can show x = a / b
Geometry Supplement 1
3. Square roots: to construct the square root of a given length a we construct the following semi-circle:
Using theorems about right triangles and circles
x
a
1
we can show that x = a .
These examples demonstrate construction methods for performing rational operations and extraction of roots, and
that means they could be used to solve linear and quadratic equations. There were also special constructions
designed to solve specific types of equations.
Collapsible Compasses
There are a couple of restrictions on these two tools if we are to strictly adhere to the postulates. First, the
straightedge is unmarked and cannot be used to measure or copy lengths; it is only used to draw lines. Second, the
compass should be collapsible. That is, you cannot copy a radius or a segment length and then transfer it to a
different location because the compass would collapse as soon as you lift it off the paper. However, we can ignore
this second restriction because Euclid showed that it is possible to transfer, or copy, a radius from one point to
another using a collapsible compass. The process, demonstrated in the diagram below, would be messy to do in
practice, so once it has been demonstrated we can accept the theorem and use a non-collapsible compass to copy a
segment length or radius, knowing that the same construction would be possible even with a collapsible compass.
Given: points A, B, and C
Goal: to construct a circle centered at A
with radius equal to BC.
1. Circle: cent = A, r = AB
2. Circle: cent = B, r = AB ⇒ point D
Question: Why is DABD equilateral?

3. Ray DA

4. Ray DB
5. Circle: cent = B, r = BC ⇒ point F
6. Circle: cent = D, r = DF ⇒ point E
7. Circle: cent = A, r = AE
Question: Why is AE = BF = BC?
D
●
●
E
●
●A
●C
●B
●F
●
Geometry Supplement 1
The Famous Construction Problems of the Greeks
There were several constructions the Greeks were unable to perform using straightedge and compass techniques,
although they can be accomplished using other techniques. Here are three that have since been proven to be
impossible:
1. Duplicating the Cube: Construct a cube whose volume is twice that of a given cube. This is the same as
2 , since if the original cube has volume V = s 3 the new cube will
have volume V = 2 s 3 , so we’d need to construct sides of length 3 2 ⋅ s .
being able to construct a length of
3
2. Squaring the Circle: Construct a square with an area equal to that a given circle. This is equivalent to
constructing a length of π.
3. Trisecting general angles: Some specific angles could be trisected, like 60º, but no method could be found
that worked for all angles.
These problems remained unsolved for 2000 years. Mathematicians were finally able to prove conclusively, using
techniques of modern abstract algebra, specifically field extensions, that these constructions are impossible. In
short, since a constructible number must be obtained by rational operations and extraction of roots, possibly
repeated to obtain rational expressions with square roots of expressions with square roots of square roots….etc., and
it was proven that:
A constructible number must be a solution of an irreducible polynomial equation whose degree is 2n for
some integer n, i.e. the degree must be a power of 2.
Therefore:
2 , but this is impossible to construct using
2 is the root of the irreducible polynomial x3 − 2 ,
1. Duplicating the cube is equivalent to constructing the number
the straightedge and compass techniques because
whose degree is not a power of 2.
3
3
2. Squaring the Circle was proven to be not constructible with straightedge and compass once π was proven to
be a transcendental number, which means that it is not a solution to any polynomial equation.
3. Trisecting angles is usually disproved by proving that it is not possible to trisect a 60° angle specifically.
After we learn some trigonometry we can show that this is equivalent to constructing the number cos 20°
and that this is not possible because that is a solution of the irreducible polynomial x 3 − 3 x − 1 .
This topic would be covered in an upper division or graduate level mathematics course in Field theory.