ERASMUS PROJECT 2015-2017

1. ERASMUS PROJECT 2015-2017 IIS LS PICCOLO OF CAPO D’ORLANDO Sicily, Italy

2. Math Machines Teacher Randazzo Tullio S.

3. Introduction to the exhibition – Pantograph section Geometrical Transformations Glissossimetria Translation Composition of two central symmetries Axial Simmetry Rotation Central Simmetry Lengthening Omotetia

4. Mechanisms for Transformations Among the various techniques to make transformations, we are going to examine the one which uses articulate systems or linkages. The machanism establishes a local correspondance between the points of two limited regions , belonging to the same plan, by physically linking them and it incorporates the properties characterizing the transformation. The study of the instrument will then let recognise the type of transformation that it makes. While the punter covers a geometrical shape drown on one of the two regions, the tracker draws the corresponding shape on the other one. Punter and tracker can be exchanged (one-to-one correspondence). The articulated system can also have two trackers each one with two degrees of liberty. No initial shape is known; the shapes that the trackers draw at the same time, or the regions that they «cover» during their movement, coincide in a transformation.

5. The Kempe Conveyor This system is made up of two articulated parallelograms having side CD in common and lying on the same plane . One of the opposite sides of CD (for ex. BA) is fixed to , the other, which is free to move, has two degrees available for movement). Originally utilized as a servomechanism to draw segments which are parallel to another given segment (BA, CD, PQ are always parallel to one another), this instrument also generates a particular geometrical transformation (biunique corrispondence between two different realms of ). Indeed, selecting a point on and setting upon it the vertex P (or Q) of the articulated system,the vertex Q (or P) automatically singles out the corresponding one in the translation. The latter is characterized in module and direction by vector BA (or AB). TRANSLATION

6. AXIAL ORTHOGONAL SYMMETRY • An articulated rhombus has two opposite vertices, bound to cursors which run within a rectilinear groove called s. • A connecting-rod has two degrees of movement: the unhindered vertices of the rhombus (P e Q) therefore describe two flat realms (limited) which are situated in opposite half planes having s as a common point of origin. P’s position univocally determines Q’s position (and viceversa). • From the simple geometry of the mechanical system one rapidly deduces that: • the line PQ is perpendicular to s; • Points P and Q are equidistant from s. • Therefore, P e Q correspond to each other in the axial orthogonal symmetry of axis s. • If (for ex.) P is bound to an assigned trajectory, Q then describes a symmetrical trajectory in relation to s.

7. CENTRAL SYMMETRY This mechanism is made up of an articulated rhombus ABCP with side AB hinged to the model plane in its central point O. The rod CB is extended to a segment BQ=CB. This machine produces a correspondence between two limited realms on the same plane in which P and Q are always aligned with O; furthermore PO=OQ. The transformation generated is a centralsymmetry with its centre at O.

8. HOMOTETHY The Scheiner Pantograph This articulated system constists of four rigid rods hinged in points A,B,C and D, selected in order to form a parallelogram. Point O is fixed on the plane on which the mechanism moves. Point P on rod BC is selected in order that the following ensues: BP/PC=OB/OA=k. Points O, D and P remain aligned during the system’s deformation. Therefore, D and P correspond to each other in a homotethy having O as its center.Considering P as corresponding to D we obtain the homotethy ratio k=OP/OD>1.If instead one considers D as corresponding to P, we obtain the homotetia ratio 1/k=OD/OP<1.

9. GLYSSOSYMMETRY This system is made up of: · an articulated rhombus and two opposite vertices. The former causes them to run in a groove called s: the other two vertices P e Q therefore correspond in the symmetry of axis s; ·a Kempe conveyor applied to Q and having one side fixed to the plane in a direction parallel to s. Point P’ (corresponding to Q in the conveyor) corresponds to P in the isometry. This is the product of the axial symmetry of axis s and of the translation singled out by a vector parallel to s: therefore, P and P’ correspond to each other in a glyssosymmetry.

10. This mechanism allows us to verifiy experimentally (in a particular case) the following theorem: The product of two central symmetries, the first having centre A, the second having centre B, is a translation. This presents a modulus equal to twice the length of segment AB, and a direction parallel to said segment. COMPOSITION OF TWOCENTRAL SYMMETRIES A symmetry with its centre at A (implemented by an articulated rhombus) brings point P in P’. A symmetry with its centre at B (implemented by an instrument of the same type) brings point P’ in P’’. The mechanisms that produce the two central symmetries are connected in series (the point of arrival of the first transformation is the point of departure of the other). A Kempe conveyor (with base CD = 2AB, parallel to AB) links P directly to P’’. The three instruments form a single articulated system with which perfect mobility is obtained (impossible if the theorem previously stated were not true).

11. The Sylvester Pantograph This machine is made up of an articulated parallelogram OABC with O as its vertex, hinged to the model plane. Two similar isosceles triangles are bonded to the two consecutive sides of the parallelogram. They are constructed in such a way that one of the equal sides coincides with one of the sides of the parallelogram so that their bases meet in its vertex B. If one moves point P, point Q also moves. However, segments PO, QO and angles POQ, PAB e QCB are always equal. Therefore, the machine effects (locally) a rotation with its centre at O and its width is equal to the angle at the vertex of the triangles. ROTATION

12. STRETCHING The Delaunay Quadrilateral BPCQ is an articulated rhombus. Points M and N (belonging to sides BP, PC respectively and selected so that PM = PN) are bonded in order to run inside a rectilinear groove r. This mechanical system (connecting rod) has two degrees of free movement. It is easy to demostrate that the line PQ is constantly (during the deformation of the system) perpendicular to r, and that the ratio between the distance of P and that of Q from r are also constant. Therefore, the flat surfaces (limited) described by P and Q in opposite half planes, having r as a common origin, correspond to each other in that particular affinity which is sometimes called stretching.

13. ABCD is an articulated rhombus. Vertices A e B are hinged to the plane. Instead, two rods of equal length (AG = GC) are linked to the vertices A and C of the rhombus and hinged at G. Therefore, the quadrilateral AGCD is a deltoid. A third rod GF is linked to hinge G so that GF = AG = GC, while around hinge E, fixed to side DC of the rhombus, another rod EF may rotate (the second extremity of which is hinged to F) so that EF = EC. Therefore, the quadrilateral CEFG is also a deltoid. The lengths of the various rods are selected so that AD : AG = AG : EF. As a result, the two deltoids are similiar and their inner angles equal. RECTILINEAR STEERING When the rhombus deforms (C and D describe circumferences having B and A as their centers respectively), point F will describe a rectilinear segment belonging to the perpendicular at AB, driven by A.

14. Thanks for Your Attention Powerpointrealised by Teacher Randazzo Tullio English Translation by Teacher Sottile Michele