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Euclidean Constructions

The notion of  geometric constructions go back at least to Greek antiquity. The classical constructions are called Euclidean constructions, but they certainly were known prior to Euclid.

The allowed instruments are idealized “mechanisms”, such as [1]

• the straightedge
• the ruler
• the collapsing compass
• the compass
• the fixed-aperture compass
• the compass with aperture bounded above
• the compass with aperture bounded below
• etc.

The collapsing compass allows to be opened to a chosen radius and used to drawn the circle, but when it is lifted off the drawing surface, it collapses as the name indicates. This means that no distance can be transferred using it. This contradicts the modern understanding of the compass usage according to which when opened to some aperture and then lifted to another location on the page the compass can be used to transfer the same distance.

Thus the “Euclidean” compass is different from what we understand under the modern compass. Therefore a natural question is whether we can make the same geometric constructions by both compass versions (together with the straightedge which usage underlies the same rules nowadays as before). The answer gives the following Compass Equivalence Theorem:

Theorem: A circle at center and radius can be congruently copied using a straightedge and collapsing compass so that a given point can serve as the center of the copy.

For the solution we can combine first two Euclid’s propositions of Book I .  Another solution we get using the steps:

• the given circle at and radius and the circle at with radius meet at points ,
• construct the circles at and both with radius , they inteesect in points ,
• the circle at with radius and the circle at with radius intersect in points and
• one of the distances or equals the radius of the original circle at

To see this, note that triangles and are congruent by the side-side-side theorem. This implies that angles and   at in both triangles are equal. Thus angles   and are also equal. By the side-angle-side theorem the triangles and are congruent, i.e. the sides and have the same length.

The straightedge was thought of as having no length markings. This because there is no postulate giving us the ability to measure the lengths. This means that you cannot use the straightedge to extend a segment twice, say, but you can use it to connect a pair of points or to extend the segment given by two distinct points.

The instruments allowed to be used in Euclidean constructions are compass and straightedge. The compass is used to establishes the equidistance, and the straightedge establishes the collinearity. Thus Euclidean geometric constructions are based on these two procedures:

1. The compass is anchored at a center point, and keeps the drawing instrument at a fixed distance from that point. Thus the points on the curve (circle) drawn by a compass are equidistant from the center point.
2. The straightedge is used only for drawing lines. It is often called ruler, but no measurements are allowed in Euclidean constructions. This instrument is used to draw the straight line passing through two given points.

Constructions are understood as a theoretical exercise, where for instance the fact that a line has zero width, and various  physical imperfections of the drawing instruments are neglected.

Thus in the spirit of the first three Euclid’s axioms (postulates):

Postulate 1. To draw a straight line from any point to any point.

Postulate 2. To produce a finite straight line continuously in a straight line.

Postulate 3. To describe a circle with any center and radius.

the following constructions are taken for granted:

Construction 1: Given two distinct points , the straight line passing through these two points is considered as being constructed.

Construction 2: Given a point and segment then the circle at and radius   is considered as being constructed.

Construction 3: The intersection point of two intersecting lines is considered as being constructed.

Construction 4: Given a circle and a secant line then their intersection points are considered as being constructed.

Construction 5: Given two intersecting circles then their intersection points are considered as being constructed.

Despite the fact that the original famous Euclid’s fifth postulate does not used the notion of parallel lines, Euclid defines this notion in his Definition 23: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

How to construct a straight line through a given point and parallel to a given one not passing trough is described in Proposition 31 of Book I . In practical constructions, however, the parallel lines are constructed using two setquares (having one right angle).

There are three classical Euclidean construction problems (i.e. compass and straightedge constructions) in Greek mathematics which were extremely important in the later development of geometry.:

1. squaring the circle,
2. doubling the cube
3. trisecting an angle

They are closely linked, and the mathematicians were able to definitely answer these problems (in the negative) only after the Galois theory was developed in 19th century.

Marked rulers (i.e. a straightedge that is notched in two places) and protractors are not allowed in the classical Euclidean constructions. Though the marked ruler is not used in Euclid’s Elements, other Greek geometers used it, for instance Hippocrates of Chios (ca. 430 BC). Pappus reports that Applonius of Perga  (ca. 262-190 BC) wrote two books on constructions using marked ruler. Pappus credo was: Whenever a construction is possible by means of compasses and straight edge, more advanced means should not be used.

The constructions using marked ruler were called neusis constructions by ancient Greeks. [2] As it was shown by Archimedes trisection of an angle is possible using a marked ruler:

Given is an angle by the intersection of two lines and which intersect at . Let be the distance between the two notches on the straightedge. Then

1. draw the circle at with radius , let , be its intersection point with and , resp., such that the angle is acute
2. put one notch of the straightedge on the line , and the other on the circle and move the straightedge until it goes through
3. thus we get a point on and a point on the circle and

Then .

Since the triangle is isosceles, , say.Then for it is an exterior angle to . In the isosceles triangle we have and . Consequently,  , as desired.

When the ruler and compass constructions did not offer a solution then also other types of construction elements were accepted, for instance

• constructions that in addition to this use conical sections (ellipses, parabolas, hyperbolas)
• constructions that in addition to this use other curves, as the so-called quadratrix of Hippias , or conchoid curve (Nicomes used it to solve the problem of trisecting the angle)

References

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