Construction with the rule and the compass

Euclide based its geometry on a system of Axiome S which ensures in particular that it is always possible to trace a right passing by two S not given and that it is always possible to trace a Cercle of center given and passing by a given point. The Euclidean Géométrie is thus the geometry of the right-hand sides and the circles, therefore of the rule and the compass . The intuition of Euclide was that any number could be built using these two instruments.

This Conjecture on the one hand will call in question the definition of a number: the rational numbers are not enough to express all the lengths since the diagonal of a square on side 1 is constructible, but corresponds to the number √2 which is not rational, in addition to engage the mathematical community in the search for impossible resolutions, like the Quadrature of the circle, the Trisection of the angle and the Duplication of the cube. The research of the constructible numbers and the constructible polygons will lead, after the development of the Algèbre and the Théorie of Welshman, on the Théorème of Gauss-Wantzel on the constructible Polygone S and to the Théorème of Wantzel for the constructible numbers.

Georg Mohr (1672) then Lorenzo Mascheroni (1797) will prove that any construction with the rule and the compass can be carried out with the compass alone.

The rule and the compass in architecture

The links which link constructions with the rule and the compass with the Architecture are very close:

The geometry offers several resources to the architect: it familiarizes it with the rule and the compass, which are especially used to him to determine the site of the buildings. (Vitruve)

With the Middle Ages, the Master architect is that which has the knowledge of the geometry, he discusses on a foot equality with the religious leaders. In a company where little can read, the plan, built with the rule and the compass, is the only simple means of communication between the architect and the workmen. The difficulty of the scale arises however since the units of length are not completely standardized. On the ground, with his compass, the parlier or Master of building site activates themselves then which establishes the link between the architect and the workmen. Those use for their construction, a compass and a rule but also, when the size of measurements is too important, the chalk line which replaces indifferently the rule (line drawn with the chalk line) and the compass. Associated with the square, the rule and the compass become then the symbol of the Architecte, main architect of the Cathédrale S or Architect of the world. Thus in the emblem of the Franc-maçonnerie they are found.

The rule and the compass in art

The role of construction to the rule and the compass in artistic works depends much on the time and the artistic currents.

In the Byzantine icons, the rule and the compass define the gun representation. The head of the saints, for example, is built on the basis of three concentric circle: for the face, the other for the contour of the head and the third for the Aureole. The rule and the compass play the same part of gun in the construction of the Mandala S in the Bouddhisme Tibetan: built initially using a rule and of a compass, they are then made out of sand of color. With the Rebirth, the Occident redécouvre the Elements of Euclide (translation of 1482). The Italian artists see then in constructions with the rule and the compass a source of Harmonie. Léonard de Vinci registers his Homme of Vitruve in a circle and a square. Albrecht To last, author of the book Instructions for measurement with the rule and the compass , built its Adam and Eve using circles and of right-hand sides. In Typography also, the artists try to find a coding harmonious of the Romance letters. Felice Feliciano (1460) seems to be the first to build letters with the rule and the compass. Francesco Torniello, Luca Pacioli and Albrecht Dürer encases the step to him. This movement perdure until 1764, date on which Fournier, in his Manuel of typography , fraudulently registered against the idea that the harmony of the letters is due to their rigorous construction. The beauty of the letters comes only from the artistic quality of their draftsman.

The development of the Perspective request a geometrical preparation of work, but the circles and the lines are not whereas a grid which makes it possible the artist to place the forms at the liking of its imagination and of its sensibilté.

About 1900, is born a movement from a new kind with Pablo Picasso and Georges Braque: the Cubism. The forms are fragmented and fall under geometrical configurations. The artists think that the beauty can spout out pure geometrical form. Wassily Kandinsky, founder of the Blaue Reiter and initiator of the Abstract art, creates in 1923 a purely geometrical work Cercles in a circle . Victor Vasarely, one of the Masters of the geometrical abstract art and father of COp Art, generalizes the use of the compass and the rule in tables which conceal geometrical figures by often giving an impression of volume.

