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GREEK CONSTRUCTIONS
Section 12.1
Section 12.1 GREEK CONSTRUCTIONS 1 / 11
Goals of studying Greek Constructions
The goals of studying method of Greek Construction is three fold:
1. To get in the habit of Axiomatic approach to proof making, and
practice documenting one’s argument by clearly quoting a small set of
Axioms;
2. To precisely Construct a variety of geometric shapes, such as lines of
various lengths, angles of various sizes, and various polygons;
3. To describe a new class of practical, and reachable real numbers,
which, in a way, form the ultimate field extension of the Rationals.
Section 12.1 GREEK CONSTRUCTIONS 2 / 11
Section 12.1: Greek Construction Basics
In this section we learn how to apply method of Greek Construction
to perform certain operations. Please learn
- to perform and document (explain by referring to the proper Axioms
used in each step of the construction),
- be sure to give a geometric proof that the object constructed is what
was intended,
- most importantly be fully aware of their existence, and be able to
use (apply) them as tools in developing the theory presented in
sections 12.2, 12.3, and 12.4.
Section 12.1 GREEK CONSTRUCTIONS 3 / 11
Theorem 12.1.2
Theorem 12.1.2 is a construction that offers the following
- bisects a given line segment, so given a length of 1 unit, one can
repeatedly construct line segments of length 12n ,
- gives construction of 60◦, 30◦ and 90◦ angles,
- right angle triangles, equilateral triangles
- lengths of
√
3 and
√
3
2 .
- Furthermore, since there is a central angle of 60◦ is constructed, with
three more steps one could construct a regular hexagon, inscribed in a
circle. (This might be useful in drawing solutions complex polynomial
equations involving powers of 6.)
Section 12.1 GREEK CONSTRUCTIONS 4 / 11
Theorem 12.1.4
This Construction bisects a given angle. Think why such a
construction is going to be important in this chapter.
First learn this construction,
then be aware of which steps of the construction of Theorem 12.2.2
are used in this construction,
Combine this theorem with the previous theorem to make a list of the
angles that you can construct using only these two theorems (keep
the list consisting of whole degrees).
Section 12.1 GREEK CONSTRUCTIONS 5 / 11
Theorem 12.1.5 Copying a line segment
This constructions copies a line segment.
Learn the steps of this construction, and be aware of its role in this
chapter.
Using this Construction one can start with a line segment of length 1,
and construct all the natural numbers.
Given two line segments a and b, this construction can copy one next
to the other; this means the result of adding two constructible
numbers is constructible; which means the collection of constructible
numbers is closed under addition.
See also Theorem 12.2.2, which claims all the integers are
Constructible.
Section 12.1 GREEK CONSTRUCTIONS 6 / 11
Theorem 12.1.6: copying angles
This Construction copies an angle on the side of another angle, or on
a given line segment.
Learn the details of this construction,
Clearly this means any two constructible angles can be added together
and the result will be constructible. See also Theorem 12.1.9.
Make a list of constructible angles that you could build using
Theorems 12.1.2, 12.1.4 and 12.1.6
Section 12.1 GREEK CONSTRUCTIONS 7 / 11
Theorem 12.1.7: drawing perpendicular to a line
Learn the steps of this construction, and be aware of its applications
throughout the theory in this chapter.
In particular, whenever we are planning to use Pythagorean identity,
of apply some facts of trigonometry, we need to have right angled
triangle.
It is important in the construction of
√
r that we have a perpendicular
line. (See Theorem 12.2.15 as visited in PS2)
Section 12.1 GREEK CONSTRUCTIONS 8 / 11
Theorem 12.1.8: drawing a perp. from a point outside the
line
Theorem 12.1.8 give the construction of a perpendicular to a line,
from a point outside the line; Learn the details, and see how Theorem
12.1.2 is used in this construction.
This Theorem is responsible for constructing a parallel line to another
line from a point outside the original line.
The consequences of this theorem is a very very important Theorem
12.1.10, which allow us to construct any ratios of constructible
numbers.
Incidentally this property is needed if we were trying to prove
Constructible numbers forms a field.
Section 12.1 GREEK CONSTRUCTIONS 9 / 11
Thales’ Similar triangles and ratios
Thales’ theorem on similar triangles plays a central role in the proofs
that follow constructions throughout chapter 12.
See definitions 11.3.7 for definition similar triangles,
and see Theorem 11.3.8 its applications.
Historically Thales is known to have used this technique to measure
the height of the pyramids (back around 600 BCE.
Also Thales was the founder of the techniques of land Surveying, and
could easily calculate the impassible distances, such as the height of a
mountain or the width of a river.
His techniques were used to measure the distance of an approaching
enemy ship, which was a crucial piece of information, needed to
estimate the charge of catapult machines used to throw rocks at the
ships and to sink them.
Thales’ results with the similar triangles are the origin if trigonometry,
and are effectively used to measure the radius of Earth, and
measuring celestial distances.
Section 12.1 GREEK CONSTRUCTIONS 10 / 11
The Fifth postulate
Note that this construction has a non-constructible element in it;
what is it?
This non-constructible item is historically important, but our textbook
wants to make it intuitive and friendly. Otherwise this known as “The
Parallel Postulate”:
- Parallel postulate Greek Construction’s fifth axiom, which suggests
“from a point outside a line there is exactly one line parallel to the
original line can be drawn.”.
Famous consequences of this postulate is that the sum of internal
degrees of a triangle are 180◦.
The absence of this postulate led to the birth of at least two types of
“non-Euclidean Geometries” known as Elliptic or Riemannian
Geometry, and hyperbolic or Lobacewskian Geometry.
