Constructions
Constructions
Constructions, in mathematics, are geometric figures that are drawn based on a set of pre-selected measurements. Much of Euclidean geometry is based on two geometric constructions: the drawing of circles and the drawing of straight lines. To draw a circle with a compass, one needs to know the location of the center and at least one point on the circle. To draw a line segment with a straightedge, one needs to know the location of its two end points. To extend a segment, one must know the location of it or a piece of it.
Three of the five postulates in Euclid’s Elements (from Greek mathematician Euclid of Alexandra [c. 325–c. 265 BC] say that these constructions are possible:
To draw a line from any point to any point.
To produce a finite straight line in a straight line.
To describe a circle with any center and distance.
The constructions based on these postulates are called straightedge and compass constructions.
The Elements does not explain why these tools have been chosen, but one may guess that it was their utter simplicity, which geometers (people who study
geometry) found, and continue to find, appealing. These tools are certainly not the only ones that the ancient Greeks employed, and they are not the only ones upon which modern draftsmen depend. Triangles, French curves, ellipsographs, T-squares, scales, pro-tractors, and other drawing aids both speed the drawing and make it more precise.
These tools are not the only ones on which contemporary geometry courses are based. Such courses will often include a protractor postulate that allows one to measure angles and to draw angles of a given size. They may include a ruler-placement postulate that allows one to measure distances and to construct segments of any length. Such postulates turn problems that were once purely geometric into problems with an arithmetic component. Nevertheless, straightedge and compass constructions are still studied.
Euclid’s first proposition is to show that, given a segment AB, he can construct an equilateral triangle ABC. (There has to be a segment. Without a segment, there will not be a triangle.) Using A as a center, he draws a circle through B. Using B as a center, he draws a circle through A. He calls either of the two points where the circles cross C. That gives him two points, so it is possible to draw segment AC. He can draw BC. Then ABC is the required triangle (Figure 1).
Once Euclid has shown that an equilateral triangle can be constructed, the ability to do so is added to his tool bag. He now can draw circles, lines, and equilateral triangles. He goes on to add the ability to draw perpendiculars, to bisect angles, to draw a line through a given point parallel to a given line, to draw equal circles, to transfer a line segment to a new location, to divide a line segment into a specified number of equal parts, and so on.
There are three constructions which, with the given tools, neither Euclid nor any of his successors were able to do. One was to trisect an arbitrary angle. Another was to draw a square whose area was equal to that of a given circle. A third was to draw the edge of a cube whose volume was double that of a given cube. “Squaring the circle”, as the second construction is called, is equivalent to drawing a segment whose length is π times that of a given segment. “Duplicating the cube” requires drawing a segment whose length is the cube root of 2 times that of the given segment.
In about 240 BC, Ancient Greek mathematician Archimedes (287–212 BC) devised a method of trisecting an arbitrary angle ABC. Figure 2 shows how he did it. Angle ABC is the given angle. ED is a movable line with ED = AB. It is placed so that E lies on BC extended; D lies on the circle; and the line passes through A. Then, ED = DB = AB, so triangles EDB and ABD are isosceles. Because the base angles of an isosceles triangle are equal and because the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles, the sizes, in terms of x, of the various angles are as marked. Angle E is, therefore, one third the size of the given angle ABC; ABC has been trisected.
Why is this ingenious but simple construction not a solution used by Euclid to solve the problem of trisecting an angle? Line ED has to be movable. It requires a straightedge with marks on it. Simple as marking a straightedge might be, the Euclidean postulates do not make provision for doing so.
Archimedes’ technique for trisecting an angle is by no means the only one that has been devised. American
KEY TERMS
Compass— A device for drawing circles having a given center and radius.
Construction— A drawing of a geometrical figure made with the use of certain specified tools.
Straightedge— An unmarked ruler of any length that can be used to draw lines through two given points.
mathematician Howard Whitley Eves (1911–2004), in his History of Mathematics, describes several others, all ingenious. He also describes techniques for squaring the circle and duplicating the cube. All the constructions he describes, however, call for tools other than a compass and straightedge.
Actually doing these constructions is not just difficult with the tools allowed; it is impossible. This was proved using algebraic arguments in the nineteenth century. Nevertheless, because the goals of the constructions are so easily stated and understood, and because the tools are so simple, people continue to work at them, not knowing, or perhaps not really caring, that their task is a Sisyphean one. (A Sisyphean task is one that is similar to the task forced upon Sisyphus, a king in Greek mythology, who was punished in the underworld by being forced to roll a boulder up a steep hill, only to see it at the top roll back down where he was again forced to roll it back up the hill—for eternity.)
The straightedge and compass are certainly simple tools, yet mathematicians have tried to get along with even simpler ones. In the tenth century, Persian mathematician and astronomer Abul Wafa (940–c. 998) based his constructions on a straightedge and a rusty compass—one that could not be adjusted. Nine centuries later, it was proved by French mathematician Jean-Victor Poncelet (1788–1867) and Swiss mathematician Jakob Steiner (1796–1863) that, except for drawing circles of a particular size, a straightedge and rusty compass could do everything a straightedge and ordinary compass could do. They went even further, replacing the rusty compass with one already-drawn circle and its center.
In 1797, Italian mathematician Lorenzo Mascheroni (1750–1800) published a book in which he showed that a compass alone could be used to do anything that one could do with a compass and straightedge together. He could not draw straight lines, of course, but he could locate the two points that would determine the undrawn line; he could find where two undrawn lines would intersect; he could locate the vertices of a pentagon; and so on. Later, his work was found to have been anticipated more than 100 years earlier by Danish mathematician Georg Mohr (1640–1697). Compass-only constructions are now known as Mohr-Mascheroni constructions.
Resources
BOOKS
Euclid. Elements. Sir Thomas L. Heath, translator. New York: Dover Publishing Co., 1956.
Gardner, Martin. Mathematical Circus. New York: Alfred A. Knopf, 1979.
Henle, Michael. Modern Geometries: Non-Euclidean, Projective, and Discrete. Upper Saddle River, NJ: Prentice Hall, 2001.
Noronha, Maria Helena. Euclidean and Non-Euclidean Geometries. Upper Saddle River, NJ: Prentice Hall, 2002.
Silvester, John R. Geometry: Ancient and Modern. Oxford, UK, and New York: Oxford University Press, 2001.
Slavin, Stephen L. Geometry: A Self-teaching Guide. Hoboken, NJ: John Wiley & Sons, 2005.
Thomas, David Allen. Modern Geometry. Pacific Grove, CA: Brooks/Cole, 2001.
J. Paul Moulton