The figure in geometry

1. The figure in geometry Rencontre de Lumigny (16 – 20 avril 2007) What is geometry ?

2. The figure which is disappeared ? Russo (Encyclopedia Universalis) Geometry is commonly defined as the science of figures in space. Ph. Nabonnand wrote in presentating of the theme of this Rencontre : The problem of the (non)-representation of geometric objects in the realm of these geometries (differential geometry, higher-dimensional spaces) comes close to the conclusions drawn by the promoters of the projective point of view tending in the research to guarantee the greatest generality attainable to pure geometry by requiring to practise geometry independent of any figurative representation (… that is in interrupting the connection between the drawn figure and the demonstration) If figures disappear from geometric activity, what then is the object of geometry?

3. An example of a figure : :Kepler and his Mysterium Cosmographicum (1596) Three levels in reading a figure : • The plane drawing • The objects in space represented by the drawing • The solar system which is represented by the figure in a rational way.

4. Archimedes by a soldier of Marcellus during the capture of Syracusa by the Romans ( - 212) He was alone with himself in studying sommething related to adrawn figure(διαγράμματος), Being lost in considerations and in thoughts, he didn’t realized neither the Romans who were everywhere around, nor the fact that the town was captured,(Plutarch, Lives of illustruous peoplechap. VII)

5. The double meaning of the concept of a figure with the Greeks : διάγραμμα (diagramme) and σχήμα (scheme) « Those [people doing geometry] who use visible figures (διάγραμμα), and construct on their base their reflections without having those figures themselves on their mind but the perfect figures (σχήμα) of witch the former are ( Platon : The Republic :chap. VI, 510 c)

6. Geometric figures Διάγραμμα Σχήμα

7. The figure (σχήμα) in Euclid’s « Elements » • Déf.14 : « A figure is that wich is contained by any boundary or boundaries » • Déf.13 : « A boundary is that wich is an extremity of anything. » • Déf.15 : « A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another » • A figure is determined by its form and its magnitude. (The area is part of the figure) Consequently the term « equal » is used for figures wich have only the same area.

8. Conservation of magnitude an variation of the form Problems of quadrature • Quadrature des Lunes • Quadrature of rectilinear figures • Quadrature of curvilinear figures

9. Conservation of the form and comparison of magnitude • Theory of proportions • Fundamental rule of similar figures

10. Gnomon A gnomon is a figure wich yields – attached to a certain figure - a figure wich is similar to the original figure

11. Figurated numbers, polygonal numbers 1,4,9

12. The central problem of greek geometry : to construct a figure with given properties • Ex. Euclide XIII, 13 to 17, construction of the five Platonic solids • The three problems of greek geometry • the duplication of the cube (to insert two mean proportional ) • the trisection of any angle • the quadrature of the circle

13. Pappus : The elder people say that there are three types of problems ; the called them plane (intersection circles and straight lines) solid (conic sections are used) and linear (other lines are used like spirals, the quadratrix, …)

14. Pappus on the question of « loci » How could it be that one of the two points wich started moving in A terminates after passing through a straight line in O whereas the other terminates in the same moment in B after passing a curved line without knowing the proportion of the segment AO to the arc AB ?

15. Descartes, La géométrie (1637) ; second book If we think of geometry as the science wich furnishes a general knowledge of the measurement of all bodies, then we have no more right to exclude the more complex curves than the simpler ones, provided the can be conceived of as described by a continuous motion or by several succesive motions, each motion being completely determined by those wich precede ; for in this way an exact knowledge of the magnitude is always obtainable.

16. The new fact proposed by the analytical representations of the XVII th century 1) Is there an analytical expression for every curve ? 2)is there to every curve an equation ?

17. Leibniz : Even if somebody draws in an arbitrary manner a line wich is partly srtaight, partly a circle and partly of quite another nature, it would be possible to find a notion or a rule or an equation wich would be common to the whole line and according to it all changes would happen. (Discours de métaphysique, § VI) • Cramer : All those points of inflection & all those curvatures, & in general all those singularities of algebraic curves (…) are expressed by the equation wich marks its nature ; the curve drawn on paper represents nothing to the eyes wich couldn’t be read off its equation if one is able to understand its language. It happens often that the equation shows us singularities wich the senses couldn’t never grasp. (Analyse des lignes courbes algébriques 1750)

18. Two directions of development 1) If it is true that the equation gives us informations that the figure doesn’t give, is the later still useful ? 2) But these methods often lead to calculations wich are long and complicated. Searching for new methods in geometry wich could compete with the analytical tools without being as clumsy and complicated.

19. Examples of the first strand • Lagrange (Mécanique analytique, préface 1788) In this work you will find no figures. The methods in it need neither constructions nor geometric or mechanic reasoning but only algebraic operations wich are regular and uniform. • Chasles (Aperçu historique sur l’origine et le développement des méthodes en géométrie (1837) The old geometry was invaded by figures(…). Because of the lake of general and abstract principles each question was treated in a concrete way with respect to the figure wich was the object of the question. This flaw of the ancient geometry give raise to the advantages of analytical geometry in comparison to it. (p.207)

20. The essential person of the second strand : Poncelet Is it possible to develop geometry in such a way that it has the same power and the same generality as the algebraic calculus ? In this work (Traité des propriétés projectives des figures) we propose the following : to enlarge the resources of simple geometry to generalize its concepts and its language wich is commonly rather restricted, to bring them closer to analytic geometry and above all to offer to it the general means wich are necessary to demonstrate and to find in an easy way that class of properties wich is common to figures when they are considered in a purely abstract way independenly of every absolute and definite quantity.