The rule and the compass in geometry

Some constructions with the rule and the compass

Parallel and perpendicular

Parallel : It is possible to trace the parallel with right-hand side (AB) passing by the point C.

To build the fourth point of a Parallelogram ABCX by tracing a Arc of circle of center C and ray BA and an arc of circle of center has and BC.

Perpendicular : In the same way, it is possible to trace the perpendicular with right-hand side (AB) passing by the point C.

Construire the symmetrical point C compared to the line (AB) . It is the point of intersection of the circle centers has and of ray AC with the circle of center B and BC .

Mediator of a segment

The principal construction of the geometry is undoubtedly the layout of the Médiatrice of a segment.

The mediator of the segment is the line D which crosses perpendicularly in its medium I .

Theorem: The mediator of a segment is the whole of the points which are at equal distance from its ends.
reciprocal Theorem: The whole of the equidistant points of the ends of a segment is the mediator of this segment.

This is seen easily by noticing that if one considers a point M mediator, the segments and are symmetrical compared to the mediator. One can also use the Théorème of Pythagore in the right-angled triangles FRIENDLY and IMB and show the equality of their Hypoténuse S.

Therefore, if one can build the mediator, one can thus determine the medium of a segment and to trace a perpendicular on a line.

For that, one opens the compass over a length higher than half the length of the segment, then one traces two circles with this radius, one centered on has , the other on B (one can be satisfied to trace only arcs of circle). The intersection of the two circles consists of two points located at equal distance from has and of B , and which thus defines well the mediator.

Bisectrix of an angle

It would be appropriate, more correctly, to speak about Bissectrice of a angular sector. It is about the Axis of symmetry of this sector.

To point the compass at the top of the angle and to trace a first Arc of circle. To mark the points of intersection of this arc with the two sides of the angle.
successively To point the compass at the points of intersection and to trace two arcs of circle of the same ray (while keeping the same spacing of the compass enters the two operations). To mark the point of intersection of these two arcs.
To connect the top of the angle and the point of intersection of the last two circles. The bisectrix appears.

It is thus possible to cut an angle in two equal shares. It is unfortunately not possible to cut out, with the rule and the compass, an angle into three share equal. It is the problem of the Trisection of the angle.

Geometry of the triangle

Regular polygons

See also: Theorem of Gauss-Wantzel, Division of a tart

A regular polygon is an inscribable polygon in a circle and of which all the sides are equal. It is easy to build with the rule and the compass the Triangle, the Carré and the Hexagone. With more difficulties, one can build the pentagon. One can rather easily double the number on sides of a constructible polygon by tracing Bissectrice S and build, for example, polygons with $2 \ times 4 = 8$ , with $2 \ times 5 = 10$ and with $2 \ times 6 = 12$ sides. Even if it is theoretically possible to trace polygons with $4 \ times 5 = 20$ , with $8 \ times 5 = 40$ and with $16 \ times 5 = 80$ sides, their realization is difficult in practice. Remain among the polygons with less than 10 sides, the Heptagone (7 sides) and the Ennéagone (9 sides) which are not constructible; It will be necessary to await Gauss, then Wantzel to make an inventory of all the constructible polygons.

Constructible numbers

For Euclide, a constructible Nombre is a number associated with a constructible length. Nowadays, a constructible number is a number obtained like constructible punctual coordinate starting from a squaring. One knows now that the whole of the constructible numbers contains the whole of the rational numbers, but is strictly included in the whole of the algebraic numbers. It is known in particular that $\ pi$ , transcendent Nombre, is not constructible (Quadrature of the circle) and that $\ sqrt 2$ is not it (Duplication of the cube)

See too

External bonds

approximate Construction To last of the pentagon with five of the same circles ray.
Construction of a wine tank in the egg form Traced of the pentagon (method of Ptolémée) and egg with the golden section.
Eight methods of construction of the pentagon to the rule and the compass.

Random links: