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.

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

Some messages through the course :•Mathematics are useful and constantly developing around us. •The questions “Why?” and “What is it useful for?” should be encouraged and deserve an answer. •The beauty of mathematical constructions•Mathematics are 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.

Mathematics and technology

• New course since winter 2001. Most students are preservice secondary school teachers.

• Purpose: Discover mathematics present in everyday technologies

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 into pieces, to dismantle them in order to discover and explain the mathematics that make them work.

• We play the game to prepare the course. We try to teach the students to do the same.

DescriptionTwo formats:• Flashs-science (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

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

The exercices

•We have spent a lot of time writing interesting exercices that make the students practice modelling and review their elementary maths

•Finding appropriate exam questions is not a trivial task

•A few examples below

A book for the course

Mathematics and technology (september 2008)C. Rousseau and Y. Saint-Aubin, Springer-Verlag

Mathématiques et technologie (october 2008)C. Rousseau and Y. Saint-Aubin, Springer-Verlag

Flashs-science

• 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)

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

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).

(a) (b)

• 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

• Voronoï diagrams (Euclidean geometry)

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)

• Robots (Rotations in 3-dimensional space, change of reference frame)

• Friezes and tilings (Symmetries – linear algebra)

• 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)• Sparing and borrowing money

Some students projects

(a list on my webpage)• Rollercoasters • The search of boundaries in a photo• Morphing IMAGES• Text compression• Mathematical morphology in treating

images

• Benford law of significant digit

• How to complete the hole in Escher’s painting “ Print Gallery”

• Polyhedra and fullerenes

Carbone 60Truncated icosahedron

• Voronoï diagrams and Delaunay triangulation in image analysis

• Sphere packings and honeycombs• The best skateboard track• Other cryptographic methods • Reed-Müller error-correcting codes• Knots and the action of enzymes on

DNA• Digital fingerprintING • Image compression : from fractals to

practical applications

• Penrose tilings• The seasons, the locus of the sunrise and

sunset at a given date, the length of day at a given date, …

• Calculation of astronomic distances, from the ancient Greeks to now

• The eclipses• The shape of sand dunes• Phyllotaxy (how to explain spirals in

sunflowers, etc.) • Population growth under constraints• Mathematical modelling of epidemics • Chaos

A remarkable property of the parabola

All rays parallel to the axis are reflected to a single point.

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

• Solar furnaces• Parabolic antennas• Radars

The corresponding property of the ellipse

Any ray issued from one focus is reflected to the other focus.

Applications: mirrors, accoustic phenomena

• Elliptic mirrors for instance behind the lamp of a cinema projector

• Accoustic phenomena: for instance Paris’ subway

Google and the PageRank algorithm

A search engine that does not order entries properly is useless.

003

100

3

10000

3

100

2

10

3

10

3

101

013

1

2

10

Where are we after two clicks?

9

100

6

10

009

100

6

10

18

50

2

19

11

9

4

3

20

18

110

6

1

6

1

2

1

Where are we after n clicks?

07.007.007.007.007.0

22.002.002.002.002.0

22.022.022.022.022.0

39.039.039.039.039.0

29.029.029.029.029.0

Why?

Order of pages

B, A, C, E, D

07.007.007.007.007.0

22.002.002.002.002.0

22.022.022.022.022.0

39.039.039.039.039.0

29.029.029.029.029.0

Image compression

The easiest way to store an image inside the memory of a computer is to store the color of each pixel.

This requires an enormous quantity of memory!

Can we do better?

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

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 our alphabet.

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.

We store in memory a program to draw the fern. Such a program on Mathematica

m=15000L[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_]:=0G[1,x_,y_]:=0.16*yF[2,x_,y_]:=x*0.85+y*0.04G[2,x_,y_]:=-x*0.04+y*0.85+1.6F[3,x_,y_]:=x*0.2-y*0.26G[3,x_,y_]:=0.23*x+0.22*y+1.6F[4,x_,y_]:=-x*0.15+y*0.28G[4,x_,y_]:=x*0.26+y*0.24+0.44x[1]:=0y[1]:=0Do[{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]

Principle for drawing the fern

The fern is the union:

•of a stalk•of three copies of the initial fern

We can reconstruct the fern from 4 affine

transformations:• the transformation T1

which sends the large fern to the fern minus two branches,

• the transformation T2 which sends the large fern on the left branch,

• the transformation T3 which sends the large fern on the right branch,

• the transformation T4 which sends the large fern on the stalk.

In order to reconstruct the fern, it suffices to store in memory this

information!Algorithm:•We take a point P on the fern. •We choose at random i in {1,2,3,4} and we plot P1 = Ti(P).•We choose at random i in {1,2,3,4} and we plot

P2 = Ti(P). •Etc...

This method is called « Iterated function systems ». It works because the fern is self similar.

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

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

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.

Example

First iterate Sixth iterate

Some exercices

The GPS (Global positioning system)fully operational since 1995

• Network of orbiting satellites whose position is known

• The receptor measures the travelling time t of 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.

• The intersection of two spheres is a circle:

• The intersection of three spheres is two points. One of them is excluded because it is non realistic.

• Hence, if we know the travelling time of the signals of three satellites to the receptor we know the position of the receptor.

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.

4 measured times 4 unknowns

• The shift between clocks

• The three coordinates of position

With this method we get a precision of 20 meters.

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

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.

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.

Signal of the GPS

Shift-register:

Example: we take (q0, q1, q2, q3 )=(1,1,0,0)

000100110101111001001101011110010011010111100100110101111000001101011110001011010111100010110101111000100101011110001001010111100010011101111000100110011110001001101111100010011010111000100110101110001001101011100010011010111

Why?

Random number generators

• Consider sequences of 0 and 1:• 0 and 1 must each appear with

probabilty ½.• All sequences of length 2 must each

appear with probability ¼.• All sequences of length n must each

appear with probability 1/2n.

• 1 appears 2n-1 times and 0 appears 2n-1 – 1 times,

• Each sequence of length 2 appears 2n-2 times except 00 which appears 2n-2 – 1 times

• Each sequence of length r appears 2n-r times except 0…0 which appears 2n-r – 1 times, for r <= n.

Theorem: in the sequences of period 2n – 1 generated by the shift-register:

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 receive 000 we decode 0

100 we decode 0010 we decode 0001 we decode 0

We have corrected 0 or 1 error.

However

If we receive 110 we decode 1

101 we decode 1011 we decode 1111 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.

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 (mod 2):u5 = u1+ u2 + u3

u6 = u2+ u3 + u4

u7 = u1+ u2 + u4

This code can correct one error. u1 erroneous: u5 and u7 incompatibles

u2 erroneous : u5, u6 and u7 incompatibles

u3 erroneous : u5 and u6 incompatibles

u4 erroneous : u6 and u7 incompatibles

u5 erroneous : u5 incompatible

u6 erroneous : u6 incompatible

u7 erroneous : u7 incompatible