Mohr–Mascheroni theorem
Template:Short description In Euclidean geometry, the Mohr–Mascheroni 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.<ref>Template:Harvnb</ref>
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 unneccessary for the purposes of construction.
History
The result was originally published by Georg Mohr in 1672,<ref>Georg Mohr, Euclides Danicus (Amsterdam: Jacob van Velsen, 1672).</ref> but his proof languished in obscurity until 1928.<ref name=Eves199>Template:Harvnb</ref><ref>Hjelmslev, J. (1928) "Om et af den danske matematiker Georg Mohr udgivet skrift Euclides Danicus, udkommet i Amsterdam i 1672" [Of a memoir Euclides Danicus published by the Danish mathematician Georg Mohr in 1672 in Amsterdam], Matematisk Tidsskrift B, pages 1–7.</ref><ref>Schogt, J. H. (1938) "Om Georg Mohr's Euclides Danicus," Matematisk Tidsskrift A, pages 34–36.</ref> The theorem was independently discovered by Lorenzo Mascheroni in 1797 and it was known as Mascheroni's Theorem until Mohr's work was rediscovered.<ref>Lorenzo Mascheroni, La Geometria del Compasso (Pavia: Pietro Galeazzi, 1797). 1901 edition.</ref>
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.<ref name="Eves199" /> In 1890, August Adler published a proof using the inversion transformation.<ref>Template:Harvnb</ref>
An algebraic approach uses the isomorphism between the Euclidean plane and the real coordinate space <math>\mathbb{R}^2</math>. In this way, a stronger version of the theorem was proven in 1990.<ref>Arnon Avron, "On strict strong constructibility with a compass alone", Journal of Geometry (1990) 38: 12.</ref> It also shows the dependence of the theorem on Archimedes' axiom (which cannot be formulated in a first-order language).
Constructive proof
Outline
To prove the Mohr–Mascheroni 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:
- Creating the line through two existing points
- Creating the circle through one point with centre another point
- Creating the point which is the intersection of two existing, non-parallel lines
- Creating the one or two points in the intersection of a line and a circle (if they intersect)
- Creating the one or two points in the intersection of two circles (if they intersect).
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.
Notation
The following notation will be used throughout this article. A circle whose center is located at point Template:Mvar and that passes through point Template:Mvar will be denoted by Template:Math. A circle with center Template:Mvar and radius specified by a number, Template:Mvar, or a line segment Template:Math will be denoted by Template:Math or Template:Math, respectively.<ref>Template:Harvnb</ref>
Some preliminary constructions
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.
Compass equivalence theorem (circle translation)
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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 Template:Mvar, Template:Mvar, and Template:Mvar, construct a circle centered at Template:Mvar with the radius Template:Mvar, using only a collapsing compass.
- Draw a circle centered at Template:Mvar and passing through Template:Mvar and vice versa (the blue circles). They will intersect at points Template:Mvar and Template:Mvar.
- Draw circles through Template:Mvar with centers at Template:Mvar and Template:Mvar (the red circles). Label their other intersection Template:Mvar.
- Draw a circle (the green circle) with center Template:Mvar passing through Template:Mvar. This is the required circle.<ref name=Eves185>Template:Harvnb</ref><ref>Template:Citation</ref>
Reflecting a point across a line
Given a line Template:Math determined by two points Template:Mvar and Template:Mvar, and an arbitrary point Template:Mvar, construct the image of Template:Mvar upon reflection across this line:
- Construct two circles: one centered at Template:Mvar and one centered at Template:Mvar, both passing through Template:Mvar.
- The other point of intersection of the two circles, Template:Mvar, is the reflection of Template:Mvar across the line Template:Math.
- If Template:Math (that is, there is a unique point of intersection of the two circles), then Template:Mvar is its own reflection and lies on the line Template:Math.
Extending the length of a line segment
Given a line Template:Math determined by two points Template:Mvar and Template:Mvar, construct the point Template:Mvar on the line such that Template:Mvar is the midpoint of line segment Template:Math.<ref name="Pedoe 1988 loc=p. 78">Template:Harvnb</ref>
- Construct point Template:Mvar as the intersection of circles Template:Math and Template:Math. (Template:Math is an equilateral triangle.)
