Set of Mohr Mascheroni

The set of Mohr-Mascheroni from the synthetic geometry states that any construction with ruler and compass can be done already with compass alone is possible. It is named after the mathematicians Georg Mohr and Lorenzo Mascheroni , who proved it independently of one another. It is thus the counterpart of the Poncelet-Steiner theorem .

history

The theorem was first proven in 1672 by Georg Mohr. His proof was forgotten, so that the theorem was proven again by Lorenzo Mascheroni in 1797. Only later was Mohr's proof rediscovered by Johannes Hjelmslev , and the theorem was named after the two mathematicians. A series of much simpler proofs followed. Most of these proofs are elementary geometrical; Jean-Claude Carrega provided an algebraic proof of the statement.

statement

The sentence says the following: If there is - starting from a given set of points - a construction method that constructs a point P by repeatedly applying the elementary constructions 1 to 5, there is also a method that constructs point P from the same starting position , but only constructions 2 and 5 are used. The elementary constructions are:

The straight line through these two points can be constructed for two (given or constructed) points.
The circle can be constructed for two points, which has its center in the first point and on whose circumference the second point lies.
The intersection point can be constructed for two straight lines (provided the straight lines are not parallel).
The intersection points (if they exist) or the point of contact can be constructed for a straight line and a circle.
The points of intersection or the point of contact can be constructed for two circles.

Proof idea

Construction of the inversion on the circle with compasses alone: The original image point P is mirrored on the inversion circle (red), resulting in the image point P '.

Most of the evidence boils down to specifying construction methods for determining the intersection of two straight lines or the intersection of a straight line with a circle using the compass alone, the straight lines being given by only two points. The following proof idea makes use of the inversion on the circle and goes back to August Adler (1863–1923). The inversion on the circle has the property that it maps straight lines and circles that do not go through the inversion center onto circles. The point of intersection can be traced back to the intersection of two circles, which can be constructed directly.

For the intersection of two straight lines you have to do the following: Given are four points A, B, C and D, the point of intersection S of the straight line AB with the straight line CD is sought. First you choose any point O, which must not lie on any of the straight lines, and draw any circle with O as the center. Then one constructs the image points A ', B', C 'and D' of the four given points with inversion on this circle and the circles through the points A ', B' and O, as well as through C ', D' and O. These Both circles are the images of the straight lines AB and CD and, apart from in O, intersect at a further point S ', the image point of the intersection point sought. The point of intersection S is obtained from this by renewed inversion.

The same procedure is used for the intersection of a straight line and a circle. It is therefore sufficient to mirror points, straight lines and circles with compasses on a given circle, for which constructions are known. For the construction of the image of a single point, it is explained under circular reflection in the paragraph With compass alone .

Constructions by Mascheroni

Napoleon's problem

Problem of Napoleon
construction solely with a compass after Mascheroni. The value , quasi the value of the diagonal of the square with the side length r, is constructed in this. Animation see

{\ displaystyle r {\ sqrt {2}}}

One of the most famous constructions by Mascheroni shows the solution to the so-called problem of Napoleon . In it, a given circular line k with its center M is subdivided into four circular arcs of equal length , using only a pair of compasses, by constructing the value of the diagonal of the square with the side length r, so to speak . ${\ displaystyle r {\ sqrt {2}}}$

The dotted lines and the colored triangle shown in the adjacent picture are not required for the solution of the task, they are only intended to illustrate the mathematical relationships.

Construction description

It begins with the drawing of the circular line k with any radius r around the center M. After setting the point A on k, the radius r is plotted three times using short circular arcs on k, resulting in points B, C and D. The Points ADC are corner points of the right triangle with the hypotenuse 2r and the small catheter r. According to Pythagoras' theorem, this results in the value for the larger leg. This length, i.e. distance AC , is now taken into the circle and a short arc around point A from C or around point D from B is drawn. The two circular arcs create the intersection point E. The points AME are corner points of the right-angled triangle with the hypotenuse and the small catheter. With the help of Pythagoras' theorem, the result for the larger leg is the value This is also the value of the sought diagonal of the square with the side length r. Finally, an arc around point D with the radius (distance ME ), which intersects the circular line k at points F and G, is required . Thus, the circular arcs MAG, MGD, MDF and MFA divide the length of the circular line k into four parts of equal length. ${\ displaystyle r {\ sqrt {3}}.}$ ${\ displaystyle r {\ sqrt {3}}}$ ${\ displaystyle r {\ sqrt {2}}.}$ ${\ displaystyle r {\ sqrt {2}}}$

...