21. Example : Poncelet’s closure theorem.

22. The inverse problem : to reduce all figures with two conics to the case of two circles

23. Poncelet (Traité des propriétés projectives des figures) : Algebra uses abstract signs wich represent absolute quantities by characters wich haven’t any value by their own. The give to those quantities every indetermination that is possible. Consequently they operate on signs of non-existence in the same way as on signs of quantities wich are always absolute, always real.

24. Indetermination in analytic geometry (C) : x2 + y2 = R2 (C’) : (x-a)2 + y2 = r2 Intersection : (d) : 2ax = R2 – r2 + a2 This is true whether the two circles cut each other in the drawing or not.

25. In pure geometry : II I IV III

26. Principle of continuity and the « figures corrélatives » The principle of continuity is presented as an extension of the method of transformation ; its idea is to consider the different figures obtained from one figure by applying to it some transformations as the same geometric object. Poncelet use the terme « figures corrélatives » proposed by Carnot. He distinguishes the correlations which don’t modify neither the order nor the reality of the elements of the figures (but modify their magnitude) from those which modify the relative position of the elements and those which make disappear or appear elements. • He introduces in that way ideal elements, imaginary elements and elements at infinity. So he is able to study the system built up by two conics in analogy to the system consisting of two circles.

27. Ideal objects, imaginary objects.  § 53. In general one can denote by the adjective imaginary every object which is rendered completely impossible or inconstructible in the correlative figure – being before in a figure absolute and real :  those which are thought to come out of the former figure by a progressive and continuous movement of certain of its parts. • The term « ideal » serves to designate the particular mode of existence of an object which – while staying real in the transformation of the primitive figure – ends up by depending in an absolute or real manner of others objects which define it in a graphical manner because those objects get imaginary. 

28. Chasles : Aperçu historique (1837) If one thinks about the procedures of algebra and of the immense advantages provided by them to geometry it becomes clear that those advantages are at least partky due to the fact that the transformations of the expressions introduced by algebra are so easy to be handled. Transformations are the secret and the mechanism of the true science and the constant object of analytical researches. It seems natural to introduce analogous transformations in pure geometry which operate directly on the given figures and their properties. Duality and homography are examples of such transformations.

29. Homographie defines a correspondence between points, straight lines and planes of one figure, and points, straight lines and planes of ano figure which conserves the descriptive properties (incidence) and the cross ratio. • Correlation reverse the relation : it defines a correspondence between points, straight lines and planes of one figure, and planes, straight lines and points (resp.) of another figure, with respect to following features • Points in a plane go to planes passing through a point, • Points on a given straight line go to planes cutting in a straight line (which is correspondent to the given line) • The cross ratio of 4 points in the first figure is equal to the cross ratio of the 4 corresponding planes Figures in the relation described are called correlated.

30. An example of correlation : the duality (Gergonne, 1826) • Geometry in two symetric parts • Exchanging the words ‘’point’’ and ‘’straight line’’ (in plane geometry) or ‘’point’’ and ‘’plane’’ (in solid geometry), ‘’points on a line’’ and ‘’lines cutting in a straight line’’etc. • The duality between geometric propositions is defined as a correspondence between words ; this correspondence between words is applicable also to the demonstration. • Chasles underlines a linguistic aspect thus preparing a structural point of view in geometry.

31. Gergonne :Considérations philosophiques sur les élémens de la science de l’étendue (1826) An example of geometry in two parts

32. Pascal’s theorem and its dual

33. The analytic expression of the principle of duality : Möbius and Plücker introduce homogenous coordinates in order to have an analytic expression for the points at infinity. The equation of a straight line in a plane, in homogenous coordinates is : ux + vy + wz = 0 In that equation, the symmetry between the variable (x, y, z) and (u, v, w) is obvious. Therefore the equation may be considered as the point wise equation of a straight line or the line wise equation of a point. In this analytical procedure provides a foundation of the principle of correlation which is more precise but less intuitive.

34. Consequences In that way one can get different figures and situations which apparently have no relation one tho the other but which are products of exactly the same reasoning : the are defined by the fact being common to a whole family of correlated figures (instead of belonging to one special figure). The figure is the representation of a general situation and that situation is defining the geometrical object. This object isn’t reducible to the ideal scheme of Platon, it is neither form nor quantity. The figure represents the object only in a context which is difined by the correlations, that is by a family of transformations under consideration.

35. Correlative figures

36. Towards the Erlangen Program (Klein, 1872) « Long before the naissance of projective geometry one has linked the properties of one figures to those of another produced by projection of the first. But projective geometry was born only in that moment in which one considered the primitive figure and all its projective imagines as completely identical and in which one announced propjective properties in a way such that their invariance under propjections became clear. This was done by taking the considerations of the group of projective transformations as a base ; in that way the difference between projective and ordinary geometry was created ».

37. Conclusion This new conception of the figure considered as the representation of a general mathematical situation implies not a disappearance but new ways of reading it. « It is not the study of the particular starting from general defined a priori, but the reconstruction of the general by particular situations that is the reconnaissance of the correlations( in the language of the XIX century) or that of the structures (in moderne language) » [R. Bkouche] The Erlangen is not the end of the figure but the starting point of a new reading of the figure. Klein himself cites his work on the icosahedron in relation to the quintic equation, explaining that « there are figures which doesn’t admit all the transformations of the group (chosen as fundamental) but only a part of them. In the sense of the group-theoretic analysis, those figures are of a particularinterest and they have remarkable properties »

38. Icosahedron by Euclid (Peyrard, p. 305)

39. Icosahedron by Klein