- Construct point Template:Math as the intersection of circles Template:Math and Template:Math. (Template:Math is an equilateral triangle.)
- Finally, construct point Template:Math as the intersection of circles Template:Math and Template:Math. (Template:Math is an equilateral triangle, and the three angles at Template:Mvar show that Template:Math are collinear.)
This construction can be repeated as often as necessary to find a point Template:Mvar so that the length of line segment Template:Math is Template:Math times the length of line segment Template:Math for any positive integer Template:Math.
Inversion in a circle
Given a circle Template:Math, for some radius Template:Mvar (in black) and a point Template:Math, construct the point Template:Mvar that is the inverse of Template:Mvar about the circle.<ref>Template:Harvnb</ref> Naturally there is no inversion for a point Template:Math.
- Draw a circle Template:Math (in red).
- Assume that the red circle intersects the black circle at Template:Mvar and Template:Mvar
- If the circles do not intersect in two points, see below for an alternative construction.
- If the circles intersect in only one point, Template:Math, it is possible to invert <math>D</math> simply by doubling the length of Template:Math (quadrupling the length of Template:Math).
- Reflect the circle center Template:Mvar across the line Template:Math:
- Construct two new circles Template:Math and Template:Math (in light blue).
- The light blue circles intersect at Template:Mvar and at another point Template:Math.
- Point Template:Mvar is the desired inverse of Template:Mvar in the black circle.
This point Template:Mvar lies on line Template:Math and satisfies Template:Math.
In the event that the above construction fails (that is, the red circle and the black circle do not intersect in two points),<ref name="Pedoe 1988 loc=p. 78"/> find a point Template:Mvar on the line Template:Math so that the length of line segment Template:Math is a positive integral multiple, say Template:Mvar, of the length of Template:Math and is greater than Template:Math. Find Template:Mvar the inverse of Template:Mvar in circle Template:Math as above (the red and black circles must now intersect in two points). The point Template:Mvar is now obtained by extending Template:Math so that Template:Math = Template:Math.
The existence of such an integer Template:Mvar relies on Archimedes' axiom. As a result, this construction may require an unbounded number of iterations depending on the ratio of Template:Math to Template:Math.
Determining the center of a circle through three points
Given three non-collinear points Template:Mvar, Template:Mvar and Template:Mvar, construct the center Template:Mvar of the circle they determine.<ref name=Pedoe123>Template:Harvnb</ref>
- Construct point Template:Mvar, the inverse of Template:Mvar in the circle Template:Math.
- Reflect Template:Mvar in the line Template:Math to the point Template:Mvar.
- Template:Mvar is the inverse of Template:Mvar in the circle Template:Math.
Intersection of two non-parallel lines
The third basic construction concerns the intersection of two non-parallel lines.
Given non-parallel lines Template:Math and Template:Math determined by points Template:Mvar, Template:Mvar, Template:Mvar, Template:Mvar, construct their point of intersection, Template:Mvar.<ref name=Pedoe123 />
- Select circle Template:Math of arbitrary radius whose center Template:Mvar does not lie on either line.
- Invert points Template:Mvar and Template:Mvar in circle Template:Math to points Template:Mvar and Template:Mvar respectively.
- The line Template:Math is inverted to the circle passing through Template:Mvar, Template:Mvar and Template:Mvar. Find the center Template:Mvar of this circle.
- Invert points Template:Mvar and Template:Mvar in circle Template:Math to points Template:Mvar and Template:Mvar respectively.
- The line Template:Math is inverted to the circle passing through Template:Mvar, Template:Mvar and Template:Mvar. Find the center Template:Mvar of this circle.
- Let Template:Math be the intersection of circles Template:Math and Template:Math.
- Template:Mvar is the inverse of Template:Mvar in the circle Template:Math.
Intersection of a line and a circle
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.
Circle center is not collinear with the line
Assume that center of the circle does not lie on the line.
Given a circle Template:Math (in black) and a line Template:Math, construct the points of intersection, Template:Mvar and Template:Mvar, between them (if they exist).<ref name=Hunger784>Template:Harvnb</ref><ref name="Eves199"/>
- Construct the point Template:Mvar, which is the reflection of point Template:Mvar across line Template:Math. (See above.)