August Adler explains in his book Theory of Geometric Constructions from 1906 a. a. three constructions by Mascheroni from his work “La geometria del compasso”, Pavia 1797, which were created solely using a compass. The first construction divides a given segment AB into any equal parts, the second determines the center of a circle and the third describes the construction blocks in pentagon, the halving of the circular arc and the construction of the pentagon and decagon side . These three circular constructions are shown and described below.

Find the middle of a route

Construction of the middle of a route solely with a compass according to Mascheroni. Can also be used to determine the center of a distance between two points, see animation .

By dividing the line AB into two equal parts , the middle is determined.

The colored triangle shown in the adjacent picture is not required for the solution of the task, it is only intended to illustrate the mathematical relationships.

Construction description

First, the distance is about the point A AB of the circular arc k ₁ with radius AB located.

The next circular arc k ₂ , around point B with radius AB , brings the points of intersection E and E '.

The circular arc k ₃ around the point E, through the point E 'with the radius | EE ' | , gives the intersection point C on k ₂ .

The arc k ₄ follows around point C, through point A with radius | CA | 'It creates the intersections D and D'.

Now one draws the circular arc k ₅ around the point D ', through the point A with the radius | D'A | .

The final circular arc k ₆ around point D, through point A with radius | DA | supplies the intersection point X, which marks the sought center of the line AB .

This construction can also be used to determine the center of a distance between two points.

Find the center of a circle

Construction of the center of a circle solely with a compass according to Mascheroni, animation see .

Construction description

It begins with the definition of the point O on the circle k ₁ , its position is freely selectable.

The arc k _{2 is} drawn around point O with an arbitrary radius, resulting in the intersection points A and B. In order for these intersection points to be obtained or usable, the radius | OA | greater than half the radius of the circle k ₁ and smaller than its diameter.

The arc k ₃ follows around the point B with the radius | BO | .

The next circular arc k ₄ , around point A with radius | AO | , brings the intersection C.

The circular arc k ₅ , around the point C and through the point O with the radius | CO | , creates the intersection points S ₁ and S ₂ .

Now draw the circular arc k ₆ around the point S ₂ through the point O with the radius | S ₂ O | .

The final circular arc k ₇ around the point S ₁ , through the point O with the radius | S ₁ O | supplies the intersection point X, which marks the sought center point of the circle k ₁ .

pentagon

Construction of the corner points of a regular pentagon, solely with a compass according to Mascheroni, animation see.

The pentagon is an example of the application of the following construction blocks:

Halving the arc of a circle

Construction of the pentagon and decagon sides

The colored pentagon shown in the adjacent picture is not part of the solution (sole use of the compass), it is only intended to serve as an illustration.

Construction description

It starts with setting the radius | AO | for the circle, which is then drawn around its center O.

From point A, four arcs with the radius | AO | Plotted on the circle, the intersection points B, C, D and E.

After Adler now follows the halving of the circular arc A, B (in the adjacent picture: circular arc OCB):

For this purpose, an arc of a circle with the radius | is first created around points D and A DB | drawn, the intersection point G is generated.

Now an arc with the radius | is created around the points C and E 1st floor | drawn, it results in the intersection point K. The subsequent circular arc with the radius | 1st floor | around point A bisects the circular arc OCB at point P ₁ . The resulting distance | P ₁ K | is the side length of the pentagon.

