In Euclidean geometry, the MohrâÂÂMascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone.
This theorem refers to geometric constructions which only involve points and circles, since it is not possible to draw straight lines without a straightedge. However, a line is considered to be determined if two distinct points on that line are given or constructed, even if the line itself is not drawn.
Although the use of a straightedge can make certain constructions significantly easier, the theorem shows that these constructions are possible even without the use of it. This means the only use of a straightedge is for the aesthetics of drawing straight lines, and is functionally unnecessary for the purposes of construction.
The result was originally published by Georg Mohr in 1672, but his proof languished in obscurity until 1928. The theorem was independently discovered by Lorenzo Mascheroni in 1797 and it was known as Mascheroni's Theorem until Mohr's work was rediscovered.
Several proofs of the result are known. Mascheroni's proof of 1797 was generally based on the idea of using reflection in a line as the major tool. Mohr's solution was different. In 1890, August Adler published a proof using the inversion transformation.
An algebraic approach uses the isomorphism between the Euclidean plane and the real coordinate space . In this way, a stronger version of the theorem was proven in 1990. It also shows the dependence of the theorem on Archimedes' axiom (which cannot be formulated in a first-order language).
To prove the MohrâÂÂMascheroni theorem, it suffices to show that each of the basic constructions of compass and straightedge is possible using a compass alone, as these are the foundations of all other constructions. All constructions can be written as a series of steps involving these five basic constructions:
Constructions (2) and (5) can be done with a compass alone. For construction (1), a line is considered to be given by any two points. It is understood that the line itself cannot be drawn without a straightedge, so the proof of the theorem lies in showing that constructions (3) and (4) are possible using only a compass. Once this is done, it follows that every compass-straightedge construction can be done under the restrictions of the theorem.
The following notation will be used throughout this article. A circle whose center is located at point and that passes through point will be denoted by . A circle with center and radius specified by a number, , or a line segment will be denoted by or , respectively.
To prove the above constructions (3) and (4), a few necessary intermediary constructions are also explained below since they are used and referenced frequently. These are also compass-only constructions.
The modern compass with its fixable aperture can be used to transfer distances directly, while a collapsible compass cannot. The compass equivalence theorem states that, while a "modern compass" appears to be a more powerful instrument, it can be simulated with a collapsing compass alone. This justifies the use of "fixed compass" moves (constructing a circle of a given radius at a different location) for the proof of this theorem.
Given points , , and , construct a circle centered at with the radius , using only a collapsing compass.
Given a line determined by two points and , and an arbitrary point , construct the image of upon reflection across this line:
Given a line determined by two points and , construct the point on the line such that is the midpoint of line segment .
This construction can be repeated as often as necessary to find a point so that the length of line segment is times the length of line segment for any positive integer .
Given a circle , for some radius (in black) and a point , construct the point that is the inverse of about the circle. Naturally there is no inversion for a point .
This point lies on line and satisfies .
In the event that the above construction fails (that is, the red circle and the black circle do not intersect in two points), find a point on the line so that the length of line segment is a positive integral multiple, say , of the length of and is greater than . Find the inverse of in circle as above (the red and black circles must now intersect in two points). The point is now obtained by extending so that = .
The existence of such an integer relies on Archimedes' axiom. As a result, this construction may require an unbounded number of iterations depending on the ratio of to .
Given three non-collinear points , and , construct the center of the circle they determine.
The third basic construction concerns the intersection of two non-parallel lines.
Given non-parallel lines and determined by points , , , , construct their point of intersection, .
The fourth basic construction concerns the intersection of a line and a circle. The construction below breaks into two cases depending upon whether the center of the circle is or is not collinear with the line.
Assume that center of the circle does not lie on the line.
Given a circle (in black) and a line , construct the points of intersection, and , between them (if they exist).
An alternate construction, using circle inversion can also be given.
Given the circle whose center lies on the line , construct the points and , the intersection points of the circle and the line.
Since all five basic constructions have been shown to be achievable with only a compass, this proves the MohrâÂÂMascheroni theorem. Any compass-straightedge construction may be achieved with the compass alone by describing their constructive steps in terms of the five basic constructions.
Dono Kijne points out that the MohrâÂÂMascheroni theorem fundamentally relies on Archimedes' axiom. As a result, any proof of MohrâÂÂMascheroni theorem must inherently involve an unbounded number of steps. This raises some questions about what constitutes a valid geometric construction.
Most geometric constructions can be thought of as "straight-line programs", a list of elementary instructions with a fixed number of steps. Under this model, the MohrâÂÂMascheroni theorem would not qualify as a valid result because it has no a priori bound on the number of iterations required.
To address this, Erwin Engeler suggested that geometric constructions be defined as "programs with loops", a list of instructions that allow conditionals and control flow. This saves the MohrâÂÂMascheroni theorem, but introduces new issues:
For example, consider straightedge-only constructions within the rational plane . If we allow an unbounded number of steps, then given any four points in general position, we can enumerate all rational points and lines in . By simply "waiting" for a line parallel to to appear, that line can then be used to construct the midpoint of . This construction does not look like an intuitively valid construction and contradicts the belief that constructing the midpoint using a straightedge is impossible.
Renaissance mathematicians Lodovico Ferrari, Gerolamo Cardano and Niccolò Fontana Tartaglia and others were able to show in the 16th century that any ruler-and-compass construction could be accomplished with a straightedge and a fixed-width compass (i.e. a rusty compass).
The compass equivalence theorem shows that in any construction, a rigid compass, which preserves distances, may be replaced with a collapsible compass, which does not preserve distances. It is possible to translate any circle in the plane with a collapsing compass using no more than three uses of the compass than with a rigid compass. In fact, Euclid's original constructions use a collapsible compass.
Motivated by Mascheroni's result, in 1822 Jean Victor Poncelet conjectured a variation on the same theme. His work paved the way for the field of projective geometry, wherein he proposed that any construction possible by straightedge and compass could be done with straightedge alone. However, the one stipulation is that no less than a single circle with its center identified must be provided. This statement, now known as the PonceletâÂÂSteiner theorem, was proved by Jakob Steiner eleven years later.
The MohrâÂÂMascheroni theorem has been generalized to higher dimensions, such as, for example, a three-dimensional variation where the straightedge is replaced with a plane, and the compass is replaced with a sphere. It has been shown that n-dimensional "straightedge and compass" constructions can still be performed even with just an ordinary two-dimensional compass.
Additionally, some research is underway to generalize the MohrâÂÂMascheroni theorem to non-Euclidean geometries.