“When will we ever use this?” This is a question that every teacher has heard at some point or at several points in time. But a better question would be, “Where has this been used this in the past?” It is important to not only look to the future, but to also look to the past. To fully understand a topic, whether it deals with science, social studies, or mathematics, its history should be explored. Specifically, to fully understand geometric constructions the history is definitely important to learn. As the world progresses and evolves so too does geometry. In high school classrooms today the role of geometry constructions has dramatically changed.
In order to understand the role of geometry today, the history of geometry must be discussed. As Marshall and Rich state in the article, The Role of History in a Mathematics Class ,
“…history has a vital role to play in today’s mathematics classrooms. It allows students and teachers to think and talk about mathematics in meaningful ways. It demythologizes mathematics by showing that it is the creation of human beings. History enriches the mathematics curriculum. It deepens the values and broadens the knowledge that students construct in mathematics class.”
This quote truly sums up the importance of relating the past to the present. Students will benefit from knowing about how mathematical topics arose and why they are still important today.
To thoroughly examine the history of geometry, we must go back to ancient Egyptian mathematics. A topic that often amazes people is the beautiful geometry in Egyptian pyramids. The mathematics and specifically geometry involved in the building of these pyramids is extensive. From Egypt, Thales brought geometric ideas and introduced them to Greece. This then led the important evolution of Greek deductive proofs. Thales is known to have come up with five theorems in geometry .
A circle is bisected by any diameter.
The base angles of an isosceles triangle are equal.
The vertical angles between two intersecting straight lines are equal.
Two triangles are congruent if they have two angles and one side equal.
An angle in a semicircle is a right angle.
However, the title of the “father of geometry” is often given to Euclid. Living around the time of 300 BC, he is most known for his book The Elements. He took the ideas of Thales and other mathematicians and put them down in an organized collection of definitions, axioms and postulates. From these basics, the rest of geometry evolves. In The Elements, the first four definitions are as follows:
A point is that which has no part.
A line is breadthless length.
The extremities of a line are points.
A straight line is a line which lies evenly with the points on itself.
Sir Thomas Heath wrote a respected translation of Euclid’s The Elements in 1926 entitled The Thirteen Books of Euclid’s Elements . This translation seems to be the most accepted version of Euclid’s writings given modifications and additions.
Since the time of Euclid there have been three famous problems which have captivated the minds and of many mathematicians. These three problems of antiquity are as follows:
In early geometry, the tools of the trade were a compass and straightedge. A compass was strictly used to make circles of a given radius. Greeks used collapsible compasses, which would automatically collapse. Nowadays, we use rigid compasses, which can hold a certain radius, but is has been shown that construction with rigid compass and straightedge is equivalent to construction with collapsible compass and straightedge. However, compasses have changed dramatically over the years. Some compasses have markings used to construct circles with a given radius. Of course, under the strict rules of Greeks, these compasses would not have been allowed.
More strictly, there were no markings on the straightedge. A straightedge was to be used only for drawing a segment between two points. There were very specific rules about what could and could not be used for mathematical drawings. These drawings, known as constructions, had to be exact. If the rules were broken, the mathematics involved in the constructions was often disregarded. When describing these concepts to students nowadays, showing pictures of ancient paintings with these tools help illustrate the importance and commonplace of geometry and these aforementioned tools.
A portion of Raphaello Sanzio’s painting The School of Athens from 16th century
The Measurers: A Flemish Image of Mathematics in the 16th century
In regards to the history of constructions, a Danish geometer, Georg Mohr, proved that any construction that could be created by using a compass and straightedge could in fact be created by a compass alone. This surprising fact published in 1672 is normally credited to the Italian mathematician, Lorenzo Mascherone from the eighteenth century. Hence, constructions created using only compasses are called Mascheroni constructions .
After Euclid, geometry continued to evolve led by Archimedes, Apollonius and others. However, the next mathematicians to make a dramatic shift in the nature of geometry were the French mathematicians, René Descartes and Pierre de Fermat, in the seventeenth century, who introduced coordinate geometry. This advance of connecting algebra to geometry directly led to other great advances in many areas of mathematics.
Non-Euclidean geometry was the next major movement. János Bolyai, following the footsteps of his father, attempted to create a new axiom to replace Euclid’s fifth axiom. Around 1824, this study led to development of a new geometry called non-Euclidean geometry. Another mathematician that made contributions to the formation of non-Euclidean geometry was Nikolai Ivanovich Lobachevsky. In 1840, Lobachevsky published Géométrie imaginaire . Because of Bolyai and Lobachevsky’s direct connection to Gauss, some believe that non-Euclidean geometry should in fact be credited to Gauss .
