Geometric Constructions

Until recently, Euclid’s name and the word geometry were synonymous. It was Euclid who first placed mathematics on an axiomatic basis. He did such a remarkable job of presenting much of the known mathematical results of his time in such an excellent format that almost all the mathematical works that preceded his were discarded. Euclid’s principal work, Elements, has been the dominating text in mathematics for twenty-three centuries. It is only in this last half of our own century that Euclid is not the primary text used by beginning students. Yet, we know almost nothing about this person who wrote the world’s most successful secular book. We suppose Euclid studied at Plato’s Academy in Athens and was an early member of the famed Museum/Library at Alexandria. Alexandria became the most important city in the Western world after the death of Alexander the Great and remained so until Caesar’s Rome dominated Cleopatra’s City. Even then, while Rome was at its height, Alexandria remained the intellectual capital of the Empire. Alexandria was a major influence for a thousand years, from the time of Euclid in 300 BC until the fall of Alexandria to the Arabs in AD 641. Greek mathematics is mostly a product of the Golden Age of Greek Science and Mathematics, which was centered at Alexandria in the third century BC. Although located in what is now Egypt, the ancient city was Greek with the full name Alexandria-near-Egypt. The first of the city’s rulers who could even speak to the Egyptians in their own language was Cleopatra, who died in 30 BC. Can there be any doubt that the very learned Cleopatra studied her geometry from Euclid’s Elements?

Through his oracle at Delos, Apollo informed the Delians that if they wanted to be rid of the plague they must construct a new cubical altar that exactly doubled the volume of the existing one. The Delian Problem then was to construct a cube having a side \( \sqrt[3]{2} \) times as long as a side of the original cube. This problem also has the somewhat misleading name The Duplication of the Cube. According to another legend, Eratosthenes reported that the problem was sent to the geometers at Plato’s Academy in Athens. Plato is reported to have said that the god had assigned the task to shame the Greeks for their neglect of mathematics and their contempt for geometry. It was not that the Greeks could not construct segments of the required length by various methods but that they could not do so using only the ruler and compass. That was their task. There is little doubt that the Greeks soon suspected the problem had no solution. However, they lacked the algebra to prove this fact. Our task is to prove the ancient Greeks necessarily failed because they were asking for the impossible. To do this, we must formulate our problems in the language of algebra.

Napoleon proposed to the French mathematicians the problem of divid-ing a circle into four congruent arcs by using the compass alone. Although not original with Napoleon, the problem has become known as Napoleon’s Problem. During his campaign in northern Italy, Napoleon had encountered the poet and geometer Lorenzo Mascheroni (1750-1800). Mascheroni was a professor at the University of Pavia, where Christopher Columbus had once been a student. Mascheroni’s most famous mathematical work is his Geometria del Compasso, published in 1797. This work, which began with an ode of some literary merit that was dedicated to Napoleon, showed that all the ruler and compass constructions can be accomplished with the euclidean compass alone. Surprisingly, any point that can be constructed with ruler and compass can be constructed without using the ruler at all. In these compass constructions, a line is considered to be constructed as soon as two points on the line are constructed. In practice, we cannot draw a line with only a compass, but we may be able to construct some particular point on the line as the intersection of circles that are drawn with the compass. As usual, we do not expect every point on a constructed line to be constructible.

A culture developed in such an entangled jungle that the culture has no motivation or use for the concept of a straightedge is necessarily limited. The ruler and the compass are such basic tools that most of us accept them without ever having considered the implication of life without them. Suppose you are faced with the problem of coming to an examination on geometric constructions only to find that in making sure that you have not forgotten your compass you have completely forgotten to bring a ruler. How can you quickly make a straightedge to get you through the examination? Ripping your clothes apart to get a thread to pull taut is of theoretical interest but is not the most practical approach. Although reflections will not play the major role here that they did in the last chapter, the idea of a reflection does provide the construction of a straightedge. You have only to fold a sheet of paper; the crease forms a straightedge.