- Under the assumption of this case, Template:Math.
- If in fact Template:Math then this construction will fail, and we have verification of collinearity.
- Construct a circle Template:Math (in red). (See above, compass equivalence.)
- The intersections of circle Template:Math and the new red circle Template:Math are points Template:Mvar and Template:Mvar.
- If the two circles are (externally) tangential then Template:Math.
- If the two circles do not intersect then neither does the circle with the line.
- Points Template:Mvar and Template:Mvar are the intersection points of circle Template:Math and the line Template:Math.
- If Template:Math then the line is tangential to the circle Template:Math.
An alternate construction, using circle inversion can also be given.<ref name=Pedoe123 />
- Invert points Template:Mvar and Template:Mvar in circle Template:Math to points Template:Mvar and Template:Mvar respectively.
- Under the assumption of this case, points Template:Mvar, Template:Mvar, and Template:Mvar are not collinear.
- Find the center Template:Mvar of the circle passing through points Template:Mvar, Template:Mvar, and Template:Mvar.
- Construct circle Template:Math, which represents the inversion of the line Template:Math into circle Template:Math.
- Template:Mvar and Template:Mvar are the intersection points of circles Template:Math and Template:Math.<ref>Pedoe carries out one more inversion at this point, but the points Template:Mvar and Template:Mvar are on the circle of inversion and so are invariant under this last unneeded inversion.</ref>
- If the two circles are (internally) tangential then Template:Math, and the line is also tangential.
Circle center is collinear with the line
Given the circle Template:Math whose center Template:Mvar lies on the line Template:Math, construct the points Template:Mvar and Template:Mvar, the intersection points of the circle and the line.<ref>Template:Harvnb</ref>
- Choose an arbitrary point Template:Mvar on the circle.
- Construct point Template:Math as the reflection of Template:Mvar across line Template:Math.
- Construct point Template:Mvar as the intersection of circles Template:Math and Template:Math. (Template:Mvar is the fourth vertex of parallelogram Template:Mvar.)
- Construct point Template:Mvar as the intersection of circles Template:Math and Template:Math. (Template:Mvar is the fourth vertex of parallelogram Template:Mvar.)
- Construct point Template:Mvar as an intersection of circles Template:Math and Template:Math. (Template:Mvar lies on Template:Math.)
- Points Template:Mvar and Template:Mvar are the intersections of circles Template:Math and Template:Math.
Conclusion
Since all five basic constructions have been shown to be achievable with only a compass, this proves the Mohr–Mascheroni theorem. Any compass-straightedge construction may be achieved with the compass alone by describing their constructive steps in terms of the five basic constructions.
Validity of the theorem
Dono Kijne points out that the Mohr–Mascheroni theorem fundamentally relies on Archimedes' axiom. As a result, any proof of Mohr–Mascheroni theorem must inherently involve an unbounded number of steps.<ref>Template:Cite thesis</ref> 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 Mohr–Mascheroni 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.<ref>Template:Cite book</ref> This saves the Mohr–Mascheroni theorem, but introduces new issues:
For example, consider straightedge-only constructions within the rational plane <math>\mathbb Q^2</math>. If we allow an unbounded number of steps, then given any four points <math>A,B,C,D\in\mathbb Q^2</math> in general position, we can enumerate all rational points and lines in <math>\mathbb Q^2</math>. By simply "waiting" for a line parallel to Template:Math to appear, that line can then be used to construct the midpoint of Template:Math. This construction does not look like an intuitively valid construction and contradicts the belief that constructing the midpoint using a straightedge is impossible.<ref>Template:Cite arXiv</ref>
Other types of restricted construction
Restrictions involving the compass
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).<ref>Template:Citation</ref>
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.
Restrictions excluding the compass
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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 Poncelet–Steiner theorem, was proved by Jakob Steiner eleven years later.
Further generalizations
The Mohr–Mascheroni 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.<ref>Template:Cite thesis</ref>
Additionally, some research is underway to generalize the Mohr–Mascheroni theorem to non-Euclidean geometries.