Even now, geometry continues to progress. In addition, how schools teach geometry has continued to change. In the past, compass and straightedge constructions were a part of the curriculum. However, in most recent years, constructions have faded out. In older textbooks, constructions were entire chapters. However, in newer textbooks, constructions are in the middle of chapters and discussed very briefly.
Instead of concentrating on paper and pencil, compass and straightedge constructions, current books tend to emphasize the use of dynamic computer software, such as Geometer’s Sketchpad. The sloppiness and inaccuracy of man-made constructions could be avoided by the use of technology. Though there are still educators that believe that using this technology is not true geometry, most realize the benefits that such software can have on comprehension.
Will true Euclidean constructions using a compass and straightedge on paper soon be a thing of the past? Will it be another lost mathematical concept like finding square roots and logarithms? Will it always been seen as an important link to the past? Will it be recognized as important but is replaced by constructions using technology?
As a link to the past, students might find constructions interesting when related to the three famous problems of antiquity of circle squaring, cube duplicating, and angle trisecting. These problems went unsolved for many years under the Greek rules of constructions. It was not until several hundred years later that they were shown to be impossible using only a compass and straightedge. The mathematics in of circle squaring, cube duplicating, and angle trisecting is interesting and can lead to good discussions.
The basics of constructions must be discussed before the complexity of the three ancient problems can be explained. The ancient Greeks’ way of representing numbers was cumbersome, with no symbol for 0 and no place-value. Perhaps as a consequence, they did arithmetic geometrically. We will use modern notation to analyze what numbers could be constructed by straightedge and compass and to study the three ancient problems.
In order to make arithmetic constructions, two segments, one of length x and the other length y, and a unit length of 1 are given. Through basic geometry and algebra, other related lengths can be created. Five arithmetic constructions are , , , , and . In order to carry out these arithmetic constructions, we must first be able to construct a parallel line.
Construct the line parallel to passing through point C.
Let E be the intersection of the parallel line and .
With a straightedge, connect points D and E. Then .
Since , by Angle-Angle Similarity. Therefore, the following proportion holds true:, . Thus, the construction of a product is possible.
Given: three lengths x, y and unit 1
With a straightedge, draw .
With a compass, construct a circle with center at A and a radius of length x. Let B be the point of intersection of this circle with .
With a compass, construct a circle with center at A and a radius of length 1. Let C be the point of intersection of this circle with .
With a straightedge, draw , with G not on .
With a compass, construct a circle with center at A and a radius of length y. Let D be the point of intersection of this circle with .
With a straightedge, construct .
Construct the line parallel to passing through point C.
Let E be the intersection of the parallel line and .
With a straightedge, connect points A and E. Then .
Since , by Angle-Angle Similarity. Therefore, the following proportion holds true:, . Thus, the construction of a quotient is possible.
Given: two lengths x and unit 1
With a straightedge, draw .
With a compass, construct a circle with center at A and a radius of length 1. Let B be the point of intersection of this circle with .
With a compass, construct a circle with center at B and a radius of length x. Let C be the point of intersection of this circle with .
Construct the midpoint D of .
With a compass, construct a circle with center at D and a radius of length .
Construct a line perpendicular to passing through point B.
Let E be the point of intersection of this perpendicular line and circle D.
With a straightedge, connect points B and E. Then .
Since and , by Angle-Angle Similarity. Therefore, the following proportion holds true: , , . Thus, the construction of a square root is possible.
These five constructions are crucial to the explanation of why the three geometric problems of antiquity are indeed impossible. Since the rules of addition, subtraction, multiplication, division, and square rooting are possible, the art of constructing numbers using such rules is possible. Numbers constructed using straightedge and compass are called constructible numbers. In terms of field theory, these numbers must lie in certain quadratic extensions of the rationals.
Since only a compass and straightedge can be used, the only constructions that can be created are segments and circles. Since an intersection point is often what is drawn, only an arc of a circle is used and not the entire circle. The construction of new points comes from the intersection of two lines, two circles, or a line and a circle. To find the coordinates of these intersections, the resulting equations would either be linear or quadratic. In either case, the equations are generally simple to solve either using basic arithmetic to solve linear equations or the quadratic formula to solve quadratic equations. Thus, the solution will be a number obtained from given numbers using the basic operations of addition, subtraction, multiplication, division, or taking the square root. All three of the impossible problems of antiquity are unsolvable under Greek construction rules because solutions would not have these characteristics. However, the proofs of showing the impossibility of these problems did not truly come about until the 19th century when geometric concepts could be related to algebraic concepts.
