Geometric Construction with the Compass Alone
Everything you can do with a ruler and a compass you can do with the compass alone.
Well, not everything. For example, you can't draw straight lines using a compass. There is no talking about it. However, you can do everything reasonable. I hope you would find this claim no less remarkable.
In what is known as the Geometry of Compass, a straight line is defined by any pair of two points. Starting with two points, other points can be constructed with compass alone. Thus in the following constructing a straight line means finding two points that belong to that line.
Remark
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There are geometries in which the ruler is never used to start with. E. g., in finite geometries that only contain a finite number of points and lines, a line is just a (finite) collection of points. On the sphere, the role of straight lines is played by the great circles. The question of geometric construction with the compass alone is not concerned with such kinds of geometries. Geometry of Compass only deals with constructions in the Euclidean plane, and its basic question could be formulated as, What ruler-and-compass constructions could be accomplished with the compass alone? |
The assertion that every ruler-and-compass construction could be accomplished with a compass is due to Lorenzo Mascheroni (1750-1800) and appeared in his 1797 tractate The Geometry of Compasses.
Interestingly, in 1928 the Danish mathematician Hjelmslev discovered in a bookshop in Copenhagen a book
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Inspired by Mascheroni's result, Jacob Steiner (1796-1863) tried to prove a similar result for a straightedge instead of a compass. In his book Geometrical Constructions Using a Straight Line and a Fixed Circle published in 1833, Steiner was able to prove that given a fixed circle and its center, all the constructions in the plane can be carried out by the straightedge alone. Using only elementary Projective Geometry it can be shown that the center of the circle is indispensable.
With regard to the Mascheroni's result, instead of checking every single construction in the plane we agree that such constructions can be accomplished with a sequence of the four basic ones:
- To draw a circle with the given center and radius
- To find the point of intersection of two circles
- To find the points of intersection of a straight line and a circle
- To find a point of intersection of two straight lines
The difficulty obviously lies with the last two problems. In the Geometry of Compass constructions may be awfully obscure even for simple problems. To avoid complicating the matters it's always useful to split a problem into a number of simpler steps. A proof to the Mascheroni result will emerge as a combination of the problems below. (However, not all of the problems are related to the proof.)
Problems (Use a compass only)
In all problems below a segment AB is given by its end points A and B.
- Construct segments 2, 3, 4, etc. times larger than AB.
- A point C is known to lie outside the straight line AB. Construct a point D symmetric to C with respect to AB.
- A circle is given by its radius R and the center O. Assume O does not lie on AB.
Find the points of intersection of the circle with the segment AB.
- Find a point C such that AC is perpendicular to AB.
- Determine whether three given points A, B, C lie on the same line.
- Given three points A, B and C. C is known to lie outside the straight line AB.
Complete the parallelogram ABCD.
- Let two points A and B belong to a circle with center O. Bisect the two arcs of the circle
defined by the points A and B.
- A circle is given by its radius R and the center O that lies on AB.
Find the points of intersection of the circle with the segment AB.
- Build a square with the side AB.
- Let the quantities a, b, c be defined as the lengths of three given segments. Find x such that
a/b = c/x. - Find the intersection point of two lines each given by a pair of points - AB and CD, respectively.
- Construct segments 2, 3, 4, etc. times smaller than AB.
- Construct the center of a given circle.
- Bisect a given line AB.
- Construct a Regular Pentagon
References
- R. Courant and H. Robbins, What is Mathematics?, Oxford University Press, 1996
- H. Dorrie, 100 Great Problems Of Elementary Mathematics, Dover Publications, NY, 1965.
- M. Gardner, Mathematical Circus, Vintage Books, NY, 1981
- R. Honsberger, Ingenuity in Mathematics, MAA, New Math Library, 1970
- A. Kostovskii, Geometrical Construction with Compasses Only, Mir Publishers, Moscow, 1986
- G. E. Martin, Geometric Constructions, Springer, 1998
- S. K. Stein, Mathematics: The Man-Made Universe, 3rd edition, Dover, 2000.
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