A course “ Mathematics and Technology ”

1. A course“ Mathematics and Technology ” Mathematics is a living science, everywhere present in science and technology The teacher should have experienced how science develops in the real world. He(she) can then show it.

2. The scientist : • He(she) asks questions; • He(she) dares to say: “I do not know.” • He(she) has an open and critical mind. Helping discovering the power of the mathematical method: • Modelling • Problem solving • Mathematical sophistication • Use of computer

3. Some messages through the course : • The questions “Why?” and “What is it useful for?” should be encouraged and deserve an answer. • The beauty of mathematical constructions • Mathematics is useful and constantly developing around us. • Mathematics is much more than numbers Most subjects treated are too advanced so that preservice teachers following the course can hope bring them directly to classroom. The purpose of the course is rather to teach them how to prepare such kind of material.

4. Mathematics and technology • New course since winter 2001. Most students are preservice secondary school teachers. • Purpose: Discover mathematics present in everyday technologies

5. Joint creation with my colleague: Yvan Saint-Aubin • Yvan is a physicist and I am a mathematician. • We knew very little of the material of the course when it was created. We now have enough material for at least two courses. • The game is to take some technologies, to dismantle them into pieces in order to discover and explain the mathematics that makes them work. • We play the game to prepare the course. We try to teach the students to do the same.

6. Description Two formats: • Science Flashes (1 hour) • More elaborate subject - 1 week = 3 hours plus 2 hours of exercices - or two weeks on one subject • The lectures are of two different types • elementary parts (subject matter for exams) • conference type lectures on advanced parts

7. Evaluation  • Two exams with open book and personal notes. Non cumulative contents • A session project on an application of mathematics (by teams of two, if possible; by larger teams (4-6) otherwise ) • A half-hour oral presentation of the project

8. The exercices • We have spent a lot of time writing interesting exercises that make the students practice modelling and review their elementary maths • Finding appropriate exam questions is not a trivial task • A few examples below

9. A book for the course Mathematics and technology, SUMAT, Springer C. Rousseau and Y. Saint-Aubin, Mathématiques et technologie, SUMAT, Springer C. Rousseau and Y. Saint-Aubin,

10. Science Flashes • Antennas and radars are parabolic. Why? (Geometric definition of conics) • Computer vision: calculating the position of one object from its position on two photos (The parametric equations of lines in 3-dimensional space)

11. Covering a territory with antennas for a mobile phone network (Euclidean geometry)

12. The corresponding exercice at the exam We fill a large planar region with nonoverlapping disks of radius r. We use two methods: in the first method we place the centers of the disks on a square network and in the second method we place them on a regular triangular network of equilateral triangles. Which method gives the denser filling? Suggestion: compute the proportion of each square covered by portions of disks in case (a) and the proportion of each triangle covered by portions of disks in case (b). (b) (a)

13. Physics : unifying the laws of reflection and refraction. The laws of nature follow optimization principles. Applications : short waves, optical fiber • A short look in the architecture of computers describing logic circuits • The regular tiling of the sphere with twelve spherical pentagons

14. More elaborate subjects • Positioning in space : GPS, GPS signal, cartography, localization of thunderstorms (Geometric locus, differential geometry, theory of finite fields) • How is a musical CD engraved: why 44100 numbers per second? (Elementary Fourier analysis) • Public key cryptography (Elementary number theory: congruences) • Error correcting codes : Hamming codes and Reed-Solomon codes (Linear algebra, finite fields) • Image compression: iterated function systems (Affine transformations of the plane) • The JPEG format (.jpg) (Elementary Fourier analysis)

15. Robots (Rotations in 3-dimensional space, change of reference frame) • Google and the Pagerank algorithm (Markov chains – linear algebra) • The skeleton and the gamma-knife surgery (Geometry) • Turing machines and DNA computers (The hierarchy of functions starting from the basic ones) • Random number generators (Finite fields) • Calculus of variations (Multi-variable calculus)

16. A remarkable property of the parabola All rays parallel to the axis are reflected to a single point.

17. Applications: the shape of many objects among which • Telescope mirrors

18. Solar furnaces • Parabolic antennas • Radars

19. The corresponding property of the ellipse Any ray issued from one focus is reflected to the other focus.

20. Applications: mirrors, accoustic phenomena • Elliptic mirrors for instance behind the lamp of a cinema projector • Accoustic phenomena: for instance Paris’ subway

21. Google and the PageRank algorithm A search engine that does not order entries properly is useless.

22. Where are we after two clicks?

23. Where are we after n clicks? Why?

24. Order of pages B, A, C, E, D

25. Image compression The easiest way to store an image inside the memory of a computer is to store the color of each pixel. This requiresan enormous quantity of memory! Can we do better?