Finally, the side length | P ₁ K | Mark five times on the circle, then the points P ₁ , P ₂ , P ₃ , P ₄ and P ₅ on the circle form a regular pentagon.

literature

Georg Mohr: Euclides danicus. Amsterdam, 1672.
Lorenzo Mascheroni: Geometry du compas. Pavia, 1797. ( online in Google book search)
Jean-Claude Carrega: Théorie des corps - La règle et le compas. Editions Hermann, Paris, 2001, ISBN 978-2-7056-1449-2 .
Norbert Hungerbühler: A short elementary proof of the Mohr-Mascheroni theorem. ( JSTOR 2974536 )

Web links

Eric W. Weisstein : Mascheroni Construction . In: MathWorld (English).

Individual evidence

↑ August Adler: Theory of geometric constructions . GJ Göschensche Verlagshandlung, Leipzig 1906, III. Section, Mascheronische Konstruktionen, § 20. Application of the principle of reciprocal radii, p. 111-112, Fig. 91., p. 301 ( archive.org [accessed June 6, 2018]).
^ ^A ^b August Adler: Theory of geometric constructions . GJ Göschensche Verlagshandlung, Leipzig 1906, III. Section, Mascheronic constructions, § 20. Application of the principle of reciprocal radii, p. 119, Fig. 96., p. 301 ( archive.org [accessed June 4, 2018]).
^ Fritz Schmidt: 200 years of the French Revolution problem and theorem of Napoleon with variations, p. 15. (PDF) In: Didaktik der Mathematik. Bayerischer Schulbuch-Verlag Munich, 1990, p. 29 , accessed on June 8, 2018 .
↑ August Adler: Theory of geometric constructions . GJ Göschensche Verlagshandlung, Leipzig 1906, III. Section, Mascheronic Constructions, § 16. Multiplying and dividing lines, pp. 97-98, Fig. 73., pp. 301 ( archive.org [accessed June 4, 2018]).
↑ August Adler: Theory of geometric constructions . GJ Göschensche Verlagshandlung, Leipzig 1906, III. Section, Mascheronic Constructions, §15. 2. The bisection of the circular arc AB, pp. 93-94, Fig. 70a. and 70b., p. 301 ( archive.org [accessed January 19, 2019]).
↑ August Adler: Theory of geometric constructions . GJ Göschensche Verlagshandlung, Leipzig 1906, III. Section, Mascheronic Constructions, §15. 3. Construction of the pentagon and decagon face, pp. 94-95, Fig. 71., pp. 301 ( archive.org [accessed January 19, 2019]).

[1] August Adler: Theory of geometric constructions . GJ Göschensche Verlagshandlung, Leipzig 1906, III. Section, Mascheronische Konstruktionen, § 20. Application of the principle of reciprocal radii, p. 111-112, Fig. 91., p. 301 ( archive.org [accessed June 6, 2018]).

[August-2] A ^b August Adler: Theory of geometric constructions . GJ Göschensche Verlagshandlung, Leipzig 1906, III. Section, Mascheronic constructions, § 20. Application of the principle of reciprocal radii, p. 119, Fig. 96., p. 301 ( archive.org [accessed June 4, 2018]).

[3] Fritz Schmidt: 200 years of the French Revolution problem and theorem of Napoleon with variations, p. 15. (PDF) In: Didaktik der Mathematik. Bayerischer Schulbuch-Verlag Munich, 1990, p. 29 , accessed on June 8, 2018 .

[4] August Adler: Theory of geometric constructions . GJ Göschensche Verlagshandlung, Leipzig 1906, III. Section, Mascheronic Constructions, § 16. Multiplying and dividing lines, pp. 97-98, Fig. 73., pp. 301 ( archive.org [accessed June 4, 2018]).

[5] August Adler: Theory of geometric constructions . GJ Göschensche Verlagshandlung, Leipzig 1906, III. Section, Mascheronic Constructions, §15. 2. The bisection of the circular arc AB, pp. 93-94, Fig. 70a. and 70b., p. 301 ( archive.org [accessed January 19, 2019]).

[6] August Adler: Theory of geometric constructions . GJ Göschensche Verlagshandlung, Leipzig 1906, III. Section, Mascheronic Constructions, §15. 3. Construction of the pentagon and decagon face, pp. 94-95, Fig. 71., pp. 301 ( archive.org [accessed January 19, 2019]).