A ruler is used to draw lines. A dividers is used to carry distances. Both are everyday tools used by a drafter. If you have seen what looked like a compass with two points and no place for a pencil, that was a dividers. As a compass is sometimes called a pair of compasses, a dividers is often called a pair of dividers. Given points A,B,C,D with A ≠ B, the dividers is used to construct the point P on \( \overrightarrow {AB} \) such that AP = CD. Therefore, with the dividers we can copy segments. Can we swing the dividers with one end on a point until the other end comes to rest on a given line? No, only because it is against the rules here; we call the tooi that does that a compass. Knowing the allowed use of the dividers, you should expect the definition below. For practice in formulating these definitions, you should write down the definition of a ruler and dividers point before you read any further.

In the early nineteenth century, Poncelet and Steiner dominated the revival of interest in pure geometry, as opposed to the methods of analytic geometry. Jacob Steiner (1796–1863) worked on his father’s farm until the age of nineteen, before going off to Berlin to become what some regard as the greatest geometer of modern times. Jean-Victor Poncelet (1788–1867) entered the French army corps of engineers just in time to take part in Napoleon’s disastrous 1812 campaign. After his capture by the Russians, Poncelet spent his time in a Moscow prison to good advantage, developing the concepts of projective geometry. Steiner also made significant contributions to this new method of geometric thinking. In 1822, Poncelet, inspired by the results of Mascheroni, gave indications of a proof that all the ruler and compass constructions could be carried out with the ruler alone, provided one circle with its center was given. Steiner published his detailed proof of this result in 1833. Both Poncelet and Steiner were ardent supporters of synthetic geometry and disliked analytic methods to the extent of attacking those who used them. Therefore, it is with apologies to both Poncelet and Steiner that we will use analytic geometry in proving the theorem that bears both their names. However, at this point, we would not consider doing it any other way.

Our task is to form a geometric construction theory that models what we can do with an inexhaustible supply of toothpicks. It should be clear that in order to do the modeling, we must first have some familiarity with what it is we are to model. Therefore, we should first play around with stacks of toothpicks just to see what we can do with them. Paper and pencil is just not satisfactory; straws or even long, thin strips of paper will serve the purpose. Reading about toothpick constructions is not the same thing as discovering for yourself some of the things that follow. Since the results of this chapter are not used later, you can go on to the next chapter now and read this chapter after you have had a chance to play with toothpicks.

A marked ruler is simply a straightedge with two marks on its edge. With an appropriate starter set, an immediate use of the marked ruler is to draw the line through two given points and mark off unit segments on the line. Since the two marks thus provide a scale in the sense of a rusty dividers, then the ruler and dividers theory can be applied to marked ruler constructions. Throughout this chapter, the two points determined by the two marks on the marked ruler in a particular position will be called R and S. We suppose the marks are one unit apart. So RS = 1, as in Figure 9.1a. For a second use of the marked ruler, we can set one mark on a given point R and rotate the marked ruler until the second mark falls on a given line s at a point S, whenever the unit circle with center R intersects the line s. See Figure 9.1b. The unit circle with center R can be taken as a poncelet circle. The Poncelet-Steiner Theorem now implies all the ruler and compass constructions are possible with the marked ruler alone. The characteristic use of the marked ruler is called verging or insertion. Given point V and two lines r and s, by verging through V with respect to r and s we determine two points R and S that are one unit apart and such that V\( \overleftrightarrow {{\text{RS}}} \) is on , R is on r, and S on s. See Figure 9.1c. Thus, in verging through V with respect to r and s, the marked ruler is placed down to pass through V with one mark on r and the other on s. Using the marked ruler in this way to solve two of the three classical construction problems goes back to the Greeks who, according to Pappus,“moved a ruler about a fixed point until by trial the intercept was found to be equal to the given length.” A verging is sometimes called by its Greek name neusis. Although Apollonius’s book Neusis on the subject is lost, J. P. Hogendijk has reconstructed the text from Arabic traces of the work.

Paperfolding, as a means of geometric construction and as opposed to origami, was introduced in 1893 by T. Sundara Row from India. In Sundara Row’s Geometric Exercises in Paper Folding it is evident that all ruler and compass constructions are possible by paperfolding. However, Sundara Row’s angle trisection is admittedly only an approximation and he mistakenly implies that constructing a cube root is impossible in general and, in particular, that the duplication of the cube cannot be accomplished by paperfolding. In his 1949 A History of Geometrie Methods, J. L. Coolidge refers to Sundara Row but limits his own folding operations to those equivalent to the use of ruler alone. The first rigorous treatment of paperfolding is apparently by R. C. Yates in 1949 in his Geometrie Tools.