26. Let’s suppose we have drawn a city: We store in memory the line segments, circle arcs, etc…, which approximate our image. We approximate our image by known geometric objects

27. To store a line segment in memory it is sufficient to store: • the two endpoints of the line segment • a program explaining to the computer how to draw a line segment with given endpoints. The geometric objects are ouralphabet.

28. How to store more complex images, for instance landscapes? • We use the same principle but we enlarge our alphabet: • We approximate our landscape by fractals, for instance the fern.

29. We store in memory a program to draw the fern. Such a program on Mathematica m=15000 L[n_]:=If[1<n<87,2,n] H[n_]:=If[86<n<94,3,L[n]] K[n_]:=If[n>93,4,H[n]] R=Table[K[Random[Integer,{1,100}]],{m}]; F[1,x_,y_]:=0 G[1,x_,y_]:=0.16*y F[2,x_,y_]:=x*0.85+y*0.04 G[2,x_,y_]:=-x*0.04+y*0.85+1.6 F[3,x_,y_]:=x*0.2-y*0.26 G[3,x_,y_]:=0.23*x+0.22*y+1.6 F[4,x_,y_]:=-x*0.15+y*0.28 G[4,x_,y_]:=x*0.26+y*0.24+0.44 x[1]:=0 y[1]:=0 Do[{x[n+1],y[n+1]}={F[R[[n]],x[n],y[n]],G[R[[n]],x[n],y[n]]},{n,1,m}] T=Table[{x[n],y[n]},{n,m}]; ListPlot[T, AspectRatio->1, Axes-> False]

30. Why does it work? Let’s look at the Sierpinski carpet: It is a union of three Sierpinski carpets. Let us start with a square and iterate a construction algorithm

32. This works with any initial set! Let’s try another one:

33. In practice • Coding: • We replace any small square by the image of a similar larger square under a homothety of ratio ½ composed with one of 8 transformations: • Identity plus 3 rotations • 4 symetries • We adjust contrast. • We make a translation of the level of grey.

34. Example Sixth iterate First iterate

35. The GPS (Global positioning system)fully operational since 1995 • Network of orbiting satellites whose position is known

36. The receptor measures the travelling timetof a signal from one satellite to the receptor. • The distance from the satellite to the receptor is d = ct c:speed of light • The points located at a distance d from a satellite are on a sphere of radius d,with center at the satellite.

37. The intersection of three spheres is two points. One of them is excluded because it is non realistic. • The intersection of two spheres is a circle: • Hence, if we know the travelling time of the signals of three satellites to the receptor we know the position of the receptor.

38. This is the theory… In practice … the satellites have atomic clocks perfectly synchronized. The receptor has a cheap clock. We have a fourth unknown: the shift between the clock of the receptor and the clocks of the satellites. We then need to “measure” the travelling time of a signal from a fourth satellite.

39. 4 unknowns 4 measured times • The shift between clocks • The three coordinates of position With this method we get a precision of 20 meters.

40. Applications of the GPS • Finding one’s way in nature • Drawing a map • Managing a fleet of vehicles • Measuring Mount Everest and observing its growth • Helping blind people • Find one’s way on the road • Landing a plane in the fog

41. GPS are a reference of time! • Electronic equipments can be synchronized with the help of GPS. • Hydro-Québec uses this method to synchronize its lightnings detectors. Once thunderstorms are localized, one can reduce the current through lines passing through zones of thunderstorms so as to minimize the risk of breakdown of the electrical network, in case one transit line receives a lightning.

42. A related exam question Meteorites regularly enter the atmosphere, rapidly heat up, disintegrate, and finally explode before hitting the surface of the Earth. This explosion generates a shock wave that travels in all directions at the speed of sound v. The shock wave is detected by seismographs installed at various locations on the surface of the Earth. If four stations (equipped with perfectly synchronized clocks) note the moment that the shock wave arrives, explain how to calculate both the position and time of the explosion.

43. Error correcting codes Principle: we lengthen the message in a redundant way. This allows to correct some errors. Example: We repeat each bit three times. We want to send 0. We send 000. If we receive000we decode0 100we decode0 010we decode0 001we decode0 We have corrected 0or 1error.

44. However If we receive110we decode1 101we decode1 011we decode 1 111 we decode 1 And the transmission is erroneous. An error correcting code is efficient if there are few errors. This code is not economical: a word of 4 bits is lengthened to 12 bits and we may only be able to correct one error.

45. We can do much better… Hamming code: We want to send a 4 bits word:u1, u2, u3, u4 We send a 7 bits word. We add: u5 = u1+ u2 + u3 u6 = u2+ u3 + u4 u7 = u1+ u2 + u4 This code can correct one error. u1erroneous:u5 andu7incompatibles u2 erroneous:u5, u6andu7incompatibles u3 erroneous:u5and u6incompatibles u4 erroneous:u6andu7incompatibles u5 erroneous:u5incompatible u6 erroneous:u6incompatible u7 erroneous:u7incompatible