The saying “squaring a circle” has been used throughout the years. The metaphor is used to describe someone trying to attempt something that is impossible. From the most ancient documents, dating back as far as 1550 BC, to more recent documents, the problem of squaring the circle has been recorded. Of the three ancient problems, the most talked about in recent years is the squaring of a circle, sometimes referred to as the quadrature of the circle. This construction entails constructing a square whose area equals that of a given circle. It was not until 1882 that Carl Louis Ferdinand von Lindemann finally proved this to be impossible .
To describe this problem in mathematical detail, assume to be given some circle with the radius measuring 1. Therefore the area of the circle is . A square with the same area would result in therefore . In order to construct a square with the same area, the length of a side of the square must be . With the constructions that we know are possible, taking the square root of a number is no problem. However creating a segment with a length of π is a problem since π cannot be created by the simple operations of addition, subtraction, multiplication, or division. It is not debated that a construction can be made ever so close to π. However, a true segment of length π cannot be constructed.
Lindemann proved that π was a transcendental number therefore proving the construction of the number π was impossible. Saying that π is transcendental is the same as saying that π is not the root of any algebraic equation with rational coefficients. Even after Lindemann proved that this construction was impossible, many people still attempted to come up with a way to create π. Many so-called proofs were presented but in the end all of them have been discredited. So in fact it is impossible to construct a square with an area equal to that of a given circle.
The next two problems of antiquity, doubling the cube and trisecting an angle, again are impossible using only a compass and an unmarked straightedge. However, many mathematicians have shown that both constructions are possible if a marked straightedge is used. But under the Greeks’ most rigorous rules, only the unmarked straightedge could be used for drawing segments. For both problems, we show that a certain cubic equation does not have rational roots. It then follows that the roots cannot lie in a quadratic extension of the rationals, and so the problem cannot be solved with straightedge and compass.
In keeping with the rules of the Greeks, doubling the cube is constructing a cube with twice the volume as a given cube, of course using only a compass and straightedge. This problem is known as doubling the cube, duplicating the cube, and the Delian problem. During the time of the Greeks this problem was the most famous. However over the years the problem of squaring the circle has overshadowed this now runner-up.
This problem has an interesting history all to itself. Of course the accuracy of these stories themselves has been questioned. The first story is that of Glaucus’ tomb that was originally a cube measuring one hundred feet in each direction. Minos was not happy with the size of the tomb and ordered it to be made double the size.
The next and more common story is that of the Delians, which is why this problem is sometimes referred to as the Delian problem. Some say that the problem of doubling the cube originated with this story. Around 430 BC there was a major plague in Athens that in the end claimed the lives of nearly one quarter of the population. During the height of the plague Athenians asked for guidance from the Oracle at Delos as to how to appease the gods so that the plague would come to an end. The Delians were guided to double the size of the altar to the god Apollo. At first the craftsmen thought to double the length of each side of the altar. However, they soon realized that this did not double the size of the altar but in fact it would create an altar eight times the size of the original. After exhausting their ideas, the Delians asked Plato for advice. He responded that the Oracle in fact wanted to embarrass the Greeks for their ignorance of mathematics, primarily their ignorance of geometry. After that time, this problem became so popular that it was studied in detail at Plato’s Academy.
Mathematicians attempted to solve the problem with no success. Finally, Hippocrates of Chios showed that the problem was simply the same as finding a solution to , where is a given segment. Furthermore line segments of and may be found such that:
which leads to…
When showing the impossibility of doubling the cube using only a compass and straightedge, this information plays an important role.
The impossibility of doubling the cube is equivalent to the impossibility of solving with only a compass and straightedge. Linking again back to the history of the Delians, the number is sometimes referred to as the Delian constant .
A cube of side length one would have a volume of . Doubling the volume would produce a new side length of so that the volume would be . In order to construct a cube with twice the volume, must have rational roots. We will show that is irreducible over the rationals, and thus its roots will not be in any quadratic extension of the rationals.
Assume that does have a rational root, where is irreducible. Then
Since and 2 is prime, 2 divides . If 2 divides p, then let .
This implies that 2 divides q also. But this is a contradiction because if that was the case then would have been reducible. Therefore, does not have a rational root and hence the solutions are not constructible.
An alternate way to show this is impossible is to use the Rational Roots